tag:blogger.com,1999:blog-40379371008190815122024-02-07T03:49:35.722+01:00Egin Matematika"Sar zaitezte matematikaren munduan, ezagutu,... usaindu, ukitu, daztatu...ez zara inoiz ere ez damutuko".
Egin Matematikahttp://www.blogger.com/profile/01348970164668368901noreply@blogger.comBlogger192125tag:blogger.com,1999:blog-4037937100819081512.post-90766357044239512352023-10-29T18:28:00.001+01:002023-10-29T18:37:28.607+01:00PROBLEMA 1 OPE-2023-EPE<p> </p><p>Jarraian doan aurkezpenean, 2023ko EPEan jarritako lehenengo problemaren ebazpen posible bat doakizue. Zenbakien Teoriako problema interesgarria:</p><p><br /></p>
<div style="text-align: center;"><iframe allowfullscreen="true" frameborder="0" height="389" mozallowfullscreen="true" src="https://docs.google.com/presentation/d/e/2PACX-1vRCElvo6bG7M-_qxVF52L-XfNbU3CrXZbe9vB5iNTcqO49YbNvbDapr1DMpB7P1TvexYxf0FfZkohZp/embed?start=false&loop=false&delayms=10000" webkitallowfullscreen="true" width="640"></iframe></div>Egin Matematikahttp://www.blogger.com/profile/01348970164668368901noreply@blogger.com0tag:blogger.com,1999:blog-4037937100819081512.post-81596708477941290602023-04-09T23:05:00.001+02:002023-04-10T00:31:57.348+02:00Ekuazio-sistema linealak (Batxi 2-USE)<p> </p><div style="text-align: justify;">
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<span face=""verdana" , sans-serif" style="font-size: large;">Matematika II ikasgaiko ekuazio linealetako sistemen gaiaren bideo batzuk uzten dizkizuet jarraian. B</span><span face=""verdana" , sans-serif" style="font-size: large;">i multzotan sailkatuta daude:</span><br /><ul><li><span face=""verdana" , sans-serif" style="font-size: large;"><span style="text-align: center;">Laburpen teorikoak eta adibide ebatziak. Gaussen metodoa, Roucheren teorema, parametroen menpeko ekuazio-sistemen eztabaida eta Cramer-en metodoa</span></span><span face=""verdana" , sans-serif" style="font-size: large;">.</span></li>
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<ul><li><span face=""verdana" , sans-serif" style="font-size: large; text-align: center;">Selektibitateko ariketa ebatziak pausoz pauso azalduta.</span></li>
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<div style="text-align: justify;"><div style="text-align: center;"><b style="color: #0b5394;"><span style="font-size: large;">LABURPEN TEORIKOAK ADIBIDEKIN</span></b></div></div>
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<tr><td style="text-align: center;"><b>GAUSSEN METODOA</b></td>
<td style="text-align: center;"><b>ROUCHÉREN TEOREMA</b></td></tr>
<tr><td style="text-align: center;"><iframe allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share" allowfullscreen="" frameborder="0" height="160" src="https://www.youtube.com/embed/hoCcMP_7D6Y?list=PL2yn2j0qRfvoumevobJMEN54JzZxvmBSB" title="EKUAZIO SISTEMAK 1 Gaussen metodoa" width="284"></iframe></td>
<td style="text-align: center;"><iframe allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share" allowfullscreen="" frameborder="0" height="160" src="https://www.youtube.com/embed/O8nbNa9VhZA?list=PL2yn2j0qRfvoumevobJMEN54JzZxvmBSB" title="EKUAZIO SISTEMAK 2 Roucheren teorema" width="284"></iframe></td></tr>
<tr><td style="text-align: center;"><b>PARAMETROEN MENPEKO SISTEMAK</b></td>
<td style="text-align: center;"><b>CRAMERREN METODOA</b></td></tr>
<tr><td style="text-align: center;"><iframe allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share" allowfullscreen="" frameborder="0" height="160" src="https://www.youtube.com/embed/ZMAp1YBupss?list=PL2yn2j0qRfvoumevobJMEN54JzZxvmBSB" title="EKUAZIO SISTEMAK 3 Parametroen menpe eztabaidatu" width="284"></iframe></td>
<td style="text-align: center;"><iframe allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share" allowfullscreen="" frameborder="0" height="160" src="https://www.youtube.com/embed/8Kx-VulGvLg?list=PL2yn2j0qRfvoumevobJMEN54JzZxvmBSB" title="EKUAZIO SISTEMAK 4 Cramerren metodoa" width="284"></iframe></td></tr>
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<span face=""verdana" , sans-serif" style="color: #0b5394; font-size: x-large;"><b>ARIKETA EBATZIAK</b></span></div>
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<tr><td style="text-align: center;"><b>2017KO EKAINA</b><b> A1</b></td>
<td style="text-align: center;"><b>2019KO EKAINA A1</b></td></tr>
<tr><td style="text-align: center;"><iframe allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share" allowfullscreen="" frameborder="0" height="160" src="https://www.youtube.com/embed/o0O_2WHBWJk?list=PL2yn2j0qRfvqv_lI_iO_TDHQ9x6IKmgbd" title="A1 EKUAZIO-SISTEMAK Ariketa 1 (2017ko EKAINA)" width="284"></iframe></td>
<td style="text-align: center;"><iframe allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share" allowfullscreen="" frameborder="0" height="160" src="https://www.youtube.com/embed/Lq-mtLmUH3s?list=PL2yn2j0qRfvqv_lI_iO_TDHQ9x6IKmgbd" title="A2 EKUAZIO SISTEMAK Ariketa 2 (2019ko EKAINA)" width="284"></iframe></td></tr>
<tr><td style="text-align: center;"><b>2020KO OHIKOA A1</b></td>
<td style="text-align: center;"><b>ARIKETA: EKUAZIO-SISTEMA</b></td></tr>
<tr><td style="text-align: center;"><iframe allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share" allowfullscreen="" frameborder="0" height="160" src="https://www.youtube.com/embed/H40G9FaDbi8?list=PL2yn2j0qRfvqv_lI_iO_TDHQ9x6IKmgbd" title="A3 EKUAZIO SISTEMAK 3 (2020ko ohikoa A1)" width="284"></iframe></td>
<td style="text-align: center;"><iframe allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share" allowfullscreen="" frameborder="0" height="160" src="https://www.youtube.com/embed/yr6UQoRAWQU?list=PL2yn2j0qRfvqv_lI_iO_TDHQ9x6IKmgbd" title="A4 EKUAZIO SISTEMAK 4" width="284"></iframe></td></tr>
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<br /><span style="color: #666666;"><span face=""verdana" , sans-serif" style="font-size: large;"><i><b>Ekuazio-sistemak, hainbat ariketa ebatziak (selektibitatea):</b></i></span><span face=""verdana" , sans-serif" style="font-size: large;"> </span></span><br />
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<span style="color: #0b5394; font-size: x-large;"><span face=""verdana" , sans-serif"><b>***** </b></span></span></div>
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</div>Egin Matematikahttp://www.blogger.com/profile/01348970164668368901noreply@blogger.com0tag:blogger.com,1999:blog-4037937100819081512.post-68805197235792711232022-07-18T01:28:00.002+02:002022-07-29T09:28:15.820+02:006174 zenbaki misteriotsua<p>1949. urtean Dattatreya Ramchandra Kaprekar (1905-1986) matematikari indiarrak 6174 zenbakiak bere baitan gordetzen zuen misterioa agerian utzi zigun, Harrezkero, zenbaki honek izen propioa du: Kaprekar konstantea.</p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEidcPoP-3AM_reUute5qjkms_UnXrrSh5kLzQmvv65HVuS28h_FZyX3HTjF6MnZlDxaP4bt_VJLjXHMhsS2Wan7V_pUpdkjn7cTWQPY1cLHsAh-2W-9xFnPl8NT9b1YSUIAuXQd02ZTfn64/s1320/KAPREKAR.PNG" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="935" data-original-width="1320" height="284" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEidcPoP-3AM_reUute5qjkms_UnXrrSh5kLzQmvv65HVuS28h_FZyX3HTjF6MnZlDxaP4bt_VJLjXHMhsS2Wan7V_pUpdkjn7cTWQPY1cLHsAh-2W-9xFnPl8NT9b1YSUIAuXQd02ZTfn64/w400-h284/KAPREKAR.PNG" width="400" /></a></div><span style="text-align: left;">6174 konstantearen zifrak berrantolatzen baditugu, alde batetik, beheranzko ordenan (7641) eta, bestetik, goranzko ordenan (1467), bi zenbaki lortuko ditugu, eta horien arteko diferentzia abiaburuko zenbakia da: </span><div><p style="text-align: center;">7641 - 1467 = 6174</p><div>Propietate deigarri hori ez da gure protagonistak bere buruarekin duena bakarrik, lau zifrako beste zenbaki batzuekin gauza bera egiteko gai baita, hau da, zurrunbilo baten erdigunera eramaten ditu, ustekabean bera amaieran agertzeko.</div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjF0lziKHQ8j5zYSBb4UHBKtZkoqmCxc5pAfaywPArRiLGR9tYIv84RMfE_FkxPs3UJa4bOMQqRH2EfzceDh7-NCah1RhWFXVgqIbArcwn0lLQij8815lbdZoTBOGWd_OREWq-UJ9O8vMSt/s1437/6174.PNG" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1005" data-original-width="1437" height="280" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjF0lziKHQ8j5zYSBb4UHBKtZkoqmCxc5pAfaywPArRiLGR9tYIv84RMfE_FkxPs3UJa4bOMQqRH2EfzceDh7-NCah1RhWFXVgqIbArcwn0lLQij8815lbdZoTBOGWd_OREWq-UJ9O8vMSt/w400-h280/6174.PNG" width="400" /></a></div><br /><div style="text-align: left;"><div>Zifra guztiak berdinak ez dituen lau zifrako edozein zenbaki aukeratuko dugu. Zifra bateko, biko eta hiruko zenbakiak barne sartuko ditugu, ezkerrean hiru, bi eta zero bat jarrita, hurrenez hurren (adibidez: 0007, 0035 eta 0826). Guztira, 10000 - 10 = 9990 zenbaki daude.</div><div><br /></div><div>Gure zenbakia aukeratu ondoren, honela jokatuko dugu:</div><div><br /></div><div>1. Zifrak beheranzko ordenan berrantolatuko ditugu, ahalik eta zenbakirik handiena lortzeko.</div><div>2. Zifrak goranzko ordenean berrantolatuko ditugu, zenbakirik txikiena lortzeko.</div><div>3. Handienetik txikiena kenduko dugu gure bigarren zenbakia eraikitzeko.</div><div>4. 6174a agertzen bada, amaitu dugu, bestela, aurreko urratsak errepikatuko ditugu gure hirugarren zenbakia eraikitzeko.</div><div>5. Gehienez 7 urratsetan 6174 zenbakia agertuko da.</div></div><div><br /></div><div>Ohar gaitezen lau zifrak berdinak izango balira urrats batean 0000ean amaituko genukeela.</div><div><br /></div><div>Adibide 1: <b>3586</b></div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhNdhChZVh8BQm5vqF_2C1o_iCCsDW5W_uEWPKay641sN-3GiWmmhyoBvxXIYrFbhhKChLeOHlFj1_1jbYiF90nWcsbEvrX_Xa-zUgv7t0_7pregfv28nGGKESgIp4nJckjsQkoMulMO60c/s1315/KAPREKAR2.PNG" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="939" data-original-width="1315" height="286" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhNdhChZVh8BQm5vqF_2C1o_iCCsDW5W_uEWPKay641sN-3GiWmmhyoBvxXIYrFbhhKChLeOHlFj1_1jbYiF90nWcsbEvrX_Xa-zUgv7t0_7pregfv28nGGKESgIp4nJckjsQkoMulMO60c/w400-h286/KAPREKAR2.PNG" width="400" /></a></div><br /><div style="text-align: center;"><br /></div><div>Adibide 2: 0035</div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiAVEPUtfJRKRP5DuRnoGdzRPKbqIvngu1ufSZiehqlAnxf-DXEPAa-4cJBahvkcZNg0WIU6Hce-sq8lfrN9VlCrzQZVco3NCBCE4Sui5237Jrw2nPwH8QNIRp1dIRGDkuuBXM9Rv8adPc3/s1319/KAPREKAR3.PNG" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="928" data-original-width="1319" height="281" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiAVEPUtfJRKRP5DuRnoGdzRPKbqIvngu1ufSZiehqlAnxf-DXEPAa-4cJBahvkcZNg0WIU6Hce-sq8lfrN9VlCrzQZVco3NCBCE4Sui5237Jrw2nPwH8QNIRp1dIRGDkuuBXM9Rv8adPc3/w400-h281/KAPREKAR3.PNG" width="400" /></a></div><div><br /></div><div>Praktikatzeko, Miguel Retegik sortutako kalkulu-orri hau utziko dizuet. Lau zifrako zenbaki bat sartuko dugu banda berdean; lehenengo zutabean, programak zifrak beheranzko ordenan jarriko ditu; bigarrenean, goranzko ordenan; eta hirugarrenean, diferentzia kalkulatuko digu.</div><div><br /></div><div style="text-align: center;"><iframe frameborder="0" height="346" scrolling="no" src="https://onedrive.live.com/embed?cid=05FC64D544B5B623&resid=5FC64D544B5B623%2173758&authkey=ABcpVJSArW8wW1M&em=2" width="550"></iframe></div><div><br /></div><div>Ez al da harrigarria? Kaprekarren prozesuak, zazpi urrats edo gutxiagotan, 6174 nukleo edo puntu finkora garamatza. Horregatik, konstante hori zenbaki magiko eta misteriotsutzat hartzen da.</div><div><br /></div><div>Kaprekar konstantea tresna didaktiko interesgarria izan daiteke. Lehen Hezkuntzan, kenketa lantzeko eta nork bere burua zuzentzeko balio diezaguke, baldin eta 6174a zazpi urrats edo gutxiagotan lortzen ez bada. Bigarren Hezkuntzan, prozesuaren barruan sar gintezke, zergatik funtzionatzen duen ikertzeko eta ustezko misterioa agerian uzten duen patroiren bat aurkitzeko.</div><div><br /></div><div>Ondoren, problema berriro planteatuko dugu eta Kaprekar prozesuaren egiaztapen bat ematen saiatuko gara (azalpenak gazteleraz daude):</div></div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiy1Ne7y2wkrOpzTdTnsG3RfizpPC7l2mD_ta3tgptOtI2_WftMbmeFamJ6EMIOxr21_lgY0A2mnmDT3oh16Z-YqWn8rUU8dYb-9XGmXxWUlfBiNOsqQZ4kLxfQTJXy5Avw7YkhRmpa5wZf/s1390/KAPREKAR5.PNG" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="791" data-original-width="1390" height="494" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiy1Ne7y2wkrOpzTdTnsG3RfizpPC7l2mD_ta3tgptOtI2_WftMbmeFamJ6EMIOxr21_lgY0A2mnmDT3oh16Z-YqWn8rUU8dYb-9XGmXxWUlfBiNOsqQZ4kLxfQTJXy5Avw7YkhRmpa5wZf/w868-h494/KAPREKAR5.PNG" width="868" /></a></div><div class="separator" style="clear: both; text-align: center;"><b style="text-align: left;"><span style="color: #3d85c6; font-family: georgia; font-size: x-large;"><u>Demostración</u></span></b></div><div class="separator" style="clear: both; text-align: left;"><b><span style="font-family: georgia;"><u><br /></u></span></b></div><div class="separator" style="clear: both; text-align: center;"><b><span style="font-family: georgia;"><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhUJQBmtty63anTD5pF_7ObwHhPNm1Vm55XpgtOLtXYDnKC1f_PLVUhSJDREX3NPjG88OpxB_DueM5-sEj2oj6mvBKNql5xWYj0weKPRM1o3qHCPgvMTGgbRI22yoXGyuFd8tgeWtv3P8DM/s1158/KAPREKAR+DEMOSTRACION+1.PNG" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1158" data-original-width="812" height="1282" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhUJQBmtty63anTD5pF_7ObwHhPNm1Vm55XpgtOLtXYDnKC1f_PLVUhSJDREX3NPjG88OpxB_DueM5-sEj2oj6mvBKNql5xWYj0weKPRM1o3qHCPgvMTGgbRI22yoXGyuFd8tgeWtv3P8DM/w898-h1282/KAPREKAR+DEMOSTRACION+1.PNG" width="898" /></a></div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj7uDaKcR7fb6RrvZ0CYR8drTYFPa0CWq6awUq2NUdU4-0ixxBYWdcWQPLg-_Agh0SJ2EIVR3IDXIHq0DJhyphenhyphenYnVq-AHkYF0jBjuSXfTKyuR1woWroiJNjfVLoj5JbZcXc9xCsaqCMq_19hD/s1249/KAPREKAR8.PNG" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1249" data-original-width="815" height="1373" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj7uDaKcR7fb6RrvZ0CYR8drTYFPa0CWq6awUq2NUdU4-0ixxBYWdcWQPLg-_Agh0SJ2EIVR3IDXIHq0DJhyphenhyphenYnVq-AHkYF0jBjuSXfTKyuR1woWroiJNjfVLoj5JbZcXc9xCsaqCMq_19hD/w897-h1373/KAPREKAR8.PNG" width="897" /></a></div></span><br /></b><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgX3IKNghKAqMq6mnJLmijFDwFH4BR7mCSNNPtIFUbi1QNg-YSKSU1st0ETbfBi6D9-5Z5cJgDqBih2L_djUZweF-mdA514qWNgxlzv3wBoPl6nYFWi59D_JY8BgMXFYf3uB4ZMo9M7PVtQ/s1185/KAPREKAR10.PNG" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1185" data-original-width="890" height="1177" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgX3IKNghKAqMq6mnJLmijFDwFH4BR7mCSNNPtIFUbi1QNg-YSKSU1st0ETbfBi6D9-5Z5cJgDqBih2L_djUZweF-mdA514qWNgxlzv3wBoPl6nYFWi59D_JY8BgMXFYf3uB4ZMo9M7PVtQ/w883-h1177/KAPREKAR10.PNG" width="883" /></a></div><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj_NajlD3yK3Oluam82sSeiGoYy9bqtgtNcmxy7gCIhxI8nrYo0PJgOCr_-fqvmK2b2FFGVU9ieaUSKoJVhiCMFvu2WQGbyJQ1u-qqo-mWKxqhqyAF4UHeib_iwV_K5NQ8vVhL-9nt8PAeH/s1211/KAPREKAR17.PNG" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1211" data-original-width="1129" height="939" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj_NajlD3yK3Oluam82sSeiGoYy9bqtgtNcmxy7gCIhxI8nrYo0PJgOCr_-fqvmK2b2FFGVU9ieaUSKoJVhiCMFvu2WQGbyJQ1u-qqo-mWKxqhqyAF4UHeib_iwV_K5NQ8vVhL-9nt8PAeH/w875-h939/KAPREKAR17.PNG" width="875" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgODqMgNYJUvWXSE9LujQ0LC4iJnytgXhxsHcU32uvr80ZuXR1EC3r1XtZUtheg6mb9-zlYjNL54CksjUMeI1wLJ1whb2enfKIYmRA05CKnny01L9Oo-sJ2iilUJjIkH6VkCtwU401rfFK8/s1243/KAPREKAR12.PNG" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1243" data-original-width="785" height="1344" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgODqMgNYJUvWXSE9LujQ0LC4iJnytgXhxsHcU32uvr80ZuXR1EC3r1XtZUtheg6mb9-zlYjNL54CksjUMeI1wLJ1whb2enfKIYmRA05CKnny01L9Oo-sJ2iilUJjIkH6VkCtwU401rfFK8/w851-h1344/KAPREKAR12.PNG" width="851" /></a></div><br /><div class="separator" style="clear: both; text-align: justify;">Azken koadroan zifrak handienetik txikienera dituztenak aukeratu ditugu ordezkari gisa. Guztiak 6174 nukleora erortzen dira, gehienez 6 urratsetan. Enuntziatuaren baldintzak betetzen dituen lau zifrako edozein zenbaki lehen iterazioan zuzenean nukleora jausiko da edo taulako 30 zenbaki horietako batekin erlazionatuko da, hau da, taulako zenbakietako batean bihurtuko da, zifrak ordena berean edo beste edozein ordenatan dituela. Lehenengo diferentzia kalkulatu ondoren, prozesuak hurrengo eskeman agertzen diren bideetako bati jarraituko dio 6174ra iritsi arte. Azkenaurreko iterazioan bi kasu daude 8532 eta 7641, azken honetara doaz prozesuan Kaprekarren konstantearen zifrak edozein ordenetan erakusten dituzten zenbakiak, erdiko kasua izan ezik (6174).</div><br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiHALUigoVuLC7i_H4wqqgHcB_6tU6aXQYE77fWTOxFSiqK6wNilSd_bwVbQQ-RbHeiQHrglF7Vog5Mn4XZxWnoao-sFmAh0GCqc_tRCyQL72OT-tPSq2eIpq-i05GUYMT_W_r_vLXsm0T1/s1126/KAPREKAR16.PNG" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1064" data-original-width="1126" height="794" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiHALUigoVuLC7i_H4wqqgHcB_6tU6aXQYE77fWTOxFSiqK6wNilSd_bwVbQQ-RbHeiQHrglF7Vog5Mn4XZxWnoao-sFmAh0GCqc_tRCyQL72OT-tPSq2eIpq-i05GUYMT_W_r_vLXsm0T1/w841-h794/KAPREKAR16.PNG" width="841" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"><br /></div>Goragoko taulak eta zenbakien erorketa adierazten duen azken irudikapena ikasleei eskuratuz gobida ditzakegu pautak eta erlazioak aurkitzera.<div><br /></div><div>Adibidez:</div><div><ul style="text-align: left;"><li>Hasierako zenbakiaren zifrek zein baldintza egiaztatu behar dute lehenengo pausoan 6174 lortzeko? Eta bi pausotan? Eta hirutan?...</li><li>Aztertu lau zifrak desberdinak eta bakoitiak dituzten zenbakiak. Zer ikusten duzu? Eta bikoitiak eta desberdinak?</li><li>Zein zenbakiren multiploak dira Kaprekar prozesuan agertzen diren zenbakiak? Kenketa hurrengo adibidean bezala egiten baduzu, errazago ikusiko duzu. Zein zenbaki atera daiteke faktore komuna amaieran? Nola orokortu daiteke propietate hau? <a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEid7sVKSIlU2rgqlAuxqF3aIOFdOK0HZrxMI_zVaLZrS-dqMAISCSKMrzDpB-qmHxlsMWD6W5JahUOlAMMePj25fVaMUoxpacZDS8lb9MpHrU3flzqFgCxfzPplHJzP8Hb0accsOkBpm0oOLwbIIYPL5Afz3lj3p3Ct4GYYj1ODuu_Yo5rKiXk7l3M2wQ/s1103/irudia.JPG" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" data-original-height="376" data-original-width="1103" height="218" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEid7sVKSIlU2rgqlAuxqF3aIOFdOK0HZrxMI_zVaLZrS-dqMAISCSKMrzDpB-qmHxlsMWD6W5JahUOlAMMePj25fVaMUoxpacZDS8lb9MpHrU3flzqFgCxfzPplHJzP8Hb0accsOkBpm0oOLwbIIYPL5Afz3lj3p3Ct4GYYj1ODuu_Yo5rKiXk7l3M2wQ/w640-h218/irudia.JPG" width="640" /></a></li><li>Bi zifrako zenbakiekin egon daiteke Kaprekar konstanterik? zergatik?</li><li>Hiru zifrako Kaprekar konstantea 495 da, jarri adibide batzuk.</li></ul><div><br /></div></div><div>Amaitzeko, Derivando kanaleko Eduardo Saenz de Cabezonen bideoa uzten dizuet:</div><div><br /></div><div class="separator" style="clear: both; text-align: center;"><iframe allowfullscreen="" class="BLOG_video_class" height="266" src="https://www.youtube.com/embed/pDXek06Bde4" width="320" youtube-src-id="pDXek06Bde4"></iframe></div><br /><div style="text-align: center;"><br /></div><div><br /></div><div><br /></div><div><div class="separator" style="clear: both; text-align: center;"><br /></div><div></div><div class="separator" style="clear: both; text-align: center;"><br /><b><span style="color: #660000; font-size: x-large;">***</span></b></div><br /></div>Egin Matematikahttp://www.blogger.com/profile/01348970164668368901noreply@blogger.com1tag:blogger.com,1999:blog-4037937100819081512.post-72298612021938828442021-10-31T00:34:00.003+02:002021-10-31T00:47:04.928+02:00PROBLEMA 3 (OPE-2021-EPE)<div style="text-align: justify;"><span style="font-size: large;">2021ko oposizioetan proba praktikoan jarritako hirugarren problema ebatziko dugu gaurko artikuluan. Geometria-problema interesgarria, ebazteko estrategia oinarrizko geometrian aurki dezakegu. Problemaren bi ebazpen proposatuko ditugu. Lehenengo metodoan triangelu zuzena zirkunferentzia batean inskribatzen da, erdibidekoak (erradioak) bi triangelu isoszeletan banatzen du hasierako triangelua eta biderkadura bektorialaren moduluaren erdia triangelu isoszele baten azalera kalkulatzeko aplikatu ondoren, angeluak kalkulatzera iritsiko gara. Bigarrenean, triangelu zuzena azalera bereko bi triangelu isoszeletan banatu eta triangelu honetan oinarrizko geometria aplikatuz angeluen balioak lortuko ditugu. Desberdintasun nagusia azalerak kalkulatzeko erabilitako erlazioan dago.</span></div><p style="text-align: justify;"></p><div><br /></div><br /><span style="color: #6fa8dc; font-family: verdana; font-size: x-large;">PROBLEMA 3 (GEOMETRIA)</span><div><br /><div style="text-align: justify;"><span style="font-size: x-large;"><div style="text-align: justify;"><span style="font-size: x-large;"><br /></span></div><span style="font-family: verdana;"><b><i><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiveSsotnPTwCZxtucAp0WIOrUvjg_8BWFO9pkyq9egFuFAmTyds87lUp4hPT_peyfU5-Eg5ci8_lJZqObsOBvs2lwxaDp_vVNagzflbxz_iaMhPPx6Swxj5OKJ4iZvCAlxh9PZ6nrgm7fr/s623/3.+PROBLEMA+irudi+11.PNG" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" data-original-height="346" data-original-width="623" height="178" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiveSsotnPTwCZxtucAp0WIOrUvjg_8BWFO9pkyq9egFuFAmTyds87lUp4hPT_peyfU5-Eg5ci8_lJZqObsOBvs2lwxaDp_vVNagzflbxz_iaMhPPx6Swxj5OKJ4iZvCAlxh9PZ6nrgm7fr/s320/3.+PROBLEMA+irudi+11.PNG" width="320" /></a></div>Kalkulatu triangelu zuzen baten angelu zorrotzak, hipotenusaren dagokion erdibidekoaren (medianaren) luzera katetoan arteko batez besteko geometrikoa dela jakinda.<br /></i></b></span></span></div><br /><br /></div><div><br /></div><div><div style="text-align: justify;"><span style="font-family: verdana; font-size: x-large;"><span style="color: #3d85c6;">Ebazpen 1</span></span></div><span style="font-family: verdana; font-size: large;"><div style="text-align: justify;"><br /></div><div style="text-align: justify;">Triangelu zuzen bat zirkunferentzia batean inskribatzen bada, triangeluaren hipotenusa diametro bat da. Bestetik, triangelu baten azalera alde-bektoreen biderkadura bektorialaren moduluaren erdia da. Problema ebazteko bi propietate hauetan oinarrituko gara.</div></span><div><div style="text-align: justify;"><br /></div><br /><div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhGOri2_VBlWox67kOTZ7MxXTJlOSC8nGXW15tEx3qmHttlvprjVlLlfp61Hi_PYlIa9TcDeX6p9cPYYgq7ZdmcW6kJ2CVPX6VJZQaK7tcx61UINOOUux7KnjxEwhb93An3HNxrO0rZLAVO/s1164/3.+PROBLEMA+irudi+14.PNG" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1164" data-original-width="965" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhGOri2_VBlWox67kOTZ7MxXTJlOSC8nGXW15tEx3qmHttlvprjVlLlfp61Hi_PYlIa9TcDeX6p9cPYYgq7ZdmcW6kJ2CVPX6VJZQaK7tcx61UINOOUux7KnjxEwhb93An3HNxrO0rZLAVO/s16000/3.+PROBLEMA+irudi+14.PNG" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><br /></div><span style="color: #3d85c6; font-family: verdana; font-size: x-large;">Ebazpen 2</span><br /><br /><span style="font-family: verdana; font-size: large;">Lehendabizi marraz dezagun triangelu zuzen bat eta hipotenusari dagokion erdibidekoa. Erdibidekoak hipotenusa bi zati berdinetan zatitzen du</span><div><div style="text-align: justify;"><br /></div><div><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhioZPDXuEEk-MerhAYDeDsZH9qAhCE_NDse0o6mUPzNKGjtZNBRHe2zGqS-KWP39SYyLrKTT6x7ECjH1X_7gzRj5jSej2TSmgAdzF-N1OeDcRLKi3iRFGNJkzY_9fdMYeCe01kr6Ta7HR5/s629/3.+PROBLEMA+irudi+3.PNG"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhioZPDXuEEk-MerhAYDeDsZH9qAhCE_NDse0o6mUPzNKGjtZNBRHe2zGqS-KWP39SYyLrKTT6x7ECjH1X_7gzRj5jSej2TSmgAdzF-N1OeDcRLKi3iRFGNJkzY_9fdMYeCe01kr6Ta7HR5/w400-h215/3.+PROBLEMA+irudi+3.PNG" /></a><br /><br /><span style="font-family: verdana; font-size: large;">Triangelu zuzen bat laukizuzen baten erdia denez, zera ikusten da:</span><div><div><br /><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjzoSavVnKPWgCt3p8bh87koP56xwvK5crFl93dix6wFFLqy1aSsvptjbSsfwL3okh3SFrDhshOXulGGu5NWunRSv4Fcn09Bodb2ISMYQtOU4XBP_yJpTtDp04MyiUNbosN0RwrV9pkNiCp/s627/3.+PROBLEMA+irudi+4.PNG"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjzoSavVnKPWgCt3p8bh87koP56xwvK5crFl93dix6wFFLqy1aSsvptjbSsfwL3okh3SFrDhshOXulGGu5NWunRSv4Fcn09Bodb2ISMYQtOU4XBP_yJpTtDp04MyiUNbosN0RwrV9pkNiCp/w400-h225/3.+PROBLEMA+irudi+4.PNG" /></a></div><div><span style="font-family: verdana; font-size: large;">AOC triangelu isoszelean zentratuko gara. Triangelu honen azalera laukizuzenaren laurdena da eta h AO aldeari dagokion altuera; orduan:</span><div class="separator" style="clear: both; text-align: center;"><br /></div><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgs4yyCcvQM3YRX9r7oMP9Vf8hF7t_hbthNj1YghGEi2J1UNgXdVK-pMbj6kn7qpMCkhvbmm9VcW4n3IWvQ4U56JXHrK6cqvxOEeYVi6J3lubJrTllpukhu31E6FYFQZdTklR9rShUcloRM/s1274/3.+PROBLEMA+irudi+13.PNG" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1274" data-original-width="910" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgs4yyCcvQM3YRX9r7oMP9Vf8hF7t_hbthNj1YghGEi2J1UNgXdVK-pMbj6kn7qpMCkhvbmm9VcW4n3IWvQ4U56JXHrK6cqvxOEeYVi6J3lubJrTllpukhu31E6FYFQZdTklR9rShUcloRM/s16000/3.+PROBLEMA+irudi+13.PNG" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"><br /></div><br /></div></div></div></div></div></div><div><br /></div><div><span style="color: #444444; font-size: large;"><b><u>Oharra:</u> bigarren metodo hau Jose Manuel Lopez Irastorzari zor diogu, problema hau proposatu nion, eta antzeko ebazpen bat bidali zidan. Ebazpen dotorea JMrena.</b></span></div><div><br /></div><div><br /></div><div><br /></div><div><br /></div><div><br /></div><div style="text-align: center;"><span style="color: #3d85c6; font-family: verdana; font-size: x-large;"><b>***</b></span></div><p></p><br /><div><br /></div><div><br /></div>Egin Matematikahttp://www.blogger.com/profile/01348970164668368901noreply@blogger.com2tag:blogger.com,1999:blog-4037937100819081512.post-35112080144311279762021-06-27T22:58:00.001+02:002021-06-29T14:46:15.918+02:00PARABOLA IRUDIKATZEN<br />
<div style="text-align: justify;"><span face="Verdana, sans-serif" style="font-size: large;"><br />Bigarren mailako funtzio polinomikoa edo funtzio koadratikoa grafikoki adieraztean guztiontzako ezaguna den parabola deritzon konika lortzen da. </span></div><div style="text-align: justify;"><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg5JvuVmTM6R7AthmWZ66z-Cbcypz8nMNo8ZUiLfWFuXVF-LkrTxAKoCO9JFfImuB7UZrIB44VDgG2ti6HXhwez0udWjccmWst_-Fg1nTbZ5Ee4N-n6CjoyU3ap_i9NUENmBcsIJBpVJzvE/s1593/PARABOLA4.PNG" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1079" data-original-width="1593" height="434" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg5JvuVmTM6R7AthmWZ66z-Cbcypz8nMNo8ZUiLfWFuXVF-LkrTxAKoCO9JFfImuB7UZrIB44VDgG2ti6HXhwez0udWjccmWst_-Fg1nTbZ5Ee4N-n6CjoyU3ap_i9NUENmBcsIJBpVJzvE/w640-h434/PARABOLA4.PNG" width="640" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div></div>
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<span face="Verdana, sans-serif" style="font-size: large;">Derrigorrezko bigarren hezkuntzan oinarrizko funtzioen gaian funtzio koadratikoa aztertzen dute gure ikasleek, eta funtzio honen adierazpen grafikoa zuzen marrazteak duen garrantzia azpimarratzen ohi dugu irakasleok. Honela, hainbat ezaugarri: ahurra-ganbila, ebaki puntuak OY eta OX ardatzekiko, simetria ardatza, erpina eta balio taula kalkulatu ondoren, parabola grafikoki adierazten dute.</span></div>
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<span face="Verdana, sans-serif" style="font-size: large;">Ezaugarri guzti hauen artean berebiziko garrantzia ematen zaio parabolaren simetria ardatza eta erpina ematen duen formulari: x = -b/2a, eta formula hau, inolako justifikaziorik izan gabe, buruz ikasten dute ikasleek. </span></div>
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<span style="font-size: large;"><span face="Verdana, sans-serif">Baina, hain beharrezko da formula hori? Badago parabola bat marrazterik aipatutako formula buruz jakin gabe? Edo, bestela, justifika dezakegu zergatik </span>x=-b/2a? Izan dezakegu funtzioaren koefizienteak ikusi bezain pronto parabolaren eskema mentala? Nola?</span></div>
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<span face="Verdana, sans-serif" style="font-size: large;">Bi puntu simetriko baino ez dugu behar. Baina, ze bi puntu dira egokienak? OX ardatzarekiko ebaki-puntuak? Eta ez badu OX ardatza ebakitzen? Nola kalkula ditzakegu bi puntu simetriko era erraz batean eta buruz?</span></div>
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<span face="Verdana, sans-serif" style="font-size: large;"><br /></span></div><div style="text-align: justify;"><b style="color: #3d85c6; font-size: xx-large;">Parabolaren berehalako bi puntu simetriko:</b></div><div style="text-align: justify;"><br /></div>
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<span face="Verdana, sans-serif" style="font-size: large;">Parabola y=c zuzen horizontalarekin mozten badugu, OY ardatzarekiko ebaki puntu bat (0,c) eta bere simetrikoa (-b/a,c) izango ditugu. Bi puntu hauen abszisen batura erdiak erpinaren abszisa emango digu: </span><span style="font-size: large;">x=-b/2a. Honela parabolaren irudi mentala izango dugu (a koefizientea positiboa bada ahurra eta negatiboa balitz ganbila).</span></div><div style="text-align: justify;"><span style="font-size: large;"><br /></span></div><div style="text-align: center;"><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg6iDbY4uEjTQ4UVt2WRkW3eW7vuxBNMCqEISOAweuQePKeBViNSRuMM23DqJKYOKDf_DnwJeSdr9MppGPKJeiPgdxVqz58VSQDOcvftQEO938O7SXHs_urfESAjDilqokS92WCbDGF5V_a/s1238/PARABOLAREN+ERPINA.PNG" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1238" data-original-width="1159" height="640" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg6iDbY4uEjTQ4UVt2WRkW3eW7vuxBNMCqEISOAweuQePKeBViNSRuMM23DqJKYOKDf_DnwJeSdr9MppGPKJeiPgdxVqz58VSQDOcvftQEO938O7SXHs_urfESAjDilqokS92WCbDGF5V_a/w600-h640/PARABOLAREN+ERPINA.PNG" width="600" /></a></div><br /><div style="text-align: left;"><b><span style="color: #990000; font-size: x-large;">ADIBIDE 1</span></b></div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEijxBUbPQCzzd1_ICNxPmlQ0PLFg9ou8EsbXh5yRSDyAeBMC2k-JTrNTkpORXr0lS0VkrcZfvnjEsQcsXP15aXOz4xuSL1WDAejNBX5ixOYS9zN9r7hGC7vMjD10fBC3moMt4J5eE3XmVuU/s1784/PARABOLAREN+ERPINA3.PNG" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="838" data-original-width="1784" height="451" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEijxBUbPQCzzd1_ICNxPmlQ0PLFg9ou8EsbXh5yRSDyAeBMC2k-JTrNTkpORXr0lS0VkrcZfvnjEsQcsXP15aXOz4xuSL1WDAejNBX5ixOYS9zN9r7hGC7vMjD10fBC3moMt4J5eE3XmVuU/w960-h451/PARABOLAREN+ERPINA3.PNG" width="960" /></a></div><br /><div class="separator" style="clear: both; font-size: x-large; text-align: center;"><br /></div><div style="font-size: x-large; text-align: center;"><br /></div><div style="font-size: x-large; text-align: left;"><b style="font-size: medium;"><span style="color: #990000; font-size: x-large;">ADIBIDE 2</span></b></div></div><div style="text-align: center;"><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiO3gTPnPOP1TUnCaCi54EB_FX8OWqD6YK4QRYGy0ItYM76MdHqtdB_DhBVen_eTKmgAP3LIwupbp-oujIDgzuweVjtNQvHZk-6l89XE4cTZFYlhX8XXb2VYOGCttkOndKm1kdEI7jZSweB/s1828/PARABOLAREN+ERPINA4.PNG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="831" data-original-width="1828" height="439" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiO3gTPnPOP1TUnCaCi54EB_FX8OWqD6YK4QRYGy0ItYM76MdHqtdB_DhBVen_eTKmgAP3LIwupbp-oujIDgzuweVjtNQvHZk-6l89XE4cTZFYlhX8XXb2VYOGCttkOndKm1kdEI7jZSweB/w969-h439/PARABOLAREN+ERPINA4.PNG" width="969" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"><br /></div><span style="font-size: large;"><br /></span><div style="text-align: justify;"><span style="font-size: large;">Funtsean goiko adibideetan erabilitako teknika, beste honekin identifika daiteke: c=0 duen parabolaren adierazpena egin ondoren, lorturiko parabola c parametroak adierazitako distantzia desplazatu bertikalki. Adibidez:</span></div><div style="text-align: justify;"><span style="font-size: large;"><br /></span></div></div><div style="text-align: justify;"><span style="font-size: large;"><b style="font-size: medium; text-align: left;"><span style="color: #990000; font-size: x-large;">ADIBIDE 3</span></b></span></div><div style="text-align: center;"><span style="font-size: large;"><b style="font-size: medium; text-align: left;"><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi_pQTZDHPhrAm_3d8AgwC5pzrVZsU3Bk-5-Ua5kdWQEX_1tlregczoLDKouqyPui5uQGGDjtk2gqfpCpZXIq_qy4avlvW5r6JJQ7hKt6f1Ey6LUmzf3_6SIm5GrCkmt6NZM7Mcst6yYMgZ/s1849/PARABOLAREN+ERPINA5.PNG" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="840" data-original-width="1849" height="464" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi_pQTZDHPhrAm_3d8AgwC5pzrVZsU3Bk-5-Ua5kdWQEX_1tlregczoLDKouqyPui5uQGGDjtk2gqfpCpZXIq_qy4avlvW5r6JJQ7hKt6f1Ey6LUmzf3_6SIm5GrCkmt6NZM7Mcst6yYMgZ/w1023-h464/PARABOLAREN+ERPINA5.PNG" width="1023" /></a></div><br /></b></span></div><div style="text-align: justify;"><span style="font-size: large;">Konturatu parabola ebakitzen duen edozein funtzio konstante y=k erabil daitekeela bi puntu simetriko aurkitzeko. Zuzen horizontal horrek parabola puntu bakar batean ebakitzen badu; hau da, ukitzailea bada, orduan naitaez ukitze-puntu horretan erpina (maximoa edo minimoa) aurkituko da. Argi dago, azkar eta buruz adierazpen grafikoa lortzeko zuzen egokiena y=c dela, adibideetan ikusi dugun bezala.</span></div><div style="text-align: justify;"><span style="font-size: large;"><br /></span></div><div style="text-align: justify;"><span style="font-size: large;"><b style="color: #3d85c6; font-size: xx-large;">Deribatuaren irizpidea:</b></span></div><div style="text-align: justify;"><span style="font-size: large;"><br /></span></div><div style="text-align: justify;"><span style="font-size: large;">Aurrerago, batxilergoan, deribatua erabil dezakegu parabolaren muturra kalkulatzeko, beharrezko baldintza aplikatuz; hau da, deribatua berdin zero eginez. Honela, berriz ere, erpinaren abszisa lortzen da x=-b/2a. Ondoren, bi puntu simetriko kalkulatzen dira funtzioa ebaluatuz (-b/2a)+n eta (-b/2a)-n balioetan. Puntu gehiago behar izanez gero, n-ri balio desberdinak emango dizkiogu.</span></div><div style="text-align: justify;"><span style="font-size: large;"><br /></span></div><div style="text-align: center;"><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgEnR2khdu_KWl3LnMsHxOv-4qJBFW91sM2chIOx5VdurllMgsxBu6ArUd78Q7UajgSPmxXjXXl5xLDUAV-l-_RHpVpkfJWA-2KmGy5QKPW-zvD8tDq20nCLQLJoWM4ywgWsqgljBAtjI-S/s1282/PARABOLAREN+ERPINA2.PNG" style="margin-left: 1em; margin-right: 1em;"><span style="font-size: large;"><img border="0" data-original-height="1217" data-original-width="1282" height="608" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgEnR2khdu_KWl3LnMsHxOv-4qJBFW91sM2chIOx5VdurllMgsxBu6ArUd78Q7UajgSPmxXjXXl5xLDUAV-l-_RHpVpkfJWA-2KmGy5QKPW-zvD8tDq20nCLQLJoWM4ywgWsqgljBAtjI-S/w640-h608/PARABOLAREN+ERPINA2.PNG" width="640" /></span></a></div><span style="font-size: large;"><br /><div style="text-align: left;"><b style="font-size: medium;"><span style="color: #990000; font-size: x-large;">ADIBIDE 4</span></b></div></span></div><div style="text-align: center;"><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhcnr7ZvLpYsRUSodC5NQBxxiLdMDJBBOl58j7apVOQLDaBHgZA3C01NFAQI9XGp6x4VZtyF9CQJ7v11NAeKBhnuXv_DZLZE-bdoV4B9K-87kFThsyKENzqmo_DpL1USdA6ORmT7JQGrdPY/s1622/PARABOLAREN+ERPINA6.PNG" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="811" data-original-width="1622" height="499" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhcnr7ZvLpYsRUSodC5NQBxxiLdMDJBBOl58j7apVOQLDaBHgZA3C01NFAQI9XGp6x4VZtyF9CQJ7v11NAeKBhnuXv_DZLZE-bdoV4B9K-87kFThsyKENzqmo_DpL1USdA6ORmT7JQGrdPY/w1001-h499/PARABOLAREN+ERPINA6.PNG" width="1001" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"><br /></div><div style="text-align: left;"><b style="color: #3d85c6; font-size: xx-large; text-align: justify;">a, b eta c koefizienteen esanahia:</b></div><div style="text-align: left;"><span style="text-align: justify;"><span style="font-size: large;"><br /></span></span></div><div style="text-align: left;"><span style="text-align: justify;"><span style="font-size: large;">a parametroaren zeinuak kurbadurari buruzko informazioa ematen digu (a>0 bada ahurra eta a<0 bada ganbila). Balio absolutuz, a koefizienteak parabolaren irekiera determinatuko du: a-ren balioa txkitzean parabola zabaldu egiten da, bigarren mailako terminoaren balioren txikitzea dakarrelako; ondorioz, a>0 den kasuan y ordenaturen balioa beherago izango dugu. b parametroarekin konbinatuz, erpinaren abszisaren kokapenean ere parte hartzen du a koefizienteak: x=-b/2a.</span></span></div><div style="text-align: left;"><span style="text-align: justify;"><span style="font-size: large;"><br /></span></span></div><div class="separator" style="clear: both; text-align: justify;"><span style="font-size: large;">a eta b koefizienteak konstate mantenduz, c parametroak bertikalki gora eta behera eramaten du parabola.</span></div><div class="separator" style="clear: both; text-align: justify;"><span style="font-size: large;"><br /></span></div><div style="text-align: left;"><span style="font-size: large;">a eta c konstante uzten badira, b koefizienteari balio desberdinak ematen bazaizkio, parabolaren erpina a eta c balioen menpe soilik dagoen beste parabola baten gainean mugitzen da. Ikus dezagun baieztapen honen egiaztapena:</span></div><div style="text-align: left;"><span style="font-size: large;"><br /></span></div><div style="text-align: center;"><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhdA_YbdjtN9ZgNKmfZUEwZa6-ZfcSL-rmLe8uBWxkLX54WAGCd1ryBhIow5ncvbNJIDs-FcHnaRb7FrWyuoBHIqbY6sXTtHhInc93XStBlp7jE_kdtZpKEuLMwBH7jWJP8ciUObmXRy7rV/s1208/PARABOLAREN+ERPINA7.PNG" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="880" data-original-width="1208" height="561" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhdA_YbdjtN9ZgNKmfZUEwZa6-ZfcSL-rmLe8uBWxkLX54WAGCd1ryBhIow5ncvbNJIDs-FcHnaRb7FrWyuoBHIqbY6sXTtHhInc93XStBlp7jE_kdtZpKEuLMwBH7jWJP8ciUObmXRy7rV/w770-h561/PARABOLAREN+ERPINA7.PNG" width="770" /></a></div><div style="text-align: justify;"><span style="font-size: large;"><br /></span></div><div style="text-align: justify;"><span style="font-size: large;">Aukeratu a eta c parametroak, eta eman balio desberdinak b koefizienteari irristatzailea erabiliz. Klikatu karratuan erpinak deskribatzen duen parabolaren ekuazioa ikusteko:</span></div><span style="font-size: large;"><iframe height="358px" scrolling="no" src="https://www.geogebra.org/material/iframe/id/Qd4E6g2d/border/888888/sfsb/true/smb/false/stb/false/stbh/false/ai/false/asb/false/sri/true/rc/false/ld/false/sdz/true/ctl/true" style="border: 0px;" title="Parabola" width="640px"> </iframe></span></div></div><div style="text-align: justify;"><br /></div><span style="font-size: large;"> </span>Egin Matematikahttp://www.blogger.com/profile/01348970164668368901noreply@blogger.com0tag:blogger.com,1999:blog-4037937100819081512.post-22809043522796157232020-08-30T19:10:00.000+02:002021-06-26T09:44:29.165+02:00Liouvilleren problema. Kuboen batura baturaren karratua denean.<p> </p><p style="text-align: justify;"><span style="font-size: large;"><span style="font-family: verdana;">Aritmetikaren meategian harribitxi disdiratsu ugari daude Matematikaren edertasunaren lekuko. </span><span style="font-family: verdana;">Gaur aurkeztuko dugun harribitxi landua Joseph Liouville (1809-1882) matematikariari zor diogu. Liouvillek Matematikaren arlo askotan egin zuen lan: Zenbakien Teoria, Topologia Diferentziala, Analisi Konplexua, ... aipatzekoa da ere, zenbaki traszendenteen existentzia probatzen lehena izan zela. Beraz, ekarpen handiak egindakoa dugu matematikari frantziarra. Jarraian aztertuko dugun problema xumeago bada ere, ez zaio dizdira falta. Zenbakien arteko erlazio ederra bezain harrigarria utzi zigun Liouvillek, altxor txiki bat.</span></span></p><p style="text-align: justify;"><span style="font-family: verdana;"><br /></span></p><p><span style="font-family: verdana; font-size: large;">Zenbaki arrunten honako propietate harrigarri hau nahiko ezaguna da:</span></p><p><span style="font-family: verdana; font-size: large;"><br /></span></p><p style="text-align: center;"><span style="color: #666666; font-family: verdana; font-size: large;"><b>"1tik hasita ondoz ondoko zenbakien kuboen batura, zenbaki hauen baturaren karratua da"</b></span></p><p style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEit6W7y6TMwWo_lpGbXO_iiNZdvGUOhYlgNKPK5lnoXLurCv1-txgAk9HW_WGji6TSxpsXdIRjRsNBawDQaJLrRd8jYO4CuH2-B1GWNeEqjLrU0_iI5Ht1yV1g4Cdr81aO8M18NKu3cKBtg/s922/ZENBAKIAK+47.PNG" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="540" data-original-width="922" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEit6W7y6TMwWo_lpGbXO_iiNZdvGUOhYlgNKPK5lnoXLurCv1-txgAk9HW_WGji6TSxpsXdIRjRsNBawDQaJLrRd8jYO4CuH2-B1GWNeEqjLrU0_iI5Ht1yV1g4Cdr81aO8M18NKu3cKBtg/s640/ZENBAKIAK+47.PNG" width="640" /></a></p><div class="separator" style="clear: both;"><br /></div><div class="separator" style="clear: both;"><span style="font-family: verdana;"><span style="font-size: large;">Propietate hau (1+2+3+...+n) aldeko karratu batean adieraz daiteke irudian ikusten den eran:</span></span></div><div class="separator" style="clear: both;"><span style="font-family: verdana;"><span style="font-size: large;"><br /></span></span></div><div class="separator" style="clear: both; text-align: center;"><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEheu9EnkENgzJ88wAY7VjL7tCsk-I4mO2it6rC_tUje5498xwtyZI23Melzv4v8rJYWxSfb73hHpRFOf8cSN51I9Ee0Q05GTb9rYN-p6d8dQRyT3ikwFBOCO8ry7V2YS3cd4MamVn5-zNsF/s1627/kuboen+batura.GIF" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1151" data-original-width="1627" height="566" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEheu9EnkENgzJ88wAY7VjL7tCsk-I4mO2it6rC_tUje5498xwtyZI23Melzv4v8rJYWxSfb73hHpRFOf8cSN51I9Ee0Q05GTb9rYN-p6d8dQRyT3ikwFBOCO8ry7V2YS3cd4MamVn5-zNsF/w800-h566/kuboen+batura.GIF" width="800" /></a></div><span style="font-family: verdana; font-size: x-large;"><br /></span></div><div class="separator" style="clear: both;"><b style="font-family: verdana;"><span style="font-size: x-large;">➤Propitatearen frogapena</span></b></div><span style="font-family: verdana;"><span><div style="font-size: x-large;"><span style="font-family: verdana;"><span><b><span><br /></span></b></span></span></div><span style="font-size: large;">Jarraian erlazio honen frogapena indukzio matematikoaz:</span></span></span><div><span style="font-family: verdana; font-size: large;"><br /></span></div><div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjbGU-2HcN1WDM2UkyjYhaPzEBEPKhIUMwyfMY8uPcYtcn4Wh_rgrlkcAoNun3B8r1t4_1hLiHyM_Tlx14aTm1R5bhypG9WmsVVGzsVHSYT3IGgzSni6zwyswEd00BQd7R9Lrw6SA_ycURM/s1174/ZENBAKIAK+25.PNG" style="margin-left: 1em; margin-right: 1em;"><span style="font-size: large;"><img border="0" data-original-height="1174" data-original-width="757" height="1250" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjbGU-2HcN1WDM2UkyjYhaPzEBEPKhIUMwyfMY8uPcYtcn4Wh_rgrlkcAoNun3B8r1t4_1hLiHyM_Tlx14aTm1R5bhypG9WmsVVGzsVHSYT3IGgzSni6zwyswEd00BQd7R9Lrw6SA_ycURM/w806-h1250/ZENBAKIAK+25.PNG" width="806" /></span></a></div><div class="separator" style="clear: both; text-align: center;"><span style="font-size: large;"><br /></span></div><div style="text-align: justify;"><span style="font-family: verdana; font-size: large;">Baina propietate hau ez da bakarrik egiaztatzen batetik hasita zenbaki arrunten segidekin, zenbaki arrunten multzoan badaude beste zenbaki familia batzuk propietate honen jabe ere direnak. Liouvillek aztertu eta aurkitu zuen metodo orokor bat zenbaki hauek lortzeko. </span></div><div><span style="font-family: verdana; font-size: large;"><br /></span></div><div><span style="font-family: verdana; font-size: x-large;"><b>➤Liouvilleren aurkikuntza</b></span></div><div><span style="font-family: verdana; font-size: large;"><br /></span></div><div style="text-align: justify;"><span style="font-family: verdana;"><b><span style="font-size: x-large;">N</span></b><span style="font-size: large;"> zenbaki osoa eta positiboaren zatitzaile bakoitzaren zatitzaile kopurua kalkulatu eta zerrendatzen badugu, zenbaki zerrenda honek propietatea betetzen du:</span></span></div><div style="text-align: justify;"><span style="font-family: verdana; font-size: large;"><br /></span></div><div><span style="font-family: verdana;"><div style="font-family: "times new roman";"><span style="font-family: verdana;"><b style="font-size: x-large;">N</b><span style="font-size: large;">ren zatitzaileak:</span><span style="font-size: x-large;"> </span><b><span><span style="font-size: x-large;">{Z₁, Z₂, Z₃, ..., Z</span><span>n</span><span style="font-size: x-large;">}</span></span></b></span></div><div style="font-family: "times new roman"; font-size: x-large;"><span style="font-family: verdana;"><br /></span></div><div style="font-family: "times new roman";"><span><span style="font-family: verdana;"><b style="font-size: x-large;"><span>Z₁</span></b><span style="font-size: large;">ren zatitzaileen kopurua: </span></span><b style="font-family: verdana; font-size: x-large;"><span>a₁</span></b></span></div><div style="font-family: "times new roman";"><span><span style="font-family: verdana;"><span style="font-size: x-large;"><b>Z₂</b></span><span style="font-size: large;">ren zatitzaileen</span></span><span style="font-family: verdana;"><span style="font-size: large;"> kopurua: </span></span><b style="font-family: verdana; font-size: x-large;"><span>a₂</span></b></span></div><div style="font-family: "times new roman";"><span><span style="font-family: verdana;"><b style="font-size: x-large;"><span>Z₃</span></b><span style="font-size: large;">ren zatitzaileen kopurua: </span></span><b style="font-family: verdana; font-size: x-large;"><span>a₃</span></b></span></div><div style="font-size: x-large;"><span>..................................</span></div><div style="font-family: "times new roman";"><span style="font-family: verdana;"><span><b><span style="font-size: large;">Z</span><span>n</span></b></span><span style="font-size: large;">ren zatitzaileen kopurua:</span><span style="font-size: x-large;"> </span><b><span style="font-size: x-large;">a</span><span>n</span></b></span></div><div style="font-family: "times new roman"; font-size: x-large;"><br /></div><div style="font-family: "times new roman";"><span style="font-family: verdana; font-size: x-large;"><b>{</b></span><b style="font-family: verdana; font-size: x-large;"><span>a₁</span></b><span style="font-family: verdana; font-size: x-large;"><b>, </b></span><b style="font-family: verdana; font-size: x-large;"><span>a₂</span></b><span style="font-family: verdana; font-size: x-large;"><b>, </b></span><b style="font-family: verdana; font-size: x-large;"><span>a₃</span></b><span style="font-family: verdana; font-size: x-large;"><b>, ..., </b></span><b style="font-family: verdana;"><span style="font-size: x-large;">a</span><span>n</span></b><span style="font-family: verdana; font-size: x-large;"><b>}</b></span><span style="font-family: verdana;"><span style="font-size: large;"> </span><span style="font-size: large;">zenbaki multzoarekin propietatea egiaztatzen da:</span></span></div><div style="font-family: "times new roman"; font-size: x-large;"><span style="font-family: verdana;"><br /></span></div><div style="font-family: "times new roman"; font-size: x-large;"><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiyILxxXMz1Rqnh1-IB281r0AIG-Jzwytzbs13PVzCZd-pADYsA4a8kUrRrSE63Vtt_E5cfZn3xOkCbsysP5C_rPZ3iNXP8SmcYCENxbsr7Gi5cTOVX1A2CVvQyYWaJnbY5bPnpxq5WgyZS/s1659/ZENBAKIAK+21.PNG" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="121" data-original-width="1659" height="38" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiyILxxXMz1Rqnh1-IB281r0AIG-Jzwytzbs13PVzCZd-pADYsA4a8kUrRrSE63Vtt_E5cfZn3xOkCbsysP5C_rPZ3iNXP8SmcYCENxbsr7Gi5cTOVX1A2CVvQyYWaJnbY5bPnpxq5WgyZS/w512-h38/ZENBAKIAK+21.PNG" width="512" /></a></div></div><div style="font-family: verdana; font-size: x-large;"><br /></div></span></div><div><span style="font-family: verdana; font-size: large;">Hona hemen adibide batzuk:</span></div><div><span style="font-family: verdana; font-size: large;"><br /></span></div><div><span style="font-family: verdana; font-size: large;"><div style="font-family: "times new roman";"><span style="font-family: verdana;"><b><u>Adibide 1</u></b></span></div><div style="font-family: "times new roman";"><span style="font-family: verdana;"><br /></span></div><div style="font-family: "times new roman";"><span style="font-family: verdana;">2·3=6ren zatitzaileak: {1, 2, 3, 6}</span></div><div style="font-family: "times new roman";"><span style="font-family: verdana;"><br /></span></div><div style="font-family: "times new roman";"><span style="font-family: verdana;">6ren zatitzaileen zerrendako zatitzaileen zatitzaile kopurua:</span></div><div style="font-family: "times new roman";"><span><span style="font-family: verdana;">1ren zatitzaileak {1}, kopurua: </span><b style="font-family: verdana;"><span>1</span></b></span></div><div style="font-family: "times new roman";"><span style="font-family: verdana;">2ren zatitzaileak {1, 2}, kopurua: <b><span>2</span></b></span></div><div style="font-family: "times new roman";"><span style="font-family: verdana;">3ren zatitzaileak {1, 3}, kopurua: <b><span>2</span></b></span></div><div style="font-family: "times new roman";"><span><span style="font-family: verdana;">6ren zatitzaileak {1, 2, 3, 6}, kopurua:</span><span style="font-family: verdana;"> </span><b style="font-family: verdana;"><span>4</span></b></span></div><div style="font-family: "times new roman";"><br /></div><div style="font-family: "times new roman";"><span style="font-family: verdana;"><b>{1, 2, 2, 4}</b></span><span style="font-family: verdana;"> <span>zenbaki multzoarekin ere propietatea betetzen da:</span></span></div><div style="font-family: "times new roman";"><span style="font-family: verdana;"><br /></span></div></span></div><div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiwfFsYMPiR-YcbgqPh2VkMO9qTGShHltfYKcSEejPPwn6Ts9Tl4lJhExuOJex_Oun7jKJoyefDTx5zoU8OsExgQFDKj8hYfNCQvpuII6AwsvVPou75uVoF9ffiZT7XWd3nuTrrIKIcFg5m/s1378/ZENBAKIAK+9.PNG" style="margin-left: 1em; margin-right: 1em;"><span style="font-size: large;"><img border="0" data-original-height="125" data-original-width="1378" height="46" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiwfFsYMPiR-YcbgqPh2VkMO9qTGShHltfYKcSEejPPwn6Ts9Tl4lJhExuOJex_Oun7jKJoyefDTx5zoU8OsExgQFDKj8hYfNCQvpuII6AwsvVPou75uVoF9ffiZT7XWd3nuTrrIKIcFg5m/w513-h46/ZENBAKIAK+9.PNG" width="513" /></span></a></div><div class="separator" style="clear: both; text-align: left;"><span style="font-size: large;"><br /></span></div><div class="separator" style="clear: both; text-align: left;"><span style="font-family: verdana; font-size: large;">1+2+2+4 aldeko karratu marrazten badugu, propietatea geometrikoki ikus dezakegu:</span></div></div><div class="separator" style="clear: both; text-align: left;"><span style="font-family: verdana; font-size: large;"><br /></span></div><div class="separator" style="clear: both; text-align: left;"><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgwi9NMfTw92ReDtqKEZUyXpHCUhCaWKmkqDSdXaWqeaPG2QKTQPWhd7MXMEAgvxd24M_UR9Z30QtVMeslB4X5esg49qiBZ-xEwF9sVRJfdt-E1KaySzfEzbAt5KFWjw_R0pnhViYhu_yT1/s674/ZENBAKIAK+10.PNG" style="margin-left: 1em; margin-right: 1em;"><span style="font-size: large;"><img border="0" data-original-height="674" data-original-width="509" height="512" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgwi9NMfTw92ReDtqKEZUyXpHCUhCaWKmkqDSdXaWqeaPG2QKTQPWhd7MXMEAgvxd24M_UR9Z30QtVMeslB4X5esg49qiBZ-xEwF9sVRJfdt-E1KaySzfEzbAt5KFWjw_R0pnhViYhu_yT1/w386-h512/ZENBAKIAK+10.PNG" width="386" /></span></a></div></div><div class="separator" style="clear: both; text-align: left;"><span style="font-size: large;"><br /></span></div><div><span style="font-family: verdana; font-size: large;"><b><u>Adibide 2</u></b></span></div><div><span style="font-family: verdana; font-size: large;"><br /></span></div><div><span style="font-family: verdana; font-size: large;"><span>2</span><span>²·3=12ren zatitzaileak :{1, 2, 3, 4, 6, 12}</span></span></div><div><span style="font-family: verdana; font-size: large;"><br /></span></div><div><span style="font-family: verdana; font-size: large;">12ren zatitzaileen zatitzaile kopurua:</span></div><div><span style="font-family: verdana; font-size: large;"><br /></span></div><div><span style="font-family: verdana; font-size: large;">1ren zatitzaileak {1}, kopurua: <b><span>1</span></b></span></div><div><span style="font-family: verdana; font-size: large;">2ren zatitzaileak {1, 2}, kopurua: <b><span>2</span></b></span></div><div><span style="font-family: verdana; font-size: large;">3ren zatitzaileak {1, 3}, kopurua: <b><span>2</span></b></span></div><div><span style="font-family: verdana; font-size: large;">4ren zatitzaileak {1, 2, 4}, kopurua: <b><span>3</span></b></span></div><div><span style="font-family: verdana; font-size: large;">6ren zatitzaileak {1, 2, 3, 6}, kopurua: <b><span>4</span></b></span></div><div><span style="font-family: verdana; font-size: large;">12ren zatitzaileak {1, 2, 3, 4, 6, 12}, kopurua: <b><span>6</span></b></span></div><div><span style="font-family: verdana; font-size: large;"><br /></span></div><div><span style="font-family: verdana; font-size: large;"><b><span>{1, 2, 2, 3, 4, 6}</span></b> <span>zenbaki multzo honekin ere propietatea betetzen da:</span></span></div><div><span style="font-family: verdana; font-size: large;"><br /></span></div><div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhAwNr3p-HuICj1cv-VXokGueDhHsw4Zsst5iSU9WvOAz_VJTSvIOKrV26eRk9RwaTqI8Q_TBTN9TOqubnATzv4G_Dql5LqVPG5oE-dDwbPFzIFsgV959I2crNBJfGjTAKGVJXvipHOEfF1/s2009/ZENBAKIAK+8.PNG" style="margin-left: 1em; margin-right: 1em;"><span style="font-size: large;"><img border="0" data-original-height="116" data-original-width="2009" height="38" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhAwNr3p-HuICj1cv-VXokGueDhHsw4Zsst5iSU9WvOAz_VJTSvIOKrV26eRk9RwaTqI8Q_TBTN9TOqubnATzv4G_Dql5LqVPG5oE-dDwbPFzIFsgV959I2crNBJfGjTAKGVJXvipHOEfF1/w640-h38/ZENBAKIAK+8.PNG" width="640" /></span></a></div><div class="separator" style="clear: both; text-align: center;"><span style="font-size: large;"><br /></span></div></div><div><div class="separator" style="clear: both; text-align: left;"><span style="font-family: verdana; font-size: large;">Propietatea 1+2+2+3+4+6 aldea duen karratuan:</span></div><div class="separator" style="clear: both; text-align: left;"><span style="font-family: verdana; font-size: large;"><br /></span></div><div class="separator" style="clear: both; text-align: left;"><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg6ToTEfxRLf4_D5Ab64_0pZZS-nB8zQIuDPHud0twplnrbNmPxnc12X0qrRYnHL8JPnQXMFgCtbSDjHSH_UrVwWYfiQ3LcWuYKcKmETr9dHj0FTQ8v7Z-zhRLxGnx7MTyGQPgAcUx59US0/s1163/ZENBAKIAK+6.PNG" style="margin-left: 1em; margin-right: 1em;"><span style="font-size: large;"><img border="0" data-original-height="1163" data-original-width="951" height="800" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg6ToTEfxRLf4_D5Ab64_0pZZS-nB8zQIuDPHud0twplnrbNmPxnc12X0qrRYnHL8JPnQXMFgCtbSDjHSH_UrVwWYfiQ3LcWuYKcKmETr9dHj0FTQ8v7Z-zhRLxGnx7MTyGQPgAcUx59US0/w654-h800/ZENBAKIAK+6.PNG" width="654" /></span></a></div></div><div class="separator" style="clear: both; text-align: left;"><span style="font-family: verdana; font-size: large;"><br /></span></div><div class="separator" style="clear: both; text-align: left;"><span style="font-family: verdana; font-size: large;"><div style="font-family: "times new roman";"><span style="font-family: verdana;"><b><u>Adibide 3</u></b></span></div><div style="font-family: "times new roman";"><span style="font-family: verdana;"><br /></span></div><div style="font-family: "times new roman";"><span style="font-family: verdana;">5³=125ren zatitzaileak :{1, 5, 25, 125}</span></div><div style="font-family: "times new roman";"><span style="font-family: verdana;"><br /></span></div><div style="font-family: "times new roman";"><span style="font-family: verdana;">125ren zatitzaileen zatitzaile kopurua:</span></div><div style="font-family: "times new roman";"><span style="font-family: verdana;"><br /></span></div><div style="font-family: "times new roman";"><span style="font-family: verdana;">1ren zatitzaileak {1}, kopurua: <b><span>1</span></b></span></div><div style="font-family: "times new roman";"><span style="font-family: verdana;">5ren zatitzaileak {1, 5}, kopurua: <b><span>2</span></b></span></div><div style="font-family: "times new roman";"><span style="font-family: verdana;">25ren zatitzaileak {1, 2, 25}, kopurua: <b><span>3</span></b></span></div><div style="font-family: "times new roman";"><span style="font-family: verdana;">125ren zatitzaileak {1, 5, 25, 125}, kopurua: <b><span>4</span></b></span></div><div style="font-family: "times new roman";"><br /></div><div style="font-family: "times new roman";"><b style="font-family: verdana;"><span>{1, 2, 3, 4}</span></b><span style="font-family: verdana;"> <span>gorago frogatu dugun moduan propietatea egiaztatzen da:</span></span></div><div style="font-family: "times new roman";"><span style="font-family: verdana;"><br /></span></div><div style="font-family: "times new roman";"><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhVTSZVfhx2fp2G61ntv1Zmnwdk_e9RyfiIYZ8cuRaZu26iQozrx1ifkMkOPn6jHs67Xw1XiXDRTLlXd2c7iebhs9IqKa2C5OVZDl6ZQFez23pV60KlbNQweyumO1IJKWqVW_OnCmwMRVNQ/s1362/ZENBAKIAK+19.PNG" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="116" data-original-width="1362" height="44" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhVTSZVfhx2fp2G61ntv1Zmnwdk_e9RyfiIYZ8cuRaZu26iQozrx1ifkMkOPn6jHs67Xw1XiXDRTLlXd2c7iebhs9IqKa2C5OVZDl6ZQFez23pV60KlbNQweyumO1IJKWqVW_OnCmwMRVNQ/w512-h44/ZENBAKIAK+19.PNG" width="512" /></a></div></div><div style="font-family: "times new roman";"><span style="font-family: verdana;">Irudi batean adieraziz:</span></div><div style="font-family: "times new roman";"><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgyk8iPy4ubdfNACfHZ_903tk6LUQdbQDwJwXIHIh28uaohGRW9s9R2uT9SLrdpvRkC5Us4uQna9EZvM9_Iicn0kRrp4OthMlXbBn2cHExXeYZ2A2dGql8yoSjp980t15HtxfB8UzPftZtV/s803/ZENBAKIAK+26.PNG" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="803" data-original-width="624" height="640" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgyk8iPy4ubdfNACfHZ_903tk6LUQdbQDwJwXIHIh28uaohGRW9s9R2uT9SLrdpvRkC5Us4uQna9EZvM9_Iicn0kRrp4OthMlXbBn2cHExXeYZ2A2dGql8yoSjp980t15HtxfB8UzPftZtV/s640/ZENBAKIAK+26.PNG" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div></div></span></div><div class="separator" style="clear: both; text-align: left;"><span style="font-family: verdana; font-size: large;"><div style="font-family: "times new roman";"><span style="font-family: verdana;"><b><u>Adibide 4</u></b></span></div><div style="font-family: "times new roman";"><span style="font-family: verdana;"><br /></span></div><div style="font-family: "times new roman";"><span style="font-family: verdana;">2⁴=16ren zatitzaileak :{1, 2, 4, 8, 16}</span></div><div style="font-family: "times new roman";"><span style="font-family: verdana;"><br /></span></div><div style="font-family: "times new roman";"><span style="font-family: verdana;">16ren zatitzaileen zatitzaile kopurua:</span></div><div style="font-family: "times new roman";"><span style="font-family: verdana;"><br /></span></div><div style="font-family: "times new roman";"><span style="font-family: verdana;">1ren zatitzaileak {1}, kopurua: <b><span>1</span></b></span></div><div style="font-family: "times new roman";"><span style="font-family: verdana;">2ren zatitzaileak {1, 2}, kopurua: <b><span>2</span></b></span></div><div style="font-family: "times new roman";"><span style="font-family: verdana;">4ren zatitzaileak {1, 2, 4}, kopurua: <b><span>3</span></b></span></div><div style="font-family: "times new roman";"><span style="font-family: verdana;">8ren zatitzaileak {1, 2, 4, 8}, kopurua: <b><span>4</span></b></span></div><div style="font-family: "times new roman";"><span style="font-family: verdana;">16ren zatitzaileak {1, 2, 4, 8, 16}, kopurua: <b><span>5</span></b></span></div><div style="font-family: "times new roman";"><br /></div><div style="font-family: "times new roman";"><span style="font-family: verdana;"><br /></span></div><div style="font-family: "times new roman";"><span style="font-family: verdana;"><b><span>{1, 2, 3, 4, 5}</span></b> <span>ondoz ondoko lehenengo bost zenbaki arrunt, beraz:</span></span></div><div style="font-family: "times new roman";"><span style="font-family: verdana;"><br /></span></div><div style="font-family: "times new roman";"><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiG7Wm8HQQgMYkbIiHkut8S8kzyN8nbajQzpm3Kj5h39LtXuDh38dUbT_GsLWKhmLOkFSOG9EILcsXY2ICS44dgeFiSlpaB3mTO4fXrUe9jSB5Ov9tOMIaaLdEr97s9QciOJQmtOXRdg439/s1694/ZENBAKIAK+12.PNG" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="115" data-original-width="1694" height="34" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiG7Wm8HQQgMYkbIiHkut8S8kzyN8nbajQzpm3Kj5h39LtXuDh38dUbT_GsLWKhmLOkFSOG9EILcsXY2ICS44dgeFiSlpaB3mTO4fXrUe9jSB5Ov9tOMIaaLdEr97s9QciOJQmtOXRdg439/w513-h34/ZENBAKIAK+12.PNG" width="513" /></a></div><span style="font-family: verdana;"><br /></span></div><div style="font-family: "times new roman";"><span style="font-family: verdana;">Propietatea geometrikoki 1+2+3+4+5 aldeko karratuan:</span></div><div style="font-family: "times new roman";"><span style="font-family: verdana;"><br /></span></div><div style="font-family: "times new roman";"><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiVx_x9X7NmD6VJQ4nXSvUMImp5QDxpfSnwuMcz5Yiuio6763FQ7k0PRjZhu3_6YZuKNfN9HKz-uJcDdblwzAYcj6pNQMZrdsuqkskkS0tWOJlvTfAcLS3tX0iE7WkfvPhZ6bw_OJjRqhIE/s1002/ZENBAKIAK+13.PNG" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1002" data-original-width="819" height="800" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiVx_x9X7NmD6VJQ4nXSvUMImp5QDxpfSnwuMcz5Yiuio6763FQ7k0PRjZhu3_6YZuKNfN9HKz-uJcDdblwzAYcj6pNQMZrdsuqkskkS0tWOJlvTfAcLS3tX0iE7WkfvPhZ6bw_OJjRqhIE/w654-h800/ZENBAKIAK+13.PNG" width="654" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div></div><div><b><u>Adibide 5</u></b></div><div><span style="font-family: verdana;"><div style="font-family: "times new roman";"><span style="font-family: verdana;"><br /></span></div><div style="font-family: "times new roman";"><span style="font-family: verdana;">3⁴=81ren zatitzaileak :{1, 3, 9, 27, 81}</span></div><div style="font-family: "times new roman";"><span style="font-family: verdana;"><br /></span></div><div style="font-family: "times new roman";"><span style="font-family: verdana;">81ren zatitzaileen zatitzaile kopurua:</span></div><div style="font-family: "times new roman";"><span style="font-family: verdana;"><br /></span></div><div style="font-family: "times new roman";"><span style="font-family: verdana;">1ren zatitzaileak {1}, kopurua: <b><span>1</span></b></span></div><div style="font-family: "times new roman";"><span style="font-family: verdana;">3ren zatitzaileak {1, 3}, kopurua: <b><span>2</span></b></span></div><div style="font-family: "times new roman";"><span style="font-family: verdana;">9ren zatitzaileak {1, 3, 9}, kopurua: <b><span>3</span></b></span></div><div style="font-family: "times new roman";"><span style="font-family: verdana;">27ren zatitzaileak {1, 3, 9, 27}, kopurua: <b><span>4</span></b></span></div><div style="font-family: "times new roman";"><span style="font-family: verdana;">81ren zatitzaileak {1, 3, 9, 27, 81}, kopurua: <b><span>5</span></b></span></div><div style="font-family: "times new roman";"><br /></div><div style="font-family: "times new roman";"><span><span style="font-family: verdana;">Aurreko adibidean lortu den zerrenda bera dugu </span><b style="font-family: verdana;"><span>{1, 2, 3, 4, 5},</span></b><span style="font-family: verdana;"> beraz:</span></span></div><div style="font-family: "times new roman";"><span style="font-family: verdana;"><br /></span></div><div style="font-family: "times new roman";"><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiG7Wm8HQQgMYkbIiHkut8S8kzyN8nbajQzpm3Kj5h39LtXuDh38dUbT_GsLWKhmLOkFSOG9EILcsXY2ICS44dgeFiSlpaB3mTO4fXrUe9jSB5Ov9tOMIaaLdEr97s9QciOJQmtOXRdg439/s1694/ZENBAKIAK+12.PNG" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="115" data-original-width="1694" height="34" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiG7Wm8HQQgMYkbIiHkut8S8kzyN8nbajQzpm3Kj5h39LtXuDh38dUbT_GsLWKhmLOkFSOG9EILcsXY2ICS44dgeFiSlpaB3mTO4fXrUe9jSB5Ov9tOMIaaLdEr97s9QciOJQmtOXRdg439/w513-h34/ZENBAKIAK+12.PNG" width="513" /></a></div></div></span></div></span></div><div class="separator" style="clear: both; text-align: left;"><span style="font-family: verdana; font-size: large;"> </span></div><div class="separator" style="clear: both; text-align: left;"><span style="font-family: verdana;"><span style="font-size: large;"> </span><span style="font-size: x-large;">➤</span></span><span style="font-size: x-large;"><b style="font-family: verdana;"><span>Zenbaki lehen baten berreketaren kasua:</span> </b><img border="0" data-original-height="70" data-original-width="73" height="36" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgo1MxRXHcxAJ9P3iJvPOJmlo0oOs7z9y8KLxv3yIZVFNhBR8nuS8Eo0ah2N-QfgcqtSevigtyyWXR9xGBe6IrHNlnp3bFW4ad-GPy12LtAdxssd8EiMh7cqS0PRefAkFIDZMugf3vc_7R1/w38-h36/ZENBAKIAK+14.PNG" width="38" /></span></div><div class="separator" style="clear: both; text-align: left;"><span style="font-family: verdana; font-size: large;"><br /></span></div><div class="separator" style="clear: both; text-align: justify;"><span style="font-size: large;"><span style="font-family: verdana;">Hasierako zenbakia </span><img border="0" data-original-height="70" data-original-width="73" height="29" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgo1MxRXHcxAJ9P3iJvPOJmlo0oOs7z9y8KLxv3yIZVFNhBR8nuS8Eo0ah2N-QfgcqtSevigtyyWXR9xGBe6IrHNlnp3bFW4ad-GPy12LtAdxssd8EiMh7cqS0PRefAkFIDZMugf3vc_7R1/w30-h29/ZENBAKIAK+14.PNG" width="30" /><span style="font-family: verdana;"> zenbaki lehen baten potentzia bat denean (3, 4 eta 5 adibideak), zenbaki horren zatitzaileek duten zatitzaile kopurua, zenbaki arrunten segidak ematen digu: 1, 2, 3,...,n, n+1 </span></span></div><div class="separator" style="clear: both; text-align: justify;"><span style="font-family: verdana; font-size: large;"><br /></span></div><div class="separator" style="clear: both; text-align: left;"><div><span style="font-size: large;"><span><span style="font-family: verdana;"><img border="0" data-original-height="70" data-original-width="73" height="29" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgo1MxRXHcxAJ9P3iJvPOJmlo0oOs7z9y8KLxv3yIZVFNhBR8nuS8Eo0ah2N-QfgcqtSevigtyyWXR9xGBe6IrHNlnp3bFW4ad-GPy12LtAdxssd8EiMh7cqS0PRefAkFIDZMugf3vc_7R1/w30-h29/ZENBAKIAK+14.PNG" width="30" /><span>ren zatitzaileak :</span><span>{</span></span></span><b style="font-family: verdana;"><i><span style="font-family: georgia;">1</span></i></b><span style="font-family: georgia;">,</span><span style="font-family: verdana;"> </span><span><span><b><i><span style="font-family: georgia;">p, p², p³, </span></i></b></span></span><b><i><span style="font-family: georgia;">p⁴, </span></i></b><span><span><b><i><span style="font-family: georgia;">...,</span></i></b><span style="font-family: verdana;"> </span></span></span><span><img border="0" data-original-height="70" data-original-width="73" height="29" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgo1MxRXHcxAJ9P3iJvPOJmlo0oOs7z9y8KLxv3yIZVFNhBR8nuS8Eo0ah2N-QfgcqtSevigtyyWXR9xGBe6IrHNlnp3bFW4ad-GPy12LtAdxssd8EiMh7cqS0PRefAkFIDZMugf3vc_7R1/w30-h29/ZENBAKIAK+14.PNG" width="30" /></span><span style="font-family: verdana;"><span>}</span></span></span></div><div><span style="font-family: verdana; font-size: large;"><br /></span></div><div><span style="font-family: verdana; font-size: large;">Zatitzaile hauen zatitzaile kopurua:</span></div><div><span style="font-family: verdana; font-size: large;"><br /></span></div><div><span style="font-family: verdana; font-size: large;"><span style="font-family: verdana;"><b><i><span>1</span></i></b>ren zatitzaileak <span>{</span></span><b><i><span>1</span></i></b><span>}</span>, kopurua: <b><span>1</span></b></span></div><div><span style="font-size: large;"><span style="font-family: verdana;"><b><i><span style="font-family: georgia;">p</span></i></b>ren zatitzaileak <span>{</span></span><span style="font-family: georgia;"><b style="font-family: verdana;"><i><span>1</span></i></b>, </span><b style="font-family: verdana;"><i><span style="font-family: georgia;">p</span></i></b><span style="font-family: verdana;">}</span><span style="font-family: verdana;">, kopurua: <b><span>2</span></b></span></span></div><div><span style="font-size: large;"><span style="font-family: verdana;"><b><i><span style="font-family: georgia;">p²</span></i></b>ren zatitzaileak <span>{</span></span><span style="font-family: georgia;"><b style="font-family: verdana;"><i><span>1</span></i></b>,</span><span style="font-family: verdana;"> </span><b style="font-family: verdana;"><i><span style="font-family: georgia;">p,</span></i></b><span style="font-family: verdana;"> </span><b><i><span style="font-family: georgia;">p²</span></i></b><span style="font-family: verdana;">}</span><span style="font-family: verdana;">, kopurua: <b><span>3</span></b></span></span></div><div><span style="font-size: large;"><span style="font-family: verdana;"><b><i><span style="font-family: georgia;">p³</span></i></b>ren zatitzaileak <span>{</span></span><span style="font-family: georgia;"><b style="font-family: verdana;"><i><span>1</span></i></b>, </span><b style="font-family: verdana;"><i><span style="font-family: georgia;">p</span></i></b><span style="font-family: georgia;">,</span><span style="font-family: verdana;"> </span><b><i><span style="font-family: georgia;">p²</span></i></b><span style="font-family: georgia;">, </span><b style="font-family: verdana;"><i><span style="font-family: georgia;">p³</span></i></b><span style="font-family: verdana;">}</span><span style="font-family: verdana;">, kopurua: <b><span>4</span></b></span></span></div><div><span style="font-family: verdana; font-size: large;">···········································································</span></div><div><span style="font-size: large;"><span style="font-family: verdana;"><img border="0" data-original-height="70" data-original-width="73" height="29" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgo1MxRXHcxAJ9P3iJvPOJmlo0oOs7z9y8KLxv3yIZVFNhBR8nuS8Eo0ah2N-QfgcqtSevigtyyWXR9xGBe6IrHNlnp3bFW4ad-GPy12LtAdxssd8EiMh7cqS0PRefAkFIDZMugf3vc_7R1/w30-h29/ZENBAKIAK+14.PNG" width="30" />ren zatitzaileak <span style="font-family: verdana;"><span>{</span></span><span style="font-family: georgia;"><b style="font-family: verdana;"><i><span>1</span></i></b>, </span><b><i><span style="font-family: georgia;">p</span></i></b><span style="font-family: georgia;">,</span><span> </span><b style="font-family: "times new roman";"><i><span style="font-family: georgia;">p²</span></i></b><span style="font-family: georgia;">, </span><b><i><span style="font-family: georgia;">p³</span></i></b></span><span style="font-family: georgia;">, ..., </span><img border="0" data-original-height="70" data-original-width="73" height="29" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgo1MxRXHcxAJ9P3iJvPOJmlo0oOs7z9y8KLxv3yIZVFNhBR8nuS8Eo0ah2N-QfgcqtSevigtyyWXR9xGBe6IrHNlnp3bFW4ad-GPy12LtAdxssd8EiMh7cqS0PRefAkFIDZMugf3vc_7R1/w30-h29/ZENBAKIAK+14.PNG" style="font-family: verdana;" width="30" /><span style="font-family: verdana;">}</span><span style="font-family: verdana;">, kopurua: <b><span>n+1</span></b></span></span></div><div><span style="font-size: large;"><br /></span></div></div><div class="separator" style="clear: both; text-align: left;"><span style="font-family: verdana;"><span style="font-size: large;"><img border="0" data-original-height="70" data-original-width="73" height="29" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgo1MxRXHcxAJ9P3iJvPOJmlo0oOs7z9y8KLxv3yIZVFNhBR8nuS8Eo0ah2N-QfgcqtSevigtyyWXR9xGBe6IrHNlnp3bFW4ad-GPy12LtAdxssd8EiMh7cqS0PRefAkFIDZMugf3vc_7R1/w30-h29/ZENBAKIAK+14.PNG" width="30" /> zenbakiaren zatitzaileen zatitzaile kopuruek lehenengo (n+1) zenbaki arrunten zerrenda eratzen dute: <b>{1, 2, 3, ...,n, n+1}</b></span></span></div><div class="separator" style="clear: both; text-align: left;"><span style="font-family: verdana;"><span style="font-size: large;"><br /></span></span></div><div class="separator" style="clear: both; text-align: left;"><span style="font-family: verdana;"><span style="font-size: large;">Beraz, zenbaki lehen baten berreketaren kasuan lehenengo (n+1) zenbaki arrunten segida lortzen da, eta hasieran indukzioz frogatu dugunez:</span></span></div><div class="separator" style="clear: both; text-align: left;"><span style="font-family: verdana; font-size: large;"><br /></span></div><div class="separator" style="clear: both; text-align: left;"><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiNUVQXu-UVzo6CVWpIp_riwhyFATHrWFD5FzVRgMdW6sNlxHrJK7uhV_joh7ssOitkqDCais7B0Qd9tLkC_dYjGMc-tw6wSptK-APM2vyrbV9aZtIan7hzEndCw8ylQ0dbsgIiS32vSeq1/s1080/ZENBAKIAK+43.PNG" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="85" data-original-width="1080" height="50" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiNUVQXu-UVzo6CVWpIp_riwhyFATHrWFD5FzVRgMdW6sNlxHrJK7uhV_joh7ssOitkqDCais7B0Qd9tLkC_dYjGMc-tw6wSptK-APM2vyrbV9aZtIan7hzEndCw8ylQ0dbsgIiS32vSeq1/w640-h50/ZENBAKIAK+43.PNG" width="640" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div></div><div class="separator" style="clear: both; text-align: left;"><span style="font-family: verdana; font-size: large;"><br /></span></div><div class="separator" style="clear: both; text-align: left;"><span style="font-family: verdana;"><b><span style="font-size: x-large;">➤Kasu orokorra</span></b></span></div><div class="separator" style="clear: both; text-align: left;"><span style="font-family: verdana; font-size: large;"><br /></span></div><div class="separator" style="clear: both; text-align: left;"><span style="font-size: large;"><span style="font-family: verdana;">Liouvillek aurkituriko lege </span><span style="font-family: verdana;">orokorr</span><span style="font-family: verdana;">a enuntziatu eta indukzioz egiaztatuko dugu,</span></span></div></div><div class="separator" style="clear: both; text-align: left;"><span style="font-size: large;"><br /></span></div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh1zvOf9-p6gj1-kVStR67qH3J-3rFsCJUIJ7FKIK913sCAcjyNDRmICOWYcJhtAJmL8c1CNk12lYtSw2GeZlazAgd5qSW1bOE-Y4iwzYL_kkfdW4ZtOCIY2jzrXgS3Aa1yvYUM0vCxybFa/s1154/ZENBAKIAK+44.PNG" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1154" data-original-width="791" height="1250" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh1zvOf9-p6gj1-kVStR67qH3J-3rFsCJUIJ7FKIK913sCAcjyNDRmICOWYcJhtAJmL8c1CNk12lYtSw2GeZlazAgd5qSW1bOE-Y4iwzYL_kkfdW4ZtOCIY2jzrXgS3Aa1yvYUM0vCxybFa/w858-h1250/ZENBAKIAK+44.PNG" width="858" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi4x1TN01D7-H8Uy5SRGCv-z0jP5oMmP6xV3AE1lxIs-dbQhDEku5B-LNwD1x54sHadkTWAN2l9Zizp7Zs9zF-AEvIz2cYxJ53iaY89QgQng_INQvbqLDyVl4VkwtvRchpzV5vfykiq-grj/s787/ZENBAKIAK+45.PNG" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="787" data-original-width="741" height="1000" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi4x1TN01D7-H8Uy5SRGCv-z0jP5oMmP6xV3AE1lxIs-dbQhDEku5B-LNwD1x54sHadkTWAN2l9Zizp7Zs9zF-AEvIz2cYxJ53iaY89QgQng_INQvbqLDyVl4VkwtvRchpzV5vfykiq-grj/w943-h1000/ZENBAKIAK+45.PNG" width="943" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: left;"><span style="font-size: large;"><br /></span></div><div class="separator" style="clear: both; text-align: left;"><b style="font-family: verdana;"><span style="font-size: x-large;">➤Zenbait kasuren azterketa. Klasean lantzeko proposamena</span></b></div><div class="separator" style="clear: both; text-align: center;"><span style="font-size: large;"><br /></span></div><div class="separator" style="clear: both; text-align: left;"><span style="font-family: verdana; font-size: large;">Liouvilleren problema ikasgelara eraman daiteke eta ikerketa bezala proposatu ikasleei. </span></div><div class="separator" style="clear: both; text-align: left;"><span style="font-family: verdana; font-size: large;"><br /></span></div><div class="separator" style="clear: both; text-align: center;"><span style="font-family: verdana; font-size: large;"><b><u>Liouvilleren problema</u></b></span></div><div class="separator" style="clear: both; text-align: center;"><span style="font-family: verdana; font-size: large;"><br /></span></div><div class="separator" style="clear: both; text-align: justify;"><span style="font-family: verdana; font-size: large;"><i>Aukeratu nahi duzun zenbakia (hasi zenbaki txikiekin) eta bere zatitzaile guztiak kalkulatu. Ondoren, zatitzaile bakoitzaren zatitzaileak kalkulatu eta idatzi zerrenda batean zatitzaile bakoitzak duen zatitzaile kopurua. Zerrenda honen balio guztiak batu eta baturaren karratua egin (1. emaitza). Zerrendako balio bakoitza ber hiru egin eta gero, kubo hauen batura kalkulatu (2. emaitza).</i></span></div><div class="separator" style="clear: both; text-align: justify;"><span style="font-family: verdana; font-size: large;"><i>Konparatu bi emaitzak. Zer ikusten duzu? Aukeratu beste zenbaki batzuk eta egiaztatu lorturiko erlazioa. Ikertu gehiago.</i></span></div><div class="separator" style="clear: both; text-align: justify;"><span style="font-family: verdana; font-size: large;"><i><br /></i></span></div><div class="separator" style="clear: both; text-align: justify;"><span style="font-family: verdana; font-size: large;">Ikerketan sakontzeko, zenbakien faktore lehenetako deskonposizioa kontuan izango dugu: zenbat faktore lehen desberdin eta hauen anizkoitasuna (berretzailea), ea pautaren bat aurkitzen dugun. Ondoren, Liouvilleren zenbakiak lortzeko metodo azkarra eta zenbaki triangeluarrekin duten erlazioa ikertuko dugu:</span></div><div class="separator" style="clear: both; text-align: left;"><span style="font-size: large;"><br /></span></div><div class="separator" style="clear: both; text-align: justify;"><span style="font-family: verdana;"><b><span style="font-size: x-large;">①</span></b><span style="font-size: large;"> </span></span><span style="font-size: large;"><img border="0" data-original-height="70" data-original-width="73" height="29" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgo1MxRXHcxAJ9P3iJvPOJmlo0oOs7z9y8KLxv3yIZVFNhBR8nuS8Eo0ah2N-QfgcqtSevigtyyWXR9xGBe6IrHNlnp3bFW4ad-GPy12LtAdxssd8EiMh7cqS0PRefAkFIDZMugf3vc_7R1/w30-h29/ZENBAKIAK+14.PNG" width="30" /><span style="font-family: verdana;"> erako zenbakiak. n berretzailea aldaraziko dugu, beheko koadro honetan ikusten den bezala. </span><span style="font-family: verdana;">Hainbat adibide aztertu ondoren, propietatea induzitu dezakete ikasleek. </span></span></div><div class="separator" style="clear: both; text-align: center;"><span style="font-size: large;"><br /></span></div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEihgL9fcGW38G-pXVNeaNQshI4kTOKYOf9xnrfAeIh5c-JtHC3KBUNNR5HO9fFuOnKqMjWC1-lUPvAgI7aK0SPkklb9Zf1GcFq8DlkMuHOhTxis-7rBp8MdYtDkdDxe4ObhIkYRoSA5i6PE/s1988/ZENBAKIAK+30.PNG" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1080" data-original-width="1988" height="544" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEihgL9fcGW38G-pXVNeaNQshI4kTOKYOf9xnrfAeIh5c-JtHC3KBUNNR5HO9fFuOnKqMjWC1-lUPvAgI7aK0SPkklb9Zf1GcFq8DlkMuHOhTxis-7rBp8MdYtDkdDxe4ObhIkYRoSA5i6PE/w1000-h544/ZENBAKIAK+30.PNG" width="1000" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: left;"><span style="font-family: verdana; text-align: justify;"><span style="font-size: large;">Erlazioa geometrikoki adierazteak problema aberasten du. Azaleren banaketa ez da bakarra, azalera bereko eta perimetro berdineko eta desberdineko hainbat poligono daude, adibidez:</span></span></div><div class="separator" style="clear: both; text-align: justify;"><span style="font-size: large;"><br /></span></div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg5FbttmzsW0jf7xwkPlsNMARohedv8-kGXPuAtczwtJ7KV4HCtQ2-Buqrv5SfhDWgYu8H1p0Gmev3iZOsfG7SZ98KttFhgleS5sD415hFUfH0Od2Bps09vw6wfuULU_mNffFqFuqi8jSr5/s1253/ZENBAKIAK+35.PNG" style="margin-left: 1em; margin-right: 1em;"><span style="font-size: large;"><img border="0" data-original-height="800" data-original-width="1253" height="511" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg5FbttmzsW0jf7xwkPlsNMARohedv8-kGXPuAtczwtJ7KV4HCtQ2-Buqrv5SfhDWgYu8H1p0Gmev3iZOsfG7SZ98KttFhgleS5sD415hFUfH0Od2Bps09vw6wfuULU_mNffFqFuqi8jSr5/w800-h511/ZENBAKIAK+35.PNG" width="800" /></span></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: left;"><span style="font-family: verdana; font-size: x-large;"><b>②</b></span><span style="font-family: verdana; font-size: large;"> Bi zenbaki lehen desberdin ditu, baten berretzailea beti 1 eta bestearena aldaraziz:</span></div><div class="separator" style="clear: both; text-align: center;"><span style="font-size: large;"><br /></span></div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj2FOfebkXfFww-M8av-PIMR2wqG0W4q0HWUbtIj_bhzelpW73C8wwTkdbX3WvdicKvODU6-t4OxIg5loHJLbcZ6Q6LRnIBT-4SLTwO-N41hdbmVpExIR1HMp0o6zCja6zNxYt74FfPLK_a/s2044/ZENBAKIAK+31.PNG" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1158" data-original-width="2044" height="568" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj2FOfebkXfFww-M8av-PIMR2wqG0W4q0HWUbtIj_bhzelpW73C8wwTkdbX3WvdicKvODU6-t4OxIg5loHJLbcZ6Q6LRnIBT-4SLTwO-N41hdbmVpExIR1HMp0o6zCja6zNxYt74FfPLK_a/w1000-h568/ZENBAKIAK+31.PNG" width="1000" /></a></div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgRaOnTbmJpFwwm6n3di7u-59BKwEXQlRd0scAKvvOp7tHL9_iLh2TbAA9DOoqwZ6xGGcoM0onCyLeg5L4ZgWS5_0z9qWzus0QeFgsbDq6IdiEsxyUa3CvVpzCyr56WajXEHz-EWP75HYK5/s2113/ZENBAKIAK+32.PNG" style="margin-left: 1em; margin-right: 1em;"><span style="font-size: large;"><img border="0" data-original-height="741" data-original-width="2113" height="350" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgRaOnTbmJpFwwm6n3di7u-59BKwEXQlRd0scAKvvOp7tHL9_iLh2TbAA9DOoqwZ6xGGcoM0onCyLeg5L4ZgWS5_0z9qWzus0QeFgsbDq6IdiEsxyUa3CvVpzCyr56WajXEHz-EWP75HYK5/w1000-h350/ZENBAKIAK+32.PNG" width="1000" /></span></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: left;"><span style="font-family: verdana; font-size: x-large;"><b>③</b></span><span style="font-family: verdana; font-size: large;"> Hainbat zenbaki lehen desberdin, denak sinpleak:</span></div><div class="separator" style="clear: both; text-align: center;"><span style="font-size: large;"><br /></span></div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiI7Yk-gRaJb2XAaz0jKAVAGPoTBQuizJCqSGS6_NinVh3iaSQul1c1G6mfz_6JpwGwQ2I6cQQCNBXHKsSms61AbI5JoJH68svTwoIZDNK1g545sLyr-HnhZlO_wYZN3oo8a4b3MNzA28lZ/s2080/ZENBAKIAK+33.PNG" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="955" data-original-width="2080" height="575" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiI7Yk-gRaJb2XAaz0jKAVAGPoTBQuizJCqSGS6_NinVh3iaSQul1c1G6mfz_6JpwGwQ2I6cQQCNBXHKsSms61AbI5JoJH68svTwoIZDNK1g545sLyr-HnhZlO_wYZN3oo8a4b3MNzA28lZ/w1250-h575/ZENBAKIAK+33.PNG" width="1250" /></a></div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjhUS_y4ryHURsKcL2D6CPZ_C-vozYZ6zFpeymfgTqIYdOXGagefpGYzWGi9t6zKJoiiVgRdy-ym8osaC1hyphenhyphenp18AWuLeI2CSqofTo_9DSlukgVAm455EJ1LLaEjOHjOgzB5VPfqOEjyF54C/s2090/ZENBAKIAK+34.PNG" style="margin-left: 1em; margin-right: 1em;"><span style="font-size: large;"><img border="0" data-original-height="977" data-original-width="2090" height="468" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjhUS_y4ryHURsKcL2D6CPZ_C-vozYZ6zFpeymfgTqIYdOXGagefpGYzWGi9t6zKJoiiVgRdy-ym8osaC1hyphenhyphenp18AWuLeI2CSqofTo_9DSlukgVAm455EJ1LLaEjOHjOgzB5VPfqOEjyF54C/w1000-h468/ZENBAKIAK+34.PNG" width="1000" /></span></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: justify;"><span style="font-family: inherit; font-size: large;">Beste adibide batzuk ere azter daitezke.</span></div><div class="separator" style="clear: both; text-align: justify;"><br /></div><div class="separator" style="clear: both; text-align: justify;"><span style="font-family: verdana;"><span style="font-size: x-large;"><b>④</b></span><span style="font-size: large;">Liouvilleren zenbaki zerrenda azkarrago lortzeko metodoa iradoki edo aipa diezaiekegu: zenbakiaren faktore lehen bakoitzaren zatitzaileen zatitzaile kopurua kalkulatu (</span></span><span style="font-size: large;"><img border="0" data-original-height="70" data-original-width="73" height="29" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgo1MxRXHcxAJ9P3iJvPOJmlo0oOs7z9y8KLxv3yIZVFNhBR8nuS8Eo0ah2N-QfgcqtSevigtyyWXR9xGBe6IrHNlnp3bFW4ad-GPy12LtAdxssd8EiMh7cqS0PRefAkFIDZMugf3vc_7R1/w30-h29/ZENBAKIAK+14.PNG" width="30" />➜<span style="font-family: verdana;">{1,2,3,...,n, n+1}) eta kopuruen zerrendak elkarren artean biderkatuz Liouvilleren zenbakiak izango ditugu, adibidez,</span></span></div><div class="separator" style="clear: both; text-align: left;"><span style="font-size: large;"><br /></span></div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh0XbzJuwWMMhDiF_36L8vG3n6ZjMKq9q6PQXopqnCnMgvNq2UDLMUJ_WcdvzsqBOJB635XtYfSTi_wCW3YKKDeLAZ9fCFjSACGUSoD1CtnipqohalHJJCbfy_25M7Nll8lzBIeDM0CEhSP/s1851/ZENBAKIAK+36.PNG" style="margin-left: 1em; margin-right: 1em;"><span style="font-size: large;"><img border="0" data-original-height="982" data-original-width="1851" height="531" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh0XbzJuwWMMhDiF_36L8vG3n6ZjMKq9q6PQXopqnCnMgvNq2UDLMUJ_WcdvzsqBOJB635XtYfSTi_wCW3YKKDeLAZ9fCFjSACGUSoD1CtnipqohalHJJCbfy_25M7Nll8lzBIeDM0CEhSP/w1000-h531/ZENBAKIAK+36.PNG" width="1000" /></span></a></div><div class="separator" style="clear: both; text-align: center;"><span style="font-size: large;"><br /></span></div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh4Lr9k6NWQFB8J11kQrCCRGBLicuUHefnKWciN68DdVI5mBZvqlBbs-cyAW0OYyyuLZ6hI8nWPyGS0QjiLEjbxofLojYap-qnukRMSlImHbAACZ8er5orf-jJ-Ne5gXIB6xVL5beEWONys/s1850/ZENBAKIAK+37.PNG" style="margin-left: 1em; margin-right: 1em;"><span style="font-size: large;"><img border="0" data-original-height="1070" data-original-width="1850" height="579" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh4Lr9k6NWQFB8J11kQrCCRGBLicuUHefnKWciN68DdVI5mBZvqlBbs-cyAW0OYyyuLZ6hI8nWPyGS0QjiLEjbxofLojYap-qnukRMSlImHbAACZ8er5orf-jJ-Ne5gXIB6xVL5beEWONys/w1000-h579/ZENBAKIAK+37.PNG" width="1000" /></span></a></div><div class="separator" style="clear: both; text-align: center;"><div class="separator" style="clear: both; text-align: center;"><span style="font-size: large;"><br /></span></div><span><div style="font-family: verdana; text-align: justify;"><span style="font-size: x-large;"><b>⑤</b></span><span style="font-size: large;"> Honetan oinarrituta ikerketan sakonduko dugu. Erlazioaren alde batean zenbaki karratuak agertzen dira, Liouvilleren zenbakien baturaren karratua. Edozein zenbaki karratu sor daiteke Liouvilleren zenbakiekin? Zeintzuk dira posibleak? 5²=25 ager daiteke? Eta 6²=36? 18²=324? 12²=144?...</span></div><div style="font-family: verdana; text-align: justify;"><span style="font-size: large;">Azter dezagu adibide bat:</span></div><div style="font-family: verdana; font-size: x-large; text-align: left;"><br /></div><div class="separator" style="clear: both; font-family: verdana; font-size: x-large; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgD5y_cUuxEHcjVETDRRZ5VhsAmOXXYeJNLNtMGe2zL6Lu51Eb7sL8bKYJ9qhqClecwlB0ud5I6MAEVxCyPjmxgT7T3VJCGKJgUgWE9p_293vENgzKwKQ6knElks28k3GtAKDIyLjEkSzhZ/s1766/ZENBAKIAK+38.PNG" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1091" data-original-width="1766" height="618" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgD5y_cUuxEHcjVETDRRZ5VhsAmOXXYeJNLNtMGe2zL6Lu51Eb7sL8bKYJ9qhqClecwlB0ud5I6MAEVxCyPjmxgT7T3VJCGKJgUgWE9p_293vENgzKwKQ6knElks28k3GtAKDIyLjEkSzhZ/w1000-h618/ZENBAKIAK+38.PNG" width="1000" /></a></div><div class="separator" style="clear: both; font-family: verdana; font-size: x-large; text-align: left;"><br /></div><div class="separator" style="clear: both; font-family: verdana; text-align: justify;"><span style="font-size: large;">Adibide gehiagorekin konproba dezakegu Liouvilleren zenbakiak zenbaki triangeluarrak eratzen dituzten segiden arteko biderkadurek sortzen dituztela eta Liouvilleren zenbakien batura beti ere zenbaki triangeluar bat edo hainbat zenbaki triangeluarren biderkadura dela. </span></div><div class="separator" style="clear: both; font-family: verdana; text-align: left;"><span style="font-size: large;"><br /></span></div><div class="separator" style="clear: both; font-family: verdana; text-align: left;"><span style="font-size: large;">Gogoratu n-garren zenbaki triangeluarra lehenengo n zenbaki arrunten batura dela:</span></div><div class="separator" style="clear: both; font-family: verdana; font-size: x-large; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhC-pUTv6xuiVw-glJMTCAnRMgFhL9eGpC69wnl84XPeUaqbN13BlcT5_G9Nzf-Q9gsJrFWCCux-F3LqvYlovqc4ByPbaQffpXSaet2r3d27KQf-4DqrIFJBeDXjfJTI88-ejCCiGAtRKme/s2091/ZENBAKIAK+40.PNG" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1005" data-original-width="2091" height="481" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhC-pUTv6xuiVw-glJMTCAnRMgFhL9eGpC69wnl84XPeUaqbN13BlcT5_G9Nzf-Q9gsJrFWCCux-F3LqvYlovqc4ByPbaQffpXSaet2r3d27KQf-4DqrIFJBeDXjfJTI88-ejCCiGAtRKme/w1000-h481/ZENBAKIAK+40.PNG" width="1000" /></a></div><div class="separator" style="clear: both; font-family: verdana; font-size: x-large; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiYERwl3n13lQUK3znWlsYnA0IP0Fe6hh8SBPMPrMNreb9apOkAC3OUNJiL2gP1T0MBPlRrKuDCHNcqJpanV6fJ0jKXyDD7rVV3TTJx9i_iTKxbtX8zxfk5PfUrc3W_lzFvObJEyYj4S9uE/s2095/ZENBAKIAK+41.PNG" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1029" data-original-width="2095" height="491" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiYERwl3n13lQUK3znWlsYnA0IP0Fe6hh8SBPMPrMNreb9apOkAC3OUNJiL2gP1T0MBPlRrKuDCHNcqJpanV6fJ0jKXyDD7rVV3TTJx9i_iTKxbtX8zxfk5PfUrc3W_lzFvObJEyYj4S9uE/w1000-h491/ZENBAKIAK+41.PNG" width="1000" /></a></div><div class="separator" style="clear: both; font-family: verdana; text-align: center;"><span style="font-size: large;"><br /></span></div><div class="separator" style="clear: both; font-family: verdana; text-align: justify;"><span style="font-size: large;">Beraz, 5²=25 eta 12²=144 ezin dira agertu Liouvilleren erlazioan, 5 eta 12=3·4 ez direlako zenbaki triangeluarrak, ezta zenbaki triangeluarren biderkadura ere. Bai, ordea, 36=6²=(T₃)² eta 18²=(3·6)²=(T₂·T₃)²</span></div><div class="separator" style="clear: both; font-family: verdana; text-align: justify;"><span style="font-size: large;">N zenbakiaren faktore lehen bakoitzaren zatitzaileen zatitzaile kopuruen batura zenbaki triangeluarra denez, Liouvileren zenbakien batura zenbaki triangeluarren biderkadura da eta zenbaki triangeluar hauen ordena faktore lehen bakoitzaren 𝝰 berretzailea gehi bat da (</span><span style="font-size: large;">𝝰</span><span style="font-size: large;">+1):</span></div><div class="separator" style="clear: both; font-family: verdana; text-align: justify;"><span style="font-size: large;"><br /></span></div><div class="separator" style="clear: both; font-family: verdana; text-align: center;"><div class="separator" style="clear: both; font-size: x-large; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgDw9XwMYyIykS_gj1UfiXDLbQL_XuOu7wNgxTofxRLoHLQHIdsvfZIBpWVe3jtBrxI3Hd9e32ue5gnvOOgfesLgdXczVqlvYSGD2IZ5QueOla5YLA5UiZIuljCUeJySN3LCSEPveFitxMv/s1990/ZENBAKIAK+42.PNG" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1021" data-original-width="1990" height="513" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgDw9XwMYyIykS_gj1UfiXDLbQL_XuOu7wNgxTofxRLoHLQHIdsvfZIBpWVe3jtBrxI3Hd9e32ue5gnvOOgfesLgdXczVqlvYSGD2IZ5QueOla5YLA5UiZIuljCUeJySN3LCSEPveFitxMv/w1000-h513/ZENBAKIAK+42.PNG" width="1000" /></a></div><div class="separator" style="clear: both; text-align: left;"><span style="font-size: large;">Jarraian hiru adibide zenbaki triangeluar eta Liouvilleren zenbakien arteko erlazioa ikusteko:</span></div><div class="separator" style="clear: both; text-align: left;"><span style="font-size: large;"><br /></span></div></div><div class="separator" style="clear: both; font-family: verdana; text-align: justify;"><span style="font-size: large;"><div class="separator" style="clear: both; font-size: medium;"><span style="font-size: large;"><b><u>Adibide 1</u></b></span></div><div class="separator" style="clear: both; font-size: medium;"><span style="font-size: large;"><i>N=144 zenbakiaren Liouvileren zenbaki zerrenda kalkulatu eta hauen batura adierazi zenbaki triangeluarrak erabiliz.</i></span></div></span></div><div class="separator" style="clear: both; font-family: verdana; text-align: justify;"><span style="font-size: large;"><b><u><br /></u></b></span></div><div class="separator" style="clear: both; font-family: verdana; text-align: justify;"><span><div class="separator" style="clear: both; text-align: left;"><span style="font-size: large;"><span><b><span>144=2⁴</span></b></span><b>·</b><b>3²</b></span></div><div class="separator" style="clear: both; text-align: left;"><span style="font-size: large;"><b>2⁴➩zatitzaleen zatitzaile kopurua ➩ {</b><b>1,2,3,4,5}</b></span></div><div class="separator" style="clear: both; text-align: left;"><span style="font-size: large;"><b>3²</b><b>➩zatitzaleen zatitzaile kopurua ➩ {</b><b><span style="color: red;">1</span>,<span style="color: #3d85c6;">2</span>,<span style="color: #6aa84f;">3</span></b><b>}</b></span></div><div class="separator" style="clear: both; text-align: left;"><span style="font-size: large;"><b>{</b><b>1,2,3,4,5} eta </b><b>{</b><b><span style="color: red;">1</span>,<span style="color: #3d85c6;">2</span>,<span style="color: #6aa84f;">3</span></b><b>} elkarren artean biderkatuz Liovilleren zenbaki zerrenda dugu:</b></span></div><div class="separator" style="clear: both; text-align: center;"><span style="font-size: large;"><b>{</b><b><span style="color: red;">1</span>,<span style="color: red;">2</span>,<span style="color: red;">3</span>,<span style="color: red;">4</span>,<span style="color: red;">5</span>,<span style="color: #3d85c6;">2</span>,<span style="color: #3d85c6;">4</span>,<span style="color: #3d85c6;">6</span>,<span style="color: #3d85c6;">8</span>,<span style="color: #3d85c6;">10</span>,<span style="color: #6aa84f;">3</span>,<span style="color: #6aa84f;">6</span>,<span style="color: #6aa84f;">9</span>,<span style="color: #6aa84f;">12</span>,<span style="color: #6aa84f;">15</span></b><b>}</b></span></div><div class="separator" style="clear: both; text-align: left;"><b><span style="font-size: large;"><br /></span></b></div><div class="separator" style="clear: both; text-align: left;"><span style="font-size: large;"><b><span style="color: red;">1</span>+<span style="color: red;">2</span>+<span style="color: red;">3</span>+<span style="color: red;">4</span>+<span style="color: red;">5</span>+<span style="color: #3d85c6;">2</span>+<span style="color: #3d85c6;">4</span>+<span style="color: #3d85c6;">6</span>+<span style="color: #3d85c6;">8</span>+<span style="color: #3d85c6;">10</span>+<span style="color: #6aa84f;">3</span>+<span style="color: #6aa84f;">6</span>+<span style="color: #6aa84f;">9</span>+<span style="color: #6aa84f;">12</span>+<span style="color: #6aa84f;">15</span></b><b>=</b><b>(1+2+3+4+5)(<span style="color: red;">1</span>+<span style="color: #3d85c6;">2</span>+<span style="color: #6aa84f;">3</span>)=</b></span></div><div class="separator" style="clear: both; text-align: left;"><span style="font-size: large;"><b><span>=T₅·T₃=</span></b><b>15·6=90</b></span></div><div class="separator" style="clear: both; text-align: left;"><span style="font-size: large;"><br /></span></div><div><div class="separator" style="clear: both; text-align: left;"><span style="font-size: large;"><b><span>1³+2</span>³+3³+4³+5³+2³+4³+6³+8³+10³</b><b>+3³</b><b>+6³</b><b>+9³</b><b>+12³</b><b>+15³</b><b>= 8.100</b></span></div><div class="separator" style="clear: both;"><div class="separator" style="clear: both; text-align: left;"><span style="font-size: large;"><span><b><span>(</span></b></span><b>1+2+3+4+5+2+4+6+8+10+3+6+9+12+15</b><span><b><span>)²=90</span>²= 8.100</b></span></span></div><div class="separator" style="clear: both; font-size: medium; text-align: center;"><br /></div><div class="separator" style="clear: both; font-size: medium; text-align: left;"><br /></div></div></div></span></div><div class="separator" style="clear: both; font-family: verdana; text-align: justify;"><span style="font-size: large;"><b><u>Adibide 2</u></b></span></div><div class="separator" style="clear: both; font-family: verdana; text-align: justify;"><span style="font-size: large;"><i>Kalkulatu batura 54 ematen duen Liovillen zenbaki zerrenda, koprobatu erlazioa egiaztatzen dela eta idatzi N zenbakiaren faktorizazio mota.</i></span></div><div class="separator" style="clear: both; font-family: verdana; text-align: justify;"><span style="font-size: large;"><br /></span></div><div class="separator" style="clear: both; font-family: verdana; text-align: justify;"><span style="font-size: large;"><b><span>54=3·3·6=T₂·T</span>₂·T₃=(1+2)(1+2)(1+2+3)=(1+2)(1+2+3+2+4+6) =1+2+3+2+4+6+2+4+6+4+8+12</b></span></div><div class="separator" style="clear: both; font-family: verdana; text-align: justify;"><span style="font-size: large;"><b><br /></b></span></div><div class="separator" style="clear: both; font-family: verdana; text-align: justify;"><span style="font-size: large;"><b><span>1³+2</span>³+3³+2³+4³+6³+2³+4³+6³+4³+8³+12³= 2.916</b></span></div><div class="separator" style="clear: both; font-family: verdana; text-align: justify;"><div class="separator" style="clear: both;"><span style="font-size: large;"><b><span>(1+2+3+2+4+6+2+4+6+4+8+12)²=54</span>²= 2.916</b></span></div><div class="separator" style="clear: both;"><span style="font-size: large;"><b><br /></b></span></div><div class="separator" style="clear: both;"><b><span style="font-size: large;">N=p·q·r² erako zenbakiari aurreko Liouvilleren zenbaki zerrenda dagokio. Adibidez: 3·7·11² ; 2·13·17² ; 5·2·3</span><span style="font-size: x-large;">² ;</span><span style="font-size: x-large;"> ...</span></b></div><div style="text-align: center;"><br /></div><div style="text-align: center;"><br /></div><div style="text-align: left;"><div class="separator" style="clear: both; text-align: justify;"><span style="font-size: large;"><b><u>Adibide 3</u></b></span></div><div class="separator" style="clear: both; text-align: justify;"><span style="font-size: large;"><i>Liouvilleren zenbakien batura zenbaki triangeluar baten potentzia izan dadin, zein izan behar da N zenbakiaren faktorizazio mota?</i></span></div><div class="separator" style="clear: both; text-align: left;"><span style="font-size: large;"><i><br /></i></span></div><div class="separator" style="clear: both; text-align: left;"><span style="font-size: large;"><i><b>Zenbaki triangeluarra ber bi:</b></i></span></div><div class="separator" style="clear: both; text-align: left;"><span style="font-size: large;"><span><i><b>(</b></i></span><b>T₂</b><span><i><b>)</b></i></span><b style="text-align: left;">²=</b><b>T₂·</b><b>T₂=(1+2)(1+2) </b><b style="text-align: left;">➩ N=p·q</b></span></div><div class="separator" style="clear: both; text-align: left;"><span style="font-size: large;"><span><i><b>(</b></i></span><span><b>T₃</b><i><b>)</b></i></span><b style="text-align: left;">²=</b><b>T₃</b><b>·</b><b>T₃</b><b>=(1+2+3)(1+2+3) </b><b style="text-align: left;">➩ N=p</b><b>²</b><b style="text-align: left;">·q</b><b>²</b></span></div><div class="separator" style="clear: both; text-align: left;"><span style="font-size: large;"><span><i><b>(</b></i></span><span><b>T₄</b><i><b>)</b></i></span><b style="text-align: left;">²=</b><b>T₄</b><b>·</b><b>T₄</b><b>=(1+2+3+4)(1+2+3+4) </b><b style="text-align: left;">➩ N=p</b><b>³</b><b style="text-align: left;">·q</b><b>³</b></span></div><div class="separator" style="clear: both; text-align: left;"><b><span style="font-size: large;">........</span></b></div><div class="separator" style="clear: both; text-align: left;"><span style="font-size: large;"><span><i><b>(</b></i></span><span><b>Tn-1</b><i><b>)</b></i></span><b style="text-align: left;">²=</b><b>T</b><b>n-1</b><b>·</b><b>T</b><b>n-1</b><b>=(1+2+...+n)(1+2+...+n) </b><b style="text-align: left;">➩ N=pⁿ</b><b style="text-align: left;">·q</b><b style="text-align: left;">ⁿ</b></span></div><div class="separator" style="clear: both; text-align: left;"><b><span style="font-size: large;"><br /></span></b></div><div class="separator" style="clear: both; text-align: left;"><i><b><span style="font-size: large;">Zenbaki triangeluarra ber hiru:</span></b></i></div><div class="separator" style="clear: both; text-align: left;"><div class="separator" style="clear: both;"><span style="font-size: large;"><span><i><b>(</b></i></span><b>T₂</b><span><i><b>)</b></i></span><b style="text-align: justify;">³</b><b>=</b><b>T₂·</b><b>T₂·</b><b>T₂</b><b>=(1+2)(1+2)</b><b>(1+2)</b><b> </b><b>➩ N=p·q·r</b></span></div><div class="separator" style="clear: both;"><span style="font-size: large;"><span><i><b>(</b></i></span><span><b>T₃</b><i><b>)</b></i></span><b style="text-align: justify;">³</b><b>=</b><b>T₃</b><b>·</b><b>T₃·</b><b>T₃</b><b>=(1+2+3)(1+2+3)</b><b>(1+2+3)</b><b> </b><b>➩ N=p</b><b>²</b><b>·q</b><b>²·r</b><b>²</b></span></div><div class="separator" style="clear: both;"><span style="font-size: large;"><span><i><b>(</b></i></span><span><b>T₄</b><i><b>)</b></i></span><b style="text-align: justify;">³</b><b>=</b><b>T₄</b><b>·</b><b>T₄·</b><b>T₄</b><b>=(1+2+3+4)(1+2+3+4)</b><b>(1+2+3+4)</b><b> </b><b>➩ N=p</b><b>³</b><b>·q</b><b>³·r</b><b>³</b></span></div><div class="separator" style="clear: both;"><b><span style="font-size: large;">........</span></b></div><div class="separator" style="clear: both;"><span><span style="font-size: x-large;"><i><b>(</b></i></span><span><b><span style="font-size: large;">T</span>n+1</b><i style="font-size: x-large;"><b>)</b></i></span><b style="font-size: x-large; text-align: justify;">³</b><b style="font-size: x-large;">=</b><b><span style="font-size: large;">T</span>n+1</b><b style="font-size: x-large;">·</b><b><span style="font-size: large;">T</span>n+1</b><b style="font-size: x-large;">·</b><b><span style="font-size: large;">T</span>n+1</b><span style="font-size: large;"><b>=</b><b>(1+2+...+n)(1+2+...+n)</b><b>(1+2+...+n)</b><b> </b><b>➩ N=pⁿ</b><b>·q</b><b>ⁿ·r</b><b>ⁿ</b></span></span></div><div class="separator" style="clear: both;"><span style="font-size: large;"><b><br /></b></span></div><div class="separator" style="clear: both; text-align: justify;"><b><span style="font-size: large;">Desberdinak diren N zenbakiaren faktore lehenen kopurua zenbaki triangeluarraren berretzaileak ematen du eta faktore lehenen berretzailea zenbaki triangeluarraren ordena ken bat da.</span></b></div></div><div style="text-align: justify;"><b style="font-size: x-large;"><br /></b></div></div><div style="text-align: center;">❃❃❃</div></div><div class="separator" style="clear: both; font-family: verdana; text-align: justify;"><span style="font-size: large;"><br /></span></div><div class="separator" style="clear: both; font-family: verdana; text-align: justify;"><span style="font-size: large;">Problema honetan erabili behar izan diren eduki matematikoak oinarrizkoak dira: zenbaki baten deskonposizioa faktore lehenetan, zatitzaileak, zatitzaile kopurua, berreturak (karratuak eta kuboak), zenbaki arrunten segida, zenbaki triangeluarrak,... Honek hainbat mailatan proposatzeko </span><span style="font-size: large;">aukera ematen digu, baina, beti ere, ikaslearen mailari dagokion egokitzapenak eginez.</span></div><div class="separator" style="clear: both; font-family: verdana; text-align: justify;"><span style="font-size: large;"> </span></div><div class="separator" style="clear: both; font-family: verdana; text-align: justify;"><span style="font-size: large;">Problemaren testuingurua guztiz matematikoa izanda ere, </span><span style="font-size: large;">hainbat kontzeptu, eduki, prozedura eta estrategia aplikatzeko aukera eskaintzen dio ikasleari; beraz, konpetentzia matematikoaren garapenarako problema egokia dugu gaurko hau. </span></div><div class="separator" style="clear: both; font-family: verdana; text-align: justify;"><br /></div><div class="separator" style="clear: both; font-family: verdana; text-align: justify;"><span style="font-size: large;">Aipatu beharra dago problemen ebazpenaren helburua ez dela emaitza zuzena lortzea besterik. Ahal den neurrian haruntzago joaten saiatuko gara: kasu partikularrak aztetu, galderak egin, konjeturak planteatu, problema zabaldu, problema berriak asmatu, hausnartu, arrazoitu, justifikatu eta orokortu; bi hitzetan: "Matematika Egin".</span></div><div class="separator" style="clear: both; font-family: verdana; text-align: justify;"><br /></div><div class="separator" style="clear: both; font-family: verdana; text-align: justify;"><span style="font-size: large;">Amaitzeko George Polya gogora ekarriko dugu prolemen ebazpenari buruzko bere esaldi esanguratsu honekin:</span></div><div class="separator" style="clear: both; font-family: verdana; text-align: justify;"><span style="font-size: large;"><br /></span></div><div class="separator" style="clear: both; font-family: verdana; text-align: center;"><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjHIrBxrTZb2jDZdCbaXgZvemqqdxKstym0tBKOnhF8blYuHScs_kOiHan4pu3tb0EBS0SttLGWiqjRg7aaCGkHZA3OvrP0UIjxlo1hf4xjvgpAYq0lUB74zP2egn6NpbfhDNRkn3GmPjMX/s1527/ZENBAKIAK+46.PNG" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="771" data-original-width="1527" height="323" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjHIrBxrTZb2jDZdCbaXgZvemqqdxKstym0tBKOnhF8blYuHScs_kOiHan4pu3tb0EBS0SttLGWiqjRg7aaCGkHZA3OvrP0UIjxlo1hf4xjvgpAYq0lUB74zP2egn6NpbfhDNRkn3GmPjMX/w640-h323/ZENBAKIAK+46.PNG" width="640" /></a></div></div><br /></span></div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: justify;"><i><b>Gaurko artikulu hau matematikaren bultzaitzaile aparta den eta laguna dudan J. M. Lopez Irastorzari eskaini nahi diot bereziki. Gure Liouville txikia, Matematikaren altxorren zaindari. </b></i></div><div class="separator" style="clear: both; text-align: justify;"><i><b>Eskerrik asko!!</b></i></div><div class="separator" style="clear: both; text-align: right;"><i><b>J. L. </b></i></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: left;"><br /></div><div><p><span style="font-family: verdana;"> </span></p></div></div>Egin Matematikahttp://www.blogger.com/profile/01348970164668368901noreply@blogger.com0tag:blogger.com,1999:blog-4037937100819081512.post-976825300139957702020-08-02T19:23:00.000+02:002021-06-26T09:44:33.136+02:00MATRIZE BITXIA<div><br /></div><br /><div><font face="verdana" size="5"><b>Honako laugarren ordenako A matrizea emanik:</b></font></div><div><font face="verdana" size="5"><b><br /></b></font></div><div style="text-align: center;"><b><br /></b></div><div style="text-align: center;"><b><img alt="" height="210" src="data:image/png;base64,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" width="608" /></b></div><div style="text-align: center;"><font face="verdana" size="5"><b><br /></b></font></div><div style="text-align: center;"><b><br /></b></div><div style="text-align: justify;"><font face="verdana" size="5"><b>Matrize honen determinantea garatu gabe, jakin al daiteke matrize hau erregularra den? Hau da, A matrizeak alderantzizko matrizea du?</b></font></div><div><font face="verdana" size="5"><br /></font></div><div><span><a name='more'></a></span><font face="verdana" size="5"><br /></font></div><div><font face="verdana" size="5"><br /></font></div><div><font color="#444444" face="verdana" size="5"><b><u>AZALPENA</u></b></font></div><div><font face="verdana" size="5"><br /></font></div><div style="text-align: justify;"><font color="#444444" face="georgia" size="5">Matrize karratu bat erregularra (alderanzgarria) da baldin alderantzizko matrizea badu. A matrize karratuak alderantzizkoa du, baldin eta bakarrik soilik Aren determinantea ≠0 bada.</font></div><div style="text-align: justify;"><font color="#444444" face="georgia" size="5"><br /></font></div><div style="text-align: justify;"><font color="#444444" face="georgia" size="5">n-garren ordenako matrize karratu baten determinantea, det(A), honela kalkulatzen da: </font></div><div style="text-align: justify;"><br /></div><div style="text-align: justify;"><span style="color: #444444; font-family: georgia; font-size: x-large;">➤Matrizeko errenkada eta zutabe desberdineko n elementu aukeratuz posibleak diren biderkadura guztiak egin (errenkada eta zutabe bakoitzeko elementu bakar bat aukeratu behar da biderkadurak osatzeko). Laugarren ordenako matrizean 4x4=16 daude.</span></div><div style="text-align: justify;"><span style="color: #444444; font-family: georgia; font-size: x-large;"><br /></span></div><div style="text-align: justify;"><span style="color: #444444; font-family: georgia; font-size: x-large;">➤Biderkadura bakoitzari (-) ikurra esleituko zaio biderkagaien azpiindizeen permutazioa bakoitia bada eta (+) ikurra bikoitia bada. Errenkadetako (zutabeetako) azpiindizeak goranzko ordenean idatzita daudela, zutabeen indizeen </span><span style="color: #444444; font-family: georgia; font-size: x-large;">(errenkadeen indizeen) </span><span style="color: #444444; font-family: georgia; font-size: x-large;">permutazioak zenbakituko dira.</span></div><div style="text-align: justify;"><span style="color: #444444; font-family: georgia; font-size: x-large;"><br /></span></div><div style="text-align: justify;"><span style="color: #444444; font-family: georgia; font-size: x-large;">➤</span><span style="color: #444444; font-family: georgia; font-size: x-large;">Biderkadura guztien baturak determinantearen balioa ematen digu.</span></div><div style="text-align: justify;"><span style="color: #444444; font-family: georgia; font-size: x-large;"><br /></span></div><div style="text-align: justify;"><span style="color: #444444; font-family: georgia; font-size: x-large;">Bestetik, </span></div><div style="text-align: justify;"><span style="color: #444444; font-family: georgia; font-size: x-large;"><br /></span></div><div style="text-align: center;"><img alt="" height="132" 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" width="409" /></div><div style="text-align: justify;"><span style="color: #444444; font-family: georgia; font-size: x-large;">Hau da,</span></div><div style="text-align: center;"><img alt="" height="133" 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" width="413" /></div><div style="text-align: justify;"><span style="color: #444444; font-family: georgia; font-size: x-large;"><br /></span></div><div style="text-align: justify;"><span style="color: #444444; font-family: georgia; font-size: x-large;">Diagonal nagusiko elementu guztiak bakoitiak direnez, biderkadura hau bakoitia da. Beste biderkadura guztiak bikoitiak dira, guztietan gutxienez zenbaki bikoiti batek parte hartzen duelako. 16 batugaietatik 15 bikoitiak eta bakar bat bakoitia. Guztien batura nahitaez bakoitia da, hau da, desberdin zero. Ondorioz, A matrizearen determinantea ez da nulua, A matrizea erregularra da.</span></div><div style="text-align: justify;"><span style="color: #444444; font-family: georgia; font-size: x-large;"><br /></span></div><div style="text-align: justify;"><span style="color: #444444; font-family: georgia; font-size: x-large;">Beste era honetan agian errazago ikus daiteke:</span></div><div style="text-align: justify;"><span style="color: #444444; font-family: georgia; font-size: x-large;"><br /></span></div><div style="text-align: justify;"><span style="color: #444444; font-family: georgia; font-size: x-large;">Matrizeko balio bakoitien lekuan 1 eta bikoitien ordez 0 jarriz, ondoko matrizea izango dugu:</span></div><div style="text-align: justify;"><span style="color: #444444; font-family: georgia; font-size: x-large;"><br /></span></div><div style="text-align: center;"><img alt="" height="190" 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" width="373" /></div><div style="text-align: justify;"><font color="#444444" face="georgia" size="5"><br /></font></div><div style="text-align: justify;"><font color="#444444" face="georgia" size="5">Matrize honen determinantea diagonal nagusiko elementuen biderkadura da.</font></div><div style="text-align: justify;"><font color="#444444" face="georgia" size="5"><br /></font></div><div style="text-align: justify;"><font color="#444444" face="georgia" size="5">B matrizearen determinantea bakoitia denez, A matrizearen determinantea ere bakoitia da, ondorioz desberdin zero,</font></div><div><font color="#444444" face="georgia" size="5"><br /></font></div><div style="text-align: center;"><img alt="" height="35" 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" width="545" /></div><div><font color="#444444" face="georgia" size="5"><br /></font></div><div><br /></div><div><font face="verdana" size="5"><br /></font></div><div style="text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><iframe allowfullscreen="" class="BLOG_video_class" height="266" src="https://www.youtube.com/embed/pMrxq1m9Lp4" width="320" youtube-src-id="pMrxq1m9Lp4"></iframe></div><div style="text-align: center;"><br /></div><div style="text-align: center;"><br /></div><div style="text-align: center;"><br /></div><div><font face="verdana" size="5"><br /></font></div><span></span><span><br /></span>Egin Matematikahttp://www.blogger.com/profile/01348970164668368901noreply@blogger.com0tag:blogger.com,1999:blog-4037937100819081512.post-12388958604169178242019-12-18T01:35:00.001+01:002023-10-24T09:50:46.274+02:00Deribatuen Erabilerak. Maximo-minimoak, Optimizazioa, Funtzioen Hazkundea, Adierazpen Grafikoak,...)<div style="text-align: center;">
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh272q8L73rK7voW31yX6erogBPHdgeOF1EARlrnCR7-8af_pnlCRyrOF8woVJtKIFaob4FE3h79vjPLJZ5pT8MEhpli1O3Ynj7XmB-hZY7-T71dnOxsz1a3xORQXWEztPbrVgMMLJo1rdV/s1600/FUNTZIOAK6.JPG" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" data-original-height="406" data-original-width="574" height="141" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh272q8L73rK7voW31yX6erogBPHdgeOF1EARlrnCR7-8af_pnlCRyrOF8woVJtKIFaob4FE3h79vjPLJZ5pT8MEhpli1O3Ynj7XmB-hZY7-T71dnOxsz1a3xORQXWEztPbrVgMMLJo1rdV/s200/FUNTZIOAK6.JPG" width="200" /></a><span face=""verdana" , sans-serif" style="font-size: large;">Batxilergoko 2. mailan funtzioen deribatuak eta hauen erabilerak gai nagusietako bat da. 17-18 urteko gazteek buru belarri ekiten diote deribatzeari eta teknika berri honen erabilerari; baina, deribatuen kontzeptua arrotza eta beraien egunerokotasunetik oso urrun ikusten dute ikasleek. Egoera honetan, zaila egiten zaie tresna honen benetako indarraz konturatzea eta deribatuen erabileraren teknika menperatzea.</span><br />
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<span face=""verdana" , sans-serif" style="font-size: large;">Ikaste-prozesu honetan ikasleak laguntzeko eta bidea, apur bat bada ere, errazteko helburuarekin, hainbat bideo grabatu ditut. Honela, ikasgela edozein ordutan eta edozein tokitan eskuragarri izango dute, beraien beharretara moldatuz.</span><br />
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<span face=""verdana" , sans-serif" style="font-size: large;">Jarraian, aipaturiko Deribatuen Erabileraren gaia prestatzeko bideoak doaz eta hautaprobetako ariketa ebatzien bilduma, ea erabilgarriak eta lagungarriak diren: </span><br />
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<span class="title">DERIBAGARRITASUNA 1</span>
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Egin Matematikahttp://www.blogger.com/profile/01348970164668368901noreply@blogger.com0tag:blogger.com,1999:blog-4037937100819081512.post-36498782198318808062019-09-01T18:48:00.001+02:002019-09-28T16:41:11.793+02:00Probabilitate problema erakargarriak. Bertrand-en kutxaren paradoxa.<br />
<span style="font-family: "verdana" , sans-serif; font-size: large;">Joseph Bertrand (1822-1900) matematikari frantziarrak "Calcul des probabilités" liburuan "Bertrand-en kutxaren paradoxa" izenarekin ezaguna den probabiliate problema proposatun zuen.</span><br />
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<span style="font-family: "verdana" , sans-serif; font-size: large;">Problema honen aldaera ugari daude, baina guztietan sortzen den paradoxak jatorri bera du: gertaeren independentzia eta gertaera baldintzatuak bereizteko zailtasunak, eta baita lagin espazioa determinatzeko ere. Askoren ustean orain gertatutako gertaera batek ezin du eraginik izan aurretik gertatutako gertaera batean;akats honi, "denboraren ardatzaren falazia" esaten zaio.</span><br />
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<span style="font-family: "verdana" , sans-serif; font-size: large;">Problemaren hainbat bertsio ikusiko ditugu jarraian, denak oso erakargarriak eta motibagarriak. Eztabaida sutsuak eta mamitsuak sor daitezke gelan edota lagunen artean proposatzen badira.</span><br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg1amuzaRwoklqDXuVg2iU5nqwgnTuVOkGIl40zxmgXydtlapxPEz5POFJ3PQlxK6hiiJya_Bn1GCVHcp5WCiawZhPfeMORe6rMsoKdaGlRqM1WxlviTBQoqT6uBnCChWVA5BRz8w8IS4w9/s1600/COFRES.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="312" data-original-width="1045" height="95" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg1amuzaRwoklqDXuVg2iU5nqwgnTuVOkGIl40zxmgXydtlapxPEz5POFJ3PQlxK6hiiJya_Bn1GCVHcp5WCiawZhPfeMORe6rMsoKdaGlRqM1WxlviTBQoqT6uBnCChWVA5BRz8w8IS4w9/s320/COFRES.JPG" width="320" /></a></div>
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<span style="color: #0b5394; font-family: "verdana" , sans-serif; font-size: large;"><b><u>1. BERTRAND-EN KUTXAREN PARADOJA</u></b></span><br />
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<span style="color: #0b5394; font-family: "verdana" , sans-serif; font-size: large;">Hiru kutxa daude bakoitzak bi kutxatxo barruan dituelarik eta kutxatxo bakoitzaren barruan txanpon bana. Lehenengo kutxan urrezko bi txanpon daude, bigarrenean urrezko txanpon bat kutxatxo batean eta zilarrezko bat bestean eta hirugarren kutxan zilarrezko bi txanpon. Zoriz kutxa bat aukeratu ondoren, zoriz ere barruko kutxatxo bat aukeratzen eta barruan dagoen txanpona ateratzean urrezkoa dela ikusten da.</span></div>
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<span style="color: #0b5394; font-family: "verdana" , sans-serif; font-size: large;">Zein probabilitatea dago kutxako beste kutxatxoan dagoen txanpona ere urrezkoa izateko?</span></div>
<span style="font-family: "verdana" , sans-serif; font-size: large;"><br /></span><span style="font-family: "verdana" , sans-serif; font-size: large;">Adi! Intuizioak sarritan kale egiten digu eta. Baldintzaren eragina ez da kontuan hartzen eta txanpon bat atera ondoren ere gertaerak ekiprobableak direla uste izaten da sarri askotan.</span><br />
<span style="font-family: "verdana" , sans-serif; font-size: large;">Hasiera batean, hiruren artean kutxa bat aukeratzen denean, probabilitate bera dago edozein aukeratzeko (1/3), gertaera ekiprobableak baitira.</span><br />
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<span style="font-family: "verdana" , sans-serif; font-size: large;">Behin kutxa bat aukeratua izan denean eta bertatik urrezko txanpona atera ondoren, egoera berri baten aurrean gaude. Informazio berri bat dugu, eta informazio honek lagin espazioa aldatu edo murriztu egiten du, zilarrezko bi txanpon dituen kutxa baztertzen baitu. Eta gainera, bi gertaerak (1. kutxatik edo 2. kutxatik aterata izatea) ekiprobableak ez direla nahiko ebidentzia dago. Badirudi atera den txanpona urrezkoa izateak, 1. kutxatik (urrezko bi txanponak dituena) aterata izatea errazagoa dela pentsatzera eramaten gaituela (kontuan izan 2. kutxan urrezko bakarra dagoela). </span><br />
<span style="font-family: "verdana" , sans-serif; font-size: large;">Izenak ipiniko dizkiegu urrezko txanponei: 1. kutxan urrezko A eta urrezko B eta 2. kutxan urrezko C eta zilarrezkoa. Atera den urrezko txanpona A, B edo C da, baina bereizgarririk ez dutenez, ezin da jakin hiru horietako zein den. Beraz, hiru kasu hauek bereizten dira:</span><br />
<span style="font-family: "verdana" , sans-serif; font-size: large;"><br /></span><span style="font-family: "verdana" , sans-serif; font-size: large;"> Urrezko A atera, barruan urrezko B</span><br />
<span style="font-family: "verdana" , sans-serif; font-size: large;"> Urrezko B atera, barruan urrezko A</span><br />
<span style="font-family: "verdana" , sans-serif; font-size: large;"> Urrezko C atera, barruan zilarrezkoa</span><br />
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<span style="font-family: "verdana" , sans-serif; font-size: large;">Hiru kasuetatik bitan barruan urrezkoa txanpona dagoenez, barruan dagoen txanpona urrezkoa izateko probabilitatea edo, berdina dena, txanpona 1. kutxatik aterata izatekoa 2/3 da eta barruko txanpona zilarrezkoa izateko 1/3.</span><br />
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<span style="font-family: "verdana" , sans-serif; font-size: large;">Problema bera beste era batean enuntziatuta, urrezko eta zilarrezko txanponen ordez bola beltzak eta zuriak erabiliz:</span><br />
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<span style="color: #0b5394; font-family: "verdana" , sans-serif; font-size: large;"><u><b>2. BERTRANDEN KUTXAREN PARADOXA (2)</b></u></span></div>
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<span style="color: #0b5394; font-family: "verdana" , sans-serif; font-size: large;">Hiru kutxa ditugu. Lehenengoak bi bola zuri, bigarrenak bat zuria eta beste bat beltza eta hirugarrenak bi beltz ditu. Zoriz kutxa bat aukeratu ondoren, be<span style="color: #666666;"></span>rtatik bola bat atera dugu. Ateratako bola beltza izan bada, zein da hirugarren kutxa aukeratu izanaren probabilitatea?</span><br />
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<span style="font-family: "verdana" , sans-serif; font-size: large;"><span style="font-family: "verdana" , sans-serif; font-size: large;">AZALPENA: </span><a href="https://eginmatematika.blogspot.com/2016/07/kutxak-bolak-eta-probabilitatea-iii.html" style="text-align: center;" target="_blank">"Kutxak, bolak eta probabilitatea (III)"</a></span></div>
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<span style="font-family: "verdana" , sans-serif; font-size: large;"><br /></span><span style="font-family: "verdana" , sans-serif; font-size: large;"><u style="color: #0b5394; text-align: justify;"><b>3. BI TXANPONAK</b></u></span> <span style="font-family: "verdana" , sans-serif; font-size: large;"><br /></span><span style="font-family: "verdana" , sans-serif; font-size: large;"><br /></span><span style="color: #0b5394; font-family: "verdana" , sans-serif; font-size: large;">Kutxa batean bi txanpon ditugu. 1. txanponak bi aldeak zuriz margotuta ditu eta 2. txanponak alde bat zuriz eta bestea gorriz. Zoriz txanpon bat hartu, airera jaurti eta zuria irten da.</span><br />
<span style="color: #0b5394; font-family: "verdana" , sans-serif; font-size: large;">Zer da probableago, ikusten ez den aldea zuria ala gorria izatea?</span><br />
<span style="color: #0b5394; font-family: "verdana" , sans-serif; font-size: large;">Bigarren aldiz txanpon bera jaurtitzen da eta berriz ere zuria irten da. Zein da ikusten ez den aldea zuria izateko probabilitatea?</span><br />
<span style="color: #0b5394; font-family: "verdana" , sans-serif; font-size: large;"><br /></span><span style="font-family: "verdana" , sans-serif; font-size: large;"><u><b>Azalpena</b></u></span><br />
<span style="font-family: "verdana" , sans-serif; font-size: large;"><u><br /></u></span><span style="font-family: "verdana" , sans-serif; font-size: large;">Gora begira irten den aldea zuria denez, beste aldearentzat bi aukera daude gorria ala zuria. Beraz, probabilitatea bera dago, 1/2. Baina, arrazoibide hau okerra da.</span><br />
<span style="font-family: "verdana" , sans-serif; font-size: large;"><br /></span><span style="font-family: "verdana" , sans-serif; font-size: large;">Egia da ezkutuan dagoen aldea zuria ala gorria izan daitekeela; hau da, lagin espazioa aurpegi honentzat: {Zuria, Gorria}, baina, gertaera hauek ez dira ekiprobableak.</span><br />
<span style="font-family: "verdana" , sans-serif; font-size: large;"><br /></span><span style="font-family: "verdana" , sans-serif; font-size: large;">Bi txanpon daude: (Z₁,Z₂), (Z₃,G); hau da, lau aurpegi. Lau aurpegietatik 3 zuriak eta bat gorria da. Ikusten den aldea zuria denez, eskutuan dagoen aldearentzat hiru aukera daude: bi zuri eta bat gorri.</span><br />
<span style="font-family: "verdana" , sans-serif; font-size: large;"><br /></span><span style="font-family: "verdana" , sans-serif; font-size: large;"><b> ➤ Z₁</b> ikusten bada, beste aldean: </span><span style="font-family: "verdana" , sans-serif; font-size: large;"><b>Z₂</b></span><br />
<span style="font-family: "verdana" , sans-serif; font-size: large;"><b> ➤ Z₂</b> </span><span style="font-family: "verdana" , sans-serif; font-size: large;">ikusten bada, beste aldean: </span><span style="font-family: "verdana" , sans-serif; font-size: large;"><b>Z₁</b></span><br />
<span style="font-family: "verdana" , sans-serif; font-size: large;"><b> ➤ Z₃</b> </span><span style="font-family: "verdana" , sans-serif; font-size: large;">ikusten bada, beste aldean: <b>G</b></span><br />
<span style="font-family: "verdana" , sans-serif; font-size: large;"><br /></span><span style="font-family: "verdana" , sans-serif; font-size: large;">Beraz, txanpona jaurti ondoren zuria irten bada, probableagoa da ikusten ez den aldea zuria izatea, honen probabilitatea gorria izatearen bikoitza da:</span><br />
<span style="font-family: "verdana" , sans-serif; font-size: large;"><br /></span><span style="font-family: "verdana" , sans-serif; font-size: large;"> P(Z) = 2/3 eta P(G) = 1/3</span><br />
<span style="font-family: "verdana" , sans-serif; font-size: large;"><br /></span><span style="font-family: "verdana" , sans-serif; font-size: large;"><span style="color: #0b5394;">➤</span>Irudi baten laguntzaz argiago ikusiko da:</span><br />
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<span style="font-family: "verdana" , sans-serif; font-size: large;"><br /></span><span style="font-family: "verdana" , sans-serif; font-size: large;"><span style="color: #0b5394;">➤</span>Kontigentzia taula baten laguntzaz:</span><br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhS3_EcWLQBFpKMzb9wvPH691fBbr9xg8HS4RcxVzl_OWy7Quy3SCXrtqvkEsbjVF-FvyAF9jYoIAaZEARi5cyfcE04RzSAonXb-I0VIG-Qo0erDlbNRwyhxMe6nPC6TfHer0OYaLmHBcB2/s1600/PROBABILITATEA4.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="512" data-original-width="1060" height="308" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhS3_EcWLQBFpKMzb9wvPH691fBbr9xg8HS4RcxVzl_OWy7Quy3SCXrtqvkEsbjVF-FvyAF9jYoIAaZEARi5cyfcE04RzSAonXb-I0VIG-Qo0erDlbNRwyhxMe6nPC6TfHer0OYaLmHBcB2/s640/PROBABILITATEA4.JPG" width="640" /></a></div>
<span style="font-family: "verdana" , sans-serif; font-size: large;"><br /></span><span style="font-family: "verdana" , sans-serif; font-size: large;">Ikusten den aldea zuria denez, lagin espazioa murriztu egin da, lehenengo zutabeari begiratu behar diogu soilik. 3 aurpegi zuri daude, eta hauetatik 2 dagozkio bi aldeak zuriz margotuta dagoen txanponari; hau da, 1. txanponari. Ondorioz, gorria baino bi aldiz probableagoa da azpiko aldea zuria izatea:</span><br />
<span style="font-family: "verdana" , sans-serif; font-size: large;"><br /></span><span style="font-family: "verdana" , sans-serif; font-size: large;">P((Z,Z)/ikusi Z)=2/3 eta P((Z,G)/ikusi Z)=1/3</span><br />
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<span style="font-family: "verdana" , sans-serif; font-size: large;"><span style="color: #0b5394;">➤</span>Bayesen teoremaz ere ebatz daiteke:</span><br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhXVUk6vD5X56A0y2J1f4qybMIRtMowqhcXH1zKegYeLsHrNcn35FervqjxH7xfEsVgHVQJ5pFWyZrCoQzRmz-FYyAwmt5NySFLCQ4IQS_qHjsQCSD_WfTCBwEdXMz6PYNgjhvd2WJBBlQv/s1600/PROBABILITATEA5.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="792" data-original-width="1367" height="370" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhXVUk6vD5X56A0y2J1f4qybMIRtMowqhcXH1zKegYeLsHrNcn35FervqjxH7xfEsVgHVQJ5pFWyZrCoQzRmz-FYyAwmt5NySFLCQ4IQS_qHjsQCSD_WfTCBwEdXMz6PYNgjhvd2WJBBlQv/s640/PROBABILITATEA5.JPG" width="640" /></a></div>
<span style="font-family: "verdana" , sans-serif; font-size: large;"><br /></span><span style="font-family: "verdana" , sans-serif; font-size: large;"><br /></span><span style="font-family: "verdana" , sans-serif; font-size: large;"><span style="font-family: "verdana" , sans-serif; font-size: large;"><u style="color: #0b5394; text-align: justify;"><b>4. HIRU TXANPONAK</b></u></span><span style="font-family: "times new roman"; font-size: small;"> </span></span><br />
<span style="font-family: "verdana" , sans-serif; font-size: large;"><br /></span><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgUoDh728RMgli2PybBxgyj11ByX2pMSREfEhmSS8Ws4RloYgs-JOJ2jSKyc0V_ximBnCS-WLoLG_uk7xKK9FtnSyqB0lrjIbU8S6eBSYbCiOG5sOagohYfmL6a6a00q8ISsP_GFT7hUmOY/s1600/txanponak.JPG" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" data-original-height="405" data-original-width="485" height="166" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgUoDh728RMgli2PybBxgyj11ByX2pMSREfEhmSS8Ws4RloYgs-JOJ2jSKyc0V_ximBnCS-WLoLG_uk7xKK9FtnSyqB0lrjIbU8S6eBSYbCiOG5sOagohYfmL6a6a00q8ISsP_GFT7hUmOY/s200/txanponak.JPG" width="200" /></a><span style="color: #0b5394; font-family: "verdana" , sans-serif; font-size: large;">Hiru txanpon ditugu, batak bi aldeetatik zuriz margotuta, beste bat bietatik gorriz eta azkena alde batetik zuriz eta bestetik gorriz. Poltsa batean sartzen ditugu eta begiratu barik zoriz bat aukeratzen dugu. Ondoren mahai gainean jartzen dugu bere aurpegietako bat bistan dagoela eta bestea ezkutuan. Demagun ikusten den aldea zuria dela.</span><br />
<span style="color: #0b5394; font-family: "verdana" , sans-serif; font-size: large;">Apustua egin behar duzu ikusten ez den aldea zuria ala gorria dela, zein aukeratuko zenuke? Zer da probableagoa, ezkutuan dagoen aurpegia zuria ala gorria izatea? </span><br />
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<span style="font-family: "verdana" , sans-serif; font-size: large;"><u><b>Azalpena</b></u></span><br />
<span style="font-family: "verdana" , sans-serif; font-size: large;"><br /></span><span style="font-family: "verdana" , sans-serif; font-size: large;">Hasieran hiru txanpon daude,eta zoriz txanpon bat hartzen da; beraz, lagin espazioa: <b>{1. txanpona, 2. txanpona, 3. txanpona}</b>. Hiru gertaera hauek ekiprobableak dira (<b>P = 1/3</b>).</span><br />
<span style="font-family: "verdana" , sans-serif; font-size: large;"><br /></span><span style="font-family: "verdana" , sans-serif; font-size: large;">Txanpon bat hartu, mahai gainean bere aurpegietako bat gora begira jarri eta zuria dela ikusterakoan, lagin espazioa aldatu egiten da, bi aldeetatik gorria den txanpona (2. txanpona) ez dela aukeratua izan dakigulako. Lagin espazioa: {1. txanpona, 3. txanpona}. Baina orain bi gertaera hauek ez dira ekiprobableak eta </span><span style="font-family: "verdana" , sans-serif; font-size: large;">bi txanponen problema ebatzi dugun eran ebatziko genuke hau ere.</span><br />
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<span style="font-family: "verdana" , sans-serif; font-size: large;"><u style="color: #0b5394; text-align: justify;"><b>5. MONTY HALL</b></u></span><br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiEDNkoOxgKnyE76bpZpmbmb4QDxBO2k2bZEhKEkzsTz8107rQnD5MMfU1yQwmY_1A065m-WACxrW5os8zRc6Ghd3sbHufkPt4w2feOkwjzlyv91PqsW3A1cOhI0FZOpz1zcpZ8YfBQLwUo/s1600/MONTY+HALL2.JPG" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" data-original-height="522" data-original-width="843" height="123" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiEDNkoOxgKnyE76bpZpmbmb4QDxBO2k2bZEhKEkzsTz8107rQnD5MMfU1yQwmY_1A065m-WACxrW5os8zRc6Ghd3sbHufkPt4w2feOkwjzlyv91PqsW3A1cOhI0FZOpz1zcpZ8YfBQLwUo/s200/MONTY+HALL2.JPG" width="200" /></a><span style="color: #0b5394; font-family: "verdana" , sans-serif; font-size: large;">Telebistako lehiaketa programa batean, lehiakideak itxita dauden hiru atetik bat aukeratu behar du, batean sari on bat dago, kotxe bat, eta beste bietan ahuntza bana.</span></div>
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<span style="color: #0b5394; font-family: "verdana" , sans-serif; font-size: large;">Ate bat aukeratutakoan, aurkezleak (ondo daki saria non dagoen) atzean ahuntza gordeta duen ate bat irekitzen du. Bi ate gelditzen dira: lehiakideak aukeratutakoa eta bestea.</span></div>
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<span style="font-family: "verdana" , sans-serif; font-size: large;">Azalpena: <a href="https://eginmatematika.blogspot.com/2019/02/probabilitate-bi-problema-monty-hall.html" target="_blank">HEMEN</a></span></div>
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<span style="font-family: "verdana" , sans-serif; font-size: large;"><br /></span><span style="font-family: "verdana" , sans-serif; font-size: large;"><u style="color: #0b5394; text-align: justify;"><b>6. PRESOAREN DILEMA</b></u></span><br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh_LUB9ueCfcBA1yY8-5Qb1gQHTqjiKvE06y7J7PxML3iwnpjIFxCPvW6ubXMBzKWiCkVV20tndowK4WkAe8sekHVTK5iqWuxvsPKU_HNj9Eljzc72TfPqp79FDxR8T7c7-fNw11ZFSeNsI/s1600/PRESO.JPG" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" data-original-height="430" data-original-width="637" height="135" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh_LUB9ueCfcBA1yY8-5Qb1gQHTqjiKvE06y7J7PxML3iwnpjIFxCPvW6ubXMBzKWiCkVV20tndowK4WkAe8sekHVTK5iqWuxvsPKU_HNj9Eljzc72TfPqp79FDxR8T7c7-fNw11ZFSeNsI/s200/PRESO.JPG" width="200" /></a><span style="font-family: "verdana" , sans-serif; font-size: large;"><span style="color: #0b5394;">Hiru pertsona (A-B-C) preso daude epaiketaren zain. Hiruretako bat hiltzera kondenatua izango da eta beste biak libre irtengo dira. Epaileak erabakia zoriz hartuko du.</span></span></div>
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<span style="color: #0b5394; font-family: "verdana" , sans-serif; font-size: large;">Epaileak A presoari C presoa libre irtengo dela esan dio eta, jarraian, galdetu dio ea nahi duen aldatu edo trukatu berarentzat hartuta dagoen erabakia B presoarekin.</span></div>
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<span style="color: #0b5394; font-family: "verdana" , sans-serif; font-size: large;">Zer egin beharko luke A presoak? Trukea onartu ala ez?</span></div>
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<span style="font-family: "verdana" , sans-serif; font-size: large;">Azalpena: <a href="https://eginmatematika.blogspot.com/2019/02/probabilitate-bi-problema-monty-hall.html" target="_blank">HEMEN</a></span></div>
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<span style="font-family: "verdana" , sans-serif; font-size: large;"><span style="font-family: "verdana" , sans-serif; font-size: large;"><u style="color: #0b5394; text-align: justify;"><b>7. ENEKOREN LAGUNEN PARADOXA</b></u></span><span style="font-family: "times new roman"; font-size: small;"> </span></span><br />
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<span style="color: #0b5394; font-family: "verdana" , sans-serif; font-size: large;">Enekok uztaila osoan Irlandan igaro ondoren, bueltan da aireportuan. Aita eta ama bere bila hurbildu dira aireporturaino.</span><br />
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj0JvQbPZcba0ybAtyIV1utiJrIffFbjn55H6s8yBvlmidyACG8fCWJlOGe8dXdT7ulS2OuLm9vUAqwlqmADvDTEmplu3xPWz4emj9Vw273j0brybvM7epHG1Xp43yGbV4w_Q5lfB6gqqXk/s1600/personas.JPG" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" data-original-height="606" data-original-width="525" height="200" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj0JvQbPZcba0ybAtyIV1utiJrIffFbjn55H6s8yBvlmidyACG8fCWJlOGe8dXdT7ulS2OuLm9vUAqwlqmADvDTEmplu3xPWz4emj9Vw273j0brybvM7epHG1Xp43yGbV4w_Q5lfB6gqqXk/s200/personas.JPG" width="173" /></a><span style="color: #0b5394;"><span style="font-family: "verdana" , sans-serif; font-size: large;"><br /></span><span style="font-family: "verdana" , sans-serif; font-size: large;">Aitak honela galdetu dio semeari: "Lagun berririk egin duzu?", Enekok: "bai, bi lagun".</span></span><br />
<span style="color: #0b5394; font-family: "verdana" , sans-serif; font-size: large;">Aitak berriz: "lagunen bat neska al da?", "bai" erantzun dio Enekok.</span><br />
<span style="color: #0b5394; font-family: "verdana" , sans-serif; font-size: large;">Orduan, bi lagunak neskak izateko probabilitatea 1/3 da. </span><span style="color: #0b5394; font-family: "verdana" , sans-serif; font-size: large;">Zergatik?</span><br />
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgcYVmkovcDq_9rr-GvKmAQIknkdhiK1X0CpRPubGWOZQ2i9Uq1hUL6-Lfk6qFJTTN58GAkX6nhmEMVCNx_LT1ak3xDcLiHKhJSB9dZ2Fe0nMWhpYnyQIui452DIbEw0CZbcwi0ApazzteY/s1600/personas2.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" data-original-height="601" data-original-width="533" height="200" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgcYVmkovcDq_9rr-GvKmAQIknkdhiK1X0CpRPubGWOZQ2i9Uq1hUL6-Lfk6qFJTTN58GAkX6nhmEMVCNx_LT1ak3xDcLiHKhJSB9dZ2Fe0nMWhpYnyQIui452DIbEw0CZbcwi0ApazzteY/s200/personas2.jpg" width="176" /></a><span style="color: #0b5394; font-family: "verdana" , sans-serif; font-size: large;"><br /></span><br />
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<span style="color: #0b5394;"><span style="font-family: "verdana" , sans-serif; font-size: large;">Ondoren, amak: "urrutitik ikusi dudan gorriz jantzita doan laguna neska al da?" Enekok: "bai".</span></span><br />
<span style="font-family: "verdana" , sans-serif; font-size: large;"><span style="color: #0b5394;">Orduan, Enekoren bi lagunak neskak izateko probabilitatea 1/2 da. </span></span><span style="color: #0b5394; font-family: "verdana" , sans-serif; font-size: large;">Zergatik?</span><br />
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<span style="font-family: "verdana" , sans-serif; font-size: large;">AZALPENA: <a href="https://eginmatematika.blogspot.com/2016/10/enekoren-lagunen-paradoxa.html" target="_blank">HEMEN</a></span></div>
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<span style="color: #0b5394; font-family: Verdana, sans-serif; font-size: x-large;"><b>*****</b></span></div>
Egin Matematikahttp://www.blogger.com/profile/01348970164668368901noreply@blogger.com0tag:blogger.com,1999:blog-4037937100819081512.post-35143568005146990992019-04-28T00:50:00.000+02:002019-09-28T16:40:29.793+02:00Geometria analitikoa espazioan<div style="text-align: justify;">
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<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEja28XqKkdj5S9xwwmLFk9eeoCziM9YviTIBi3i6P_Fvg7_3R2OUOeNBZywlCpzHVIxURtYcRlpO5bw0e3JroxoD3ky8LhScD-gY_xcdVmE4vZ8juD5M4WA1coStcEKcZL-9afPlwgGEI-R/s1600/ERREFERENTZIA+SISTEMA.JPG" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="578" data-original-width="571" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEja28XqKkdj5S9xwwmLFk9eeoCziM9YviTIBi3i6P_Fvg7_3R2OUOeNBZywlCpzHVIxURtYcRlpO5bw0e3JroxoD3ky8LhScD-gY_xcdVmE4vZ8juD5M4WA1coStcEKcZL-9afPlwgGEI-R/s320/ERREFERENTZIA+SISTEMA.JPG" width="316" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;"><b style="color: #0b5394; font-size: medium;">BEKTORE-ESPAZIO EUKLIDEARRA<br />ETA ESPAZIO AFINA</b></td></tr>
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<span style="font-family: "verdana" , sans-serif; font-size: large;">Matematika II ikasgaiko epazioko geometria analitikoa (Bektoreak-Zuzenak-Planoak-Angeluak-Distantziak) lantzeko lagungarri izango delakoan hainbat azalpen bideo doazkizue jarraian. </span><br />
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<span style="font-family: "verdana" , sans-serif; font-size: large;">Bideoak bi multzotan sailkatuta daude:</span><br />
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<li><span style="font-family: "verdana" , sans-serif; font-size: large;"><span style="text-align: center;">T1, T2,... ➢ Laburpen teorikoak adibideekin. Bideo hauetan jakin beharreko eduki teorikoak azaltzen dira hainbat adibidez osatuak. P</span></span><span style="font-family: "verdana" , sans-serif; font-size: large;">roposatutako ordenean ikustea gomendatzen da.</span></li>
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<li><span style="font-family: "verdana" , sans-serif; font-size: large; text-align: center;">A1, A2,... ➢ Ariketa ebatziak pausoz pauso azalduta irudien laguntzaz. Selektibitateko ariketa batzuk ere ebazten dira.</span></li>
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<span style="color: #0b5394; font-family: "verdana" , sans-serif; font-size: x-large;"><b>LABURPEN TEORIKOAK ETA ADIBIDEAK</b></span></div>
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<table align="center" background="" border="5" bordercolor="grey" cellpadding="4" center="" table=""><tbody>
<tr><td style="text-align: center;"><b>(T1) BIDERKADURA ESKALARRA</b></td>
<td style="text-align: center;"><b>(T2) BIDERKADURA BEKTORIALA</b></td></tr>
<tr><td><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" frameborder="0" height="160" src="https://www.youtube.com/embed/g3IF8kacNbI" width="284"></iframe></td>
<td><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" frameborder="0" height="160" src="https://www.youtube.com/embed/VR_iSXtwVqU" width="284"></iframe></td></tr>
<tr><td style="text-align: center;"><b>(T3) BIDERKADURA MISTOA</b></td>
<td style="text-align: center;"><b>(T4) ZUZENAREN EKUAZIOAK</b></td></tr>
<tr><td><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" frameborder="0" height="160" src="https://www.youtube.com/embed/8QTtY4nsukI" width="284"></iframe></td>
<td><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" frameborder="0" height="160" src="https://www.youtube.com/embed/-bQlG3UfaV8" width="284"></iframe></td></tr>
<tr><td><div style="text-align: center;">
<b>(T5) PLANOAREN EKUAZIOAK</b></div>
</td>
<td><div style="text-align: center;">
<b>(T6) ZUZENAK: POSIZIO ERLATIBOAK</b></div>
</td></tr>
<tr><td><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" frameborder="0" height="160" src="https://www.youtube.com/embed/UkmGug1fsVI" width="284"></iframe></td>
<td><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" frameborder="0" height="160" src="https://www.youtube.com/embed/Yt_ujrJY6kY" width="284"></iframe></td></tr>
<tr><td><div style="text-align: center;">
<b>(T7) PLANOAK: POSIZIO ERLATIBOAK</b></div>
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<td><div style="text-align: center;">
<b>(T8) ZUZENA-PLANOA, POSIZIO ERLATIBOAK</b></div>
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<tr><td><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" frameborder="0" height="160" src="https://www.youtube.com/embed/buVeyq7xXwo" width="284"></iframe></td>
<td><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" frameborder="0" height="160" src="https://www.youtube.com/embed/Y5VA8CucXLo" width="284"></iframe></td></tr>
<tr><td><div style="text-align: center;">
<b>(T9)</b></div>
</td>
<td><div style="text-align: center;">
<b>(T10)</b></div>
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<tr><td><br /></td>
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<span style="color: #0b5394; font-family: "verdana" , sans-serif; font-size: x-large;"><b>ARIKETA EBATZIAK</b></span></div>
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<tr><td style="text-align: center;"><b>(A1) Lau puntu planokideak</b><br />
<b>izateko baldintza </b></td>
<td style="text-align: center;"><b>(A2) Zuzenen eta planoen ekuazioak.</b><br />
<b>Zuzenen arteko posizio erlatiboa.</b></td></tr>
<tr><td><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" frameborder="0" height="160" src="https://www.youtube.com/embed/1PplcFv0LlQ?list=PL2yn2j0qRfvq_tE4OpTBVUOkZ1fP17zei" width="284"></iframe></td>
<td><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" frameborder="0" height="160" src="https://www.youtube.com/embed/snG9i1Wqj_s?list=PL2yn2j0qRfvq_tE4OpTBVUOkZ1fP17zei" width="284"></iframe></td></tr>
<tr><td style="text-align: center;"><b>(A3) Zuzena eta planoa, posizio erlatiboa </b><br />
<b>parametro baten menpe</b></td>
<td style="text-align: center;"><b>(A4) Zuzen batekiko paraleloa den zuzena</b></td></tr>
<tr><td><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" frameborder="0" height="160" src="https://www.youtube.com/embed/ELMTzVdnRKA" width="284"></iframe></td>
<td><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" frameborder="0" height="160" src="https://www.youtube.com/embed/nWG2GNXMtDo" width="284"></iframe></td></tr>
<tr><td><div style="text-align: center;">
<b>(A5) Plano batekiko paraleloa den planoa</b></div>
</td>
<td><div style="text-align: center;">
<b>(A6) Zuzenaren ekuazio inplizituetatik</b><br />
<b>ekuazio parametrikoak.</b></div>
</td></tr>
<tr><td><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" frameborder="0" height="160" src="https://www.youtube.com/embed/iAj6_V1xGVY" width="284"></iframe></td>
<td><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" frameborder="0" height="160" src="https://www.youtube.com/embed/oUnzIMm6ke4" width="284"></iframe></td></tr>
<tr><td><div style="text-align: center;">
<b>(A7) Puntu baten puntu simetrikoa</b><br />
<b>plano batekiko</b></div>
</td>
<td><div style="text-align: center;">
<b>(A8) Puntu baten puntu simetrikoa</b><br />
<b>zuzen batekiko</b></div>
</td></tr>
<tr><td><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" frameborder="0" height="160" src="https://www.youtube.com/embed/QBuZddhs5PU?list=PL2yn2j0qRfvq_tE4OpTBVUOkZ1fP17zei" width="284"></iframe></td>
<td><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" frameborder="0" height="160" src="https://www.youtube.com/embed/SCyBT4Mu73U?list=PL2yn2j0qRfvq_tE4OpTBVUOkZ1fP17zei" width="284"></iframe></td></tr>
<tr><td><div style="text-align: center;">
<b>(A9) Zuzena eta planoa, posizioa</b><br />
<b>parametro baten menpe.</b></div>
</td>
<td><div style="text-align: center;">
<b>(A10</b>) <b>Zuzena eta planoa, posizioa</b><br />
<b>parametro baten menpe.</b></div>
</td></tr>
<tr><td><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" frameborder="0" height="160" src="https://www.youtube.com/embed/7C9DaMeKOns?list=PL2yn2j0qRfvq_tE4OpTBVUOkZ1fP17zei" width="284"></iframe></td>
<td><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" frameborder="0" height="160" src="https://www.youtube.com/embed/wSdm2Gmj2Yk?list=PL2yn2j0qRfvq_tE4OpTBVUOkZ1fP17zei" width="284"></iframe></td></tr>
<tr><td><div style="text-align: center;">
<b>(A11)</b></div>
</td>
<td><div style="text-align: center;">
<b>(A12</b>)</div>
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<tr><td><br /></td>
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<span style="color: #666666;"><span style="font-family: "verdana" , sans-serif; font-size: large;"><i><b>Geometriako hainbat ariketa ebatziak (selektibitatea) 20:</b></i></span><span style="font-family: "verdana" , sans-serif; font-size: large;"> </span></span><br />
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<span style="color: #0b5394; font-size: x-large;"><span style="font-family: "verdana" , sans-serif;"><b>***** </b></span></span></div>
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Egin Matematikahttp://www.blogger.com/profile/01348970164668368901noreply@blogger.com0tag:blogger.com,1999:blog-4037937100819081512.post-2601808768570663492019-03-18T23:57:00.000+01:002019-03-29T23:57:00.345+01:00Aurpegi kopurua berdinduz<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhdYTjCGuZyxel4sk9OfXazeiWt1q7EEng2wCIutDBDQQ04XWmqnZ7UzlUCRYEG2Bag_O-jx5lmpvoZCI5Y_N9QvCN5xKIHccToa6LnHzGbbXYPmbcEXOywuOe1lsNi2IXqhWFS1vr6XJrS/s1600/txanponak.png" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" data-original-height="900" data-original-width="1280" height="140" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhdYTjCGuZyxel4sk9OfXazeiWt1q7EEng2wCIutDBDQQ04XWmqnZ7UzlUCRYEG2Bag_O-jx5lmpvoZCI5Y_N9QvCN5xKIHccToa6LnHzGbbXYPmbcEXOywuOe1lsNi2IXqhWFS1vr6XJrS/s200/txanponak.png" width="200" /></a><span style="color: #666666; font-family: "verdana" , sans-serif; font-size: large;"></span></div>
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<span style="color: #666666; font-family: "verdana" , sans-serif; font-size: large;">Mahai gainean 101 txanpon jarri ditut, horietatik 15 txanponek aurpegia gora begira dute eta beste 86-k gurutzea. </span><br />
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<span style="color: #666666; font-family: "verdana" , sans-serif; font-size: large;">Begiak estalita dituzula, ondorengo erronka proposatzen dizut:</span></div>
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<span style="color: #666666; font-family: "verdana" , sans-serif; font-size: large;">Mahaiko txanpon guztiak bi multzotan banatu, halako moldez non bi multzoetan aurpegia gora begira duten txanpon kopurua berdina den (hauxe da egiaztatu behar den baldintza bakarra). Gogoratu 15 txanpon aurpegia gora eta 86 gurutzea gora begira daudela eta txanponak ez dituzula ikusten, bakarrik mugitu eta nahi baduzu biratu ere egin ditzakezula. Ukimenez ez dago jakiterik aurpegia ala gurutzea den.</span></div>
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<span style="color: #666666; font-family: "verdana" , sans-serif; font-size: large; text-align: justify;">Ikusten duzunez, enuntziatua erraza da, eta lehenengo irakurketan ezinezko lana dirudien arren, posiblea da, egin daiteke, ez dago tranparik.</span></div>
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<span style="color: #666666; font-family: "verdana" , sans-serif; font-size: large; text-align: justify;">Ez konplikatu, erraz pentsatu eta listo!</span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiBnd11XTUIVAEYlNZftEanky3GDKeV-1ytrLssvN5bBNrAK7l9HC3Mk7C8UF3rUOqqG0e3lPx1N8EW-XzkYY3-PwF1S-r-yPSy0ljBWCGnGa3WxD-N1hyphenhyphenuNPDz_Paq2G6qtcvgd-0-GVb6/s1600/emoticono+pensativo.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="322" data-original-width="308" height="200" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiBnd11XTUIVAEYlNZftEanky3GDKeV-1ytrLssvN5bBNrAK7l9HC3Mk7C8UF3rUOqqG0e3lPx1N8EW-XzkYY3-PwF1S-r-yPSy0ljBWCGnGa3WxD-N1hyphenhyphenuNPDz_Paq2G6qtcvgd-0-GVb6/s200/emoticono+pensativo.JPG" width="191" /></a></div>
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<span style="font-family: "arial" , "helvetica" , sans-serif; font-size: large;">Jarraian ebazpena doakizue, baina begiratu baino lehenago eman beste buelta bat problemari. </span><br />
<span style="font-family: "arial" , "helvetica" , sans-serif; font-size: large;">Kontuan izan hasierako txanpon kopurua edozein izan daitekela, eta ematen diguten informazio garrantzitsuena gora begira dagoen txanpon eta gurutze kopurua dela.</span><br />
<span style="font-family: "arial" , "helvetica" , sans-serif; font-size: large;">Orduan, bi multzotan banatzen dugunean, zein izango da multzo bakoitzaren txanpon kopurua?</span><br />
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<span style="font-family: "arial" , "helvetica" , sans-serif; font-size: large;"><b style="font-family: "Times New Roman"; font-size: medium;"><span style="background-color: lime; color: #0b5394; font-size: x-large;">AZALPENA</span></b></span></div>
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<iframe allowfullscreen="true" frameborder="0" height="389" mozallowfullscreen="true" src="https://docs.google.com/presentation/d/e/2PACX-1vQ-PBNqQO6kxOPYjxHiZQDOZn5p4MP8mJZJ6DVkIKC7iqzcUrwrY8MEjD5F8jUudHBXiOie2FeqxwuA/embed?start=false&loop=false&delayms=5000" webkitallowfullscreen="true" width="640"></iframe>
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<b><span style="background-color: #fce5cd; color: #0b5394; font-size: x-large;"><br /></span></b><span style="color: #0b5394; font-size: x-large;"><span style="background-color: #fce5cd;">*****</span></span></h2>
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Egin Matematikahttp://www.blogger.com/profile/01348970164668368901noreply@blogger.com2tag:blogger.com,1999:blog-4037937100819081512.post-15146036649019765062019-02-18T00:05:00.000+01:002020-11-07T17:13:40.773+01:00Probabilitate bi problema: Monty Hall eta Presoaren Dilema.<br />
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<span style="font-family: "verdana" , sans-serif; font-size: large;">Egunotan probabilitatea izan dugu protagonista batxilergoko klase orduetan. Intutuizioa axiomatikaren gainetik jarriz, hainbat problema paradoxiko eta anti-intuitibo ebazteari ekin diogu. Horien artean, gaurko artikulu honetan aztertuko ditugun "Monty Hall-en problema" eta "Presoaren dilema". Monty Hall-en problema bigarren aldiz proposatzen dut nire blogean, aurreko sarrera batean izan genuen ere aztergai: <a href="https://eginmatematika.blogspot.com/2016/10/monty-hall-en-problema.html" target="_blank">HEMEN </a></span><br />
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<span style="font-family: "verdana" , sans-serif; font-size: large;"><b style="font-size: xx-large;">"MONTY HALL"</b></span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhbZa-YqBaqqz44FmSOgQCmOb3zQcwzaK2fSF31_RpXgbcV2IFZrMy750wW1FI7QbhzT1WDHICyx7aufiFMYP0e_TU925WIDvBbNT35ztvLXJ3vt5fBE-HzHVYwxK9rBheJhZc2CGYwJksN/s1600/MONTY+HALL2.JPG" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" data-original-height="522" data-original-width="843" height="196" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhbZa-YqBaqqz44FmSOgQCmOb3zQcwzaK2fSF31_RpXgbcV2IFZrMy750wW1FI7QbhzT1WDHICyx7aufiFMYP0e_TU925WIDvBbNT35ztvLXJ3vt5fBE-HzHVYwxK9rBheJhZc2CGYwJksN/s320/MONTY+HALL2.JPG" width="320" /></a><span style="color: #666666; font-family: "verdana" , sans-serif; font-size: large;"><b>Telebistako lehiaketa programa batean, lehiakideak itxita dauden hiru atetik bat aukeratu behar du, batean sari on bat dago, kotxe bat, eta beste bietan ahuntza bana.</b></span></div>
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<span style="color: #666666; font-family: "verdana" , sans-serif; font-size: large;"><b>Ate bat aukeratutakoan, aurkezleak (ondo daki saria non dagoen) atzean ahuntza gordeta duen ate bat irekitzen du. Bi ate gelditzen dira: lehiakideak aukeratutakoa eta bestea.</b></span></div>
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<span style="color: #666666; font-family: "verdana" , sans-serif; font-size: large;"><b>Jarraian, aukera ematen dio lehiakideari aldatzeko: lehen aukeratutako atea utzi eta bestea hautatu.</b></span></div>
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<span style="color: #666666; font-family: "verdana" , sans-serif; font-size: large;"><b>Zuk zer egingo zenuke? aldatu ala ez?</b></span></div>
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<span style="font-family: "verdana" , sans-serif; font-size: large;">Intuizioari eutsiz azter dezagun problema hau lasaitasunez.</span><br />
<span style="font-family: "verdana" , sans-serif; font-size: large;"><br /></span><span style="font-family: "verdana" , sans-serif; font-size: large;">Zoriz hiru ateetatik bat hautatzen dugunean, </span><span style="font-family: "verdana" , sans-serif; font-size: large;">kotxe berrian etxera itzultzeko izango dugun probabilitatea 1/3 da. I</span><span style="font-family: "verdana" , sans-serif; font-size: large;">go al daiteke probabilitate hau atez ez aldatzea erabakitzen bada?</span><br />
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<span style="font-family: "verdana" , sans-serif; font-size: large;">Lehenengo irakurketan pentsa liteke atez aldatzeak ez digula inolako bentajarik ekarriko, aldatu ala ez kotxea irabazteko probabilitatea 1/2 dela. Baina, 1/2 bada, igo egin da; orduan, aurkezleak ahuntza duen ate bat zabaltzeagatik hasierako 1/3ko probabilitatea igo egin da 1/2ra? Nola? Benetan hala da? Zer dela eta 1/2? Bi ate gelditzen direlako? Ez al genekien beste bi ateetako batean, gutxienez batean, ahuntz bat zegoela? Inporta al da ate hori zabalik egotea gure atea ez uztea erabaki badugu? Ate aldaketarik ez badago, irabazteko probabilitatean ere ez da aldaketarik emango.</span><br />
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<span style="font-family: "verdana" , sans-serif; font-size: large;">Bildu beste bi ateak eta hartu bloke bat moduan. Argi dago bloke honetan kotxea egoteko probabilitatea 2/3 dela, eta </span><span style="font-family: "verdana" , sans-serif; font-size: large;">aukera izanez gero, </span><span style="font-family: "verdana" , sans-serif; font-size: large;">bloke osoa hautatuko genukeela. Bloke honetako sari gabeko ate bat irekitzen denean, blokeak jarraitzen du izaten 2/3ko probabilitatea, baina orain guztia ireki gabe dagoen atera pasa da. Aurkezleak aldatzeko proposamena egiten digunean, bloke osoa eskaintzen digu (irekita dagoen atea ez dugu ezertarako nahi, ez guk, ez berak). Ondorioz, atez aldatzea komeni zaigu, irabazteko probabilitatea bikoiztu egiten baita.</span><br />
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<span style="font-family: "verdana" , sans-serif; font-size: large;">Hobeto ulertzeko alda dezagun problemaren enuntziatua. Hiru sobre daude, batean premio on bat (kotxea) eta beste bietan ezer ez (edo ahuntza bana). Zoriz sobre bat aukeratu dugu eta aurkezleak beste bietatik sari gabeko sobre bat zabaltzen du. Hutsik dagoenez, oparitu egiten dizu. Jarraian, zuk aukeratutako sobrea ireki gabe dagoenarekin aldatzeko proposamena luzatzen dizu. Ez zenuke aldatuko? Funtsean, hasierako bi sobreak eskaini dizkizu. Ireki gabeko sobrean 2/3ko probabilitatea metatu da; beraz, aldatzea komeni zaizu.</span><br />
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<span style="font-family: "verdana" , sans-serif; font-size: large;">Jokalari baten estrategia atez aldatzea bada, ondorengo zuhaitz diagraman labur daiteke irabazteko duen probabilitatea:</span><br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhwEBoSC9xGo27pJ2PvVu51UoFLMRAqeToiOfpAlMJojMszP_Mhhwgyseb5P76ZAkSRy93JJDL2Z4uhMsBavY1vFKvlsTx6mX7YNI98tcaJjzTtP501tAWY_AZw_SUXL6uGpeo1o7NdFzQX/s1600/MONTY+HALL-ZUHAITZA1.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1212" data-original-width="1600" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhwEBoSC9xGo27pJ2PvVu51UoFLMRAqeToiOfpAlMJojMszP_Mhhwgyseb5P76ZAkSRy93JJDL2Z4uhMsBavY1vFKvlsTx6mX7YNI98tcaJjzTtP501tAWY_AZw_SUXL6uGpeo1o7NdFzQX/s1600/MONTY+HALL-ZUHAITZA1.JPG" /></a></div>
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<span style="font-family: "verdana" , sans-serif; font-size: large;">Estrategia atez ez aldatzea denean:</span><br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgVkG2EY4DO_DHK6qQ1Eonh7TQn2xtPAMA09UErBYHqGmUBPfSXt-KA7X4C4s8FsPdTmQR4miZv7HPcn8RGBHKnkgHD4uVgEqj06bavMmiDp4re2sRubpQ9vB4LuOQT4EnhShyBBjIdQxBZ/s1600/MONTY+HALL-ZUHAITZA2.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1210" data-original-width="1600" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgVkG2EY4DO_DHK6qQ1Eonh7TQn2xtPAMA09UErBYHqGmUBPfSXt-KA7X4C4s8FsPdTmQR4miZv7HPcn8RGBHKnkgHD4uVgEqj06bavMmiDp4re2sRubpQ9vB4LuOQT4EnhShyBBjIdQxBZ/s1600/MONTY+HALL-ZUHAITZA2.JPG" /></a></div>
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<span style="font-family: "verdana" , sans-serif; font-size: large;"><br /></span><span style="font-family: "verdana" , sans-serif; font-size: large;">Aurkezpenean honetan hainbat azalpen proposatzen dira:</span><br />
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<iframe allowfullscreen="true" frameborder="0" height="389" mozallowfullscreen="true" src="https://docs.google.com/presentation/d/e/2PACX-1vRrbL2ZuSP3f4QIsPFIVca3gLZUxMCl4CzZzq2zVIhEDJH-1D1kTUhoVPPtLnskc6Abp1tPkMsNGMji/embed?start=false&loop=false&delayms=5000" webkitallowfullscreen="true" width="480"></iframe>
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<span style="font-family: "verdana" , sans-serif;"><b><span style="color: #0b5394; font-size: x-large;">*****</span></b></span></div>
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<span style="font-family: "verdana" , sans-serif; font-size: large;"><b style="font-size: xx-large;">"PRESOAREN DILEMA"</b></span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhxurDMGeCfOsfb9zBcjQcnzcYWiZDIFIjAR8kjBV6csN_-qn05qXsqEDrNkKQmoPPfW5jvoZD7KGiLCMBDK-abwiKQyWCjb-HxtPmVWOSX0qTmFeeBP5vvXMTwYLFi4j7PvvQu1WNcR4Wr/s1600/PRESO.JPG" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" data-original-height="430" data-original-width="637" height="216" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhxurDMGeCfOsfb9zBcjQcnzcYWiZDIFIjAR8kjBV6csN_-qn05qXsqEDrNkKQmoPPfW5jvoZD7KGiLCMBDK-abwiKQyWCjb-HxtPmVWOSX0qTmFeeBP5vvXMTwYLFi4j7PvvQu1WNcR4Wr/s320/PRESO.JPG" width="320" /></a><span style="font-family: "verdana" , sans-serif; font-size: large;"><span style="color: #666666;"><b>Hiru pertsona (A-B-C) preso daude epaiketaren zain. Hiruretako bat hiltzera kondenatua izango da eta beste biak libre irtengo dira. Epaileak erabakia zoriz hartuko du.</b></span></span><br />
<span style="color: #666666; font-family: "verdana" , sans-serif; font-size: large;"><b>Epaileak A presoari C presoa libre irtengo dela esan dio eta, jarraian, galdetu dio ea nahi duen aldatu edo trukatu berarentzat hartuta dagoen erabakia B presoarekin.</b></span><br />
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<span style="color: #666666; font-family: "verdana" , sans-serif; font-size: large;"><b>Zer egin beharko luke A presoak? Trukea onartu ala ez?</b></span><br />
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<span style="font-family: "verdana" , sans-serif; font-size: large;">Problema hau "Monty Hall"-en problemarekin uztartu daiteke honako identifikazio hauek eginez: </span><br />
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<span style="font-family: "verdana" , sans-serif; font-size: large;">Aske geratzea <span style="color: #0b5394;">➪</span> Ahuntza irabaztea</span></div>
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<span style="font-family: "verdana" , sans-serif; font-size: large;">Hiltzera kondenatuta izatea <span style="color: #0b5394;">➪</span> kotxea irabaztea</span></div>
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<span style="font-family: "verdana" , sans-serif; font-size: large;">C presoa aske geratuko dela jakitea <span style="color: #0b5394;">➪</span> Ahuntza duen ate bat zabaltzea</span></div>
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<span style="font-family: "verdana" , sans-serif; font-size: large;">Beraz, oraingo honetan ez aldatzea komeni zaio, ateen probleman ahuntza irabazteko estrategia bera, honela hasieran aske geratzeko zeukan probabilitatea 2/3 mantenduko du. Aldatzen badu probabilitatea 1/3 izatera pasatuko litzateke.</span><br />
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<span style="font-family: "verdana" , sans-serif; font-size: large;"><b style="font-size: medium;"><span style="color: #0b5394; font-size: x-large;">*****</span></b></span></div>
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<br />Egin Matematikahttp://www.blogger.com/profile/01348970164668368901noreply@blogger.com0tag:blogger.com,1999:blog-4037937100819081512.post-21393938853846556972019-02-10T19:59:00.002+01:002019-02-19T22:50:50.696+01:00Probabilitate problema. Oinarrizko proportzioaren falazia.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhjKwdyH30pgqybd0I73F5ueFOu6gfrksdAA72Dz_d_N76LAn1FXSOnX9XRvEqWIO_C_I9MqfNk4VaxGoupFuISoW6YeDnuk4oS4BD8lOSSbMzt2jGet8yOXhHSl7CU6tsnvnX6eopJLXY3/s1600/ROBO+1.JPG" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="356" data-original-width="357" height="197" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhjKwdyH30pgqybd0I73F5ueFOu6gfrksdAA72Dz_d_N76LAn1FXSOnX9XRvEqWIO_C_I9MqfNk4VaxGoupFuISoW6YeDnuk4oS4BD8lOSSbMzt2jGet8yOXhHSl7CU6tsnvnX6eopJLXY3/s200/ROBO+1.JPG" width="200" /></a><span style="font-family: "verdana" , sans-serif; font-size: large;"></span></div>
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<b><span style="color: #666666;"><span style="font-family: "verdana" , sans-serif; font-size: large;">Hiri bateko biztanleen % 90 zuriak dira eta % 10 beltzak. Hiri horretan d</span><span style="font-family: "verdana" , sans-serif; font-size: large;">elitu bat gertatu da, gauean etxe batean lapurretan egin dute.</span></span></b><br />
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<b><span style="color: #666666;"><span style="font-family: "verdana" , sans-serif; font-size: large;">Auzokide batek lapurra gizon beltz bat izan dela deklaratu du. Deklarazio hau kontuan harturik, p</span><span style="font-family: "verdana" , sans-serif; font-size: large;">oliziak gizon beltz bat atxilotu eta erruduntzat hartu du.</span></span></b></div>
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<span style="color: #666666; font-family: "verdana" , sans-serif; font-size: large;"><b>Salaketari indarra eman nahian, lapurreta gertatu zen ordu berean eta baldintza berdinetan proba bat egiten diote lekukoari, eta honek lapurraren azal kolorea % 80 aldiz asmatzen du.</b></span></div>
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<span style="color: #666666; font-family: "verdana" , sans-serif; font-size: large;"><b>Badago lapurreta leporatzerik lekukoak esandakoarekin? Hau da, lekukoak lapurra beltza dela esan badu, zein da benetan lapurra beltza izateko probabilitatea?</b></span><br />
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<span style="font-size: large;">🔷🔷🔷</span></div>
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<span style="font-family: "arial" , "helvetica" , sans-serif; font-size: large;">Probabilitate problema paradoxikoa dugu gaurko hau, eta oinarrizko proportzioaren falazia izenarekin ezagutzen da.</span><br />
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Joera handia izaten dugu eskaturiko probabilitatea lekukoak proban izandako % 80rekin identifikatzeko. Baina ez da hori, kontuan izan behar dugu lekukoak esaten duenak baldintzatzen duela benetan lapurra beltza izatearen probabilitatea. Bestelako proportzio bat erabili beharra dago, non lekukoak esandakoa kontuan izaten den.</span></span><br />
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<span style="font-family: "arial" , "helvetica" , sans-serif; font-size: large;">Jarraian azalpen bat doakizue:</span><br />
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<span style="font-family: "arial" , "helvetica" , sans-serif; font-size: medium;"><span style="font-size: large;">🔷</span><b><span style="font-size: x-large;">AZALPENA</span></b><span style="font-size: large;">🔷</span></span><br />
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<span style="color: #0b5394; font-family: "verdana" , sans-serif; font-size: x-large;"><b>*****</b></span></div>
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Egin Matematikahttp://www.blogger.com/profile/01348970164668368901noreply@blogger.com0tag:blogger.com,1999:blog-4037937100819081512.post-66972125961345114792019-02-06T18:28:00.000+01:002019-02-19T22:51:11.870+01:00Probabilitatea eta banaketa justua.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgaqGbAHgRLpT0iZr4IzWj3w988zXpCWhjxaKyAJzXgfhKBwrxpTAO4mTTs-I8kkv3ycC-O5gkH_1arHIVQYF58MV1TM5MDjKRTQ0OVp3377CSQs1P-w1TyoH19btOF5VbG6ZO2QSsEkHQo/s1600/cubilete+dados.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="270" data-original-width="517" height="104" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgaqGbAHgRLpT0iZr4IzWj3w988zXpCWhjxaKyAJzXgfhKBwrxpTAO4mTTs-I8kkv3ycC-O5gkH_1arHIVQYF58MV1TM5MDjKRTQ0OVp3377CSQs1P-w1TyoH19btOF5VbG6ZO2QSsEkHQo/s200/cubilete+dados.JPG" width="200" /></a></div>
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg8qXJACvxmHkWd8JDDPRgeXgWnVVm0GsTCR8u2zZxdR3wCOLGtHqVkd8KZakCx8qQrzw9dbX3AGciaDHgc4z2eNUl8L1DUDson4Wn9OS_3H3Bk_L2bPTYBxRi-BzR7keFI4SQAcKhqvM-I/s1600/EURO3.JPG" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" data-original-height="219" data-original-width="217" height="200" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg8qXJACvxmHkWd8JDDPRgeXgWnVVm0GsTCR8u2zZxdR3wCOLGtHqVkd8KZakCx8qQrzw9dbX3AGciaDHgc4z2eNUl8L1DUDson4Wn9OS_3H3Bk_L2bPTYBxRi-BzR7keFI4SQAcKhqvM-I/s200/EURO3.JPG" width="198" /></a><span style="font-family: "verdana" , sans-serif; font-size: large;">Martxel eta Erika, matematikako irakaslearen erretolikaz asperturik, gogoz ari dira jolasten. Zorizko joko bat asmatu dute, 6 puntu lortzen dituena 32 € irabaziko ditu (bakoitzak 16 € ipini ditu). Erikak 5 puntu eta Martxelek 3 dituztenean txirrinak jo du eta jokoa eten egin behar izan dute.</span><br />
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<span style="font-family: "verdana" , sans-serif; font-size: large;">Martxelek zera proposatu dio Erikari: “32 euroak lorturiko puntuen arabera banatuko ditugu, 4 € puntu bakoitzeko, 20 zuretzat eta 12 niretzat, ados?” </span><br />
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<span style="font-family: "verdana" , sans-serif; font-size: large;">Erika probabilitate klaseetan adi-adi egon denez, badaki proposamena ez dela justua eta ez du onartzen. Zergatik ez du onartu? Nola egin behar da banaketa?</span><br />
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<span style="color: #666666; font-family: "arial" , "helvetica" , sans-serif; font-size: medium;">(Problema hau Fibonaccik (1180-1250) proposatu zuen “Liber Abaci” liburuan eta Luca Paciolik (1445-1514) gaizki ebatzi zuen)</span></div>
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<b><span style="color: #0b5394; font-size: large;">AZALPENA</span></b></div>
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Egin Matematikahttp://www.blogger.com/profile/01348970164668368901noreply@blogger.com0tag:blogger.com,1999:blog-4037937100819081512.post-43289784733256710742019-01-01T18:31:00.000+01:002020-11-11T18:34:11.662+01:00Karratuen jokoa eta probabilitatea<br />
<b><span style="color: #6fa8dc; font-size: x-large;">PROBABILITATE PROBLEMA</span></b><br />
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<span style="font-size: large;">Ezkutuan eta zoriz irudiko puntuak zenbakitu dira 1etik 20ra. A, B eta C jokalariek txandaka 4 zenbaki aukeratuko dituzte, 1etik 20ra, lau puntuak lotzerakoan karratu bat lortu nahian. A hasiko da, ondoren B eta azkenik C (hirurek esango dute konbinazio bat, baina aurretik esandako konbinazioa, lau zenbakiak, errepikatu gabe).</span><br />
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<span style="font-size: large;">a) Zenbat karratu eraiki daitezke lau puntu lotuz?</span></div>
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<span style="font-size: large;">b) Zein da lehenengo txandan jokalari bakoitzak aukeratutako zenbaki konbinazioarekin karratu bat lortzeko duen probabilitatea?</span></div>
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<span style="font-size: large;">c) Zein da zortzigarren konbinazioa esatean lehenengo aldiz asmatzeko probabilitatea?</span></div>
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<span style="font-size: large;">d) Esandako lehenengo 7 konbinazioetan inork ez duela asmatu jakinda, zein da 8. konbinazioa esatean asmatzeko probabilitatea?</span></div>
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<span style="font-size: large;">e) Zenbat konbinazio oker esan behar dira hurrengoko konbinazioa esatean asmatzeko probabilitatea %1 izan dadin?</span><br />
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<span style="color: #3d85c6; font-size: x-large;"><b>EBAZPENA</b></span></div>
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Egin Matematikahttp://www.blogger.com/profile/01348970164668368901noreply@blogger.com0tag:blogger.com,1999:blog-4037937100819081512.post-40039740539865841092018-12-17T00:29:00.001+01:002018-12-19T12:17:10.249+01:00Dorrea eta estrategia irabazlea<br />
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<span style="font-family: Verdana, sans-serif; font-size: large;">Hemen doakizue Adrian Paenzak bere problemetako liburu batean proposaturiko estrategiazo joko polit bat.</span><br />
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<span style="font-family: Verdana, sans-serif; font-size: large;">Xake taula, dorre bat eta apur bat pentsatzea besterik ez dugu behar.</span><br />
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<span style="color: #666666; font-family: Verdana, sans-serif; font-size: large;"><b>Hauek dira jokoaren arauak:</b></span><br />
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<span style="font-family: Verdana, sans-serif; font-size: large;"><b><span style="color: #783f04;">➤</span><span style="color: #666666;"> Bi jokalari: A eta B.</span></b></span><br />
<span style="font-family: Verdana, sans-serif; font-size: large;"><b><span style="color: #783f04;">➤</span><span style="color: #666666;"> A jokalariak aurreneko mugimendua egingo du, ondoren B-k mugituko du.</span></b></span><br />
<span style="font-family: Verdana, sans-serif; font-size: large;"><b><span style="color: #783f04;">➤</span><span style="color: #666666;"> Hasieran dorrea taulako behe-ezkerreko laukian dago.</span></b></span><br />
<span style="font-family: Verdana, sans-serif; font-size: large;"><b><span style="color: #783f04;">➤</span><span style="color: #666666;"> Baimendutako mugimenduak: gora eta eskuinera.</span></b></span><br />
<span style="font-family: Verdana, sans-serif; font-size: large;"><b><span style="color: #783f04;">➤</span><span style="color: #666666;"> Mugimendu bakoitzean nahi beste lauki gora ala eskumara mugitzea aukeratu behar da.</span></b></span><br />
<span style="font-family: Verdana, sans-serif; font-size: large;"><b><span style="color: #783f04;">➤</span><span style="color: #666666;"> Helburua goi-eskuineko laukira iristea.</span></b></span><br />
<span style="font-family: Verdana, sans-serif; font-size: large;"><b><span style="color: #783f04;">➤</span><span style="color: #666666;"> Txandaka mugituko dute dorrea: A jokalaria hasiko da.</span></b></span><br />
<span style="font-family: Verdana, sans-serif; font-size: large;"><b><span style="color: #783f04;">➤</span><span style="color: #666666;"> Bere txandan goi-eskuinean dagoen lauki beltzera iristen denak jokoa irabaziko du.</span></b></span><br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiA78E8iF1csZg8Z1rH9bRgfsjtJXfvkYWPVj5pfLUvCB4_rutbpAoM-QwmJmf4qB30f_0xonDIs425kHrkCYhmmoo-w_6fQFElZ5MKpuphqANLNB-6AHnRWJc6KjXrPJFZn6eVqqKpKA15/s1600/tablero+ajedrez4.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1058" data-original-width="1056" height="640" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiA78E8iF1csZg8Z1rH9bRgfsjtJXfvkYWPVj5pfLUvCB4_rutbpAoM-QwmJmf4qB30f_0xonDIs425kHrkCYhmmoo-w_6fQFElZ5MKpuphqANLNB-6AHnRWJc6KjXrPJFZn6eVqqKpKA15/s640/tablero+ajedrez4.JPG" width="638" /></a></div>
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<span style="color: #666666; font-family: Verdana, sans-serif; font-size: large;"><b>Hauxe jakin nahi da:</b></span></div>
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<span style="font-family: Verdana, sans-serif; font-size: large;"><b><span style="color: #783f04;">➤ </span><span style="color: #666666;">Ba al dago beti irabazteko estrategiarik? </span></b></span></div>
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<span style="font-family: Verdana, sans-serif; font-size: large;"><b><span style="color: #783f04;">➤ </span><span style="color: #666666;">Hala bada, nork dauka estrategia irabazlea, A lehenengo jokalariak ala B bigarrenak?</span></b></span></div>
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<span style="font-family: Verdana, sans-serif;"><span style="font-size: x-large;"><span style="color: #783f04;">♜♜♜♜♜</span> </span></span></div>
<span style="font-family: Verdana, sans-serif; font-size: large;"><br /></span>Egin Matematikahttp://www.blogger.com/profile/01348970164668368901noreply@blogger.com5tag:blogger.com,1999:blog-4037937100819081512.post-73536997580992002332018-12-09T18:05:00.000+01:002019-02-06T18:29:27.492+01:00Integralak: Zatikako Metodoa. Nola aukeratu u eta dv? Beharrezkoa al da formula?<br />
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiMprNmVoqzNyAFC1zCm3k2ENKtxNlZZQsEVoih0KMQEfDxzSCvoGenmtCPAZUs59T2fZaoaOLuzojWELE4Cv3a4NPgP40Oj1hY1ovf2WUPlyGpwxKo0ciHv3PbP6lSszpT5IQnFQmdtDlV/s1600/Integralaren+ikurra.png" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" data-original-height="640" data-original-width="524" height="200" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiMprNmVoqzNyAFC1zCm3k2ENKtxNlZZQsEVoih0KMQEfDxzSCvoGenmtCPAZUs59T2fZaoaOLuzojWELE4Cv3a4NPgP40Oj1hY1ovf2WUPlyGpwxKo0ciHv3PbP6lSszpT5IQnFQmdtDlV/s200/Integralaren+ikurra.png" width="163" /></a><span style="font-family: "verdana" , sans-serif; font-size: large;"></span><br />
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<span style="font-family: "verdana" , sans-serif; font-size: large;">Batxilergoko 2. mailan ikasleak kalkulu integralean murgiltzen dira buru belarri funtzio deribatua integratuz jatorrizkoen bila. Jatorrizko funtzioak aurkitzeko hainbat metodo edota teknika ikasten dute, horien artean, metodo bitxi bat, zatikako metodoa deiturikoa. </span></div>
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<span style="font-family: "verdana" , sans-serif; font-size: large;">Metodo hau eraginkortasunez erabiltzeko pauta eta adibide batzuk (formulaz eta formula erabili barik) emango ditugu jarraian. </span></div>
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<span style="font-family: "verdana" , sans-serif; font-size: large;">➤Nola aukeratu <b>u</b> eta <b>dv</b> zatikako metodoan? </span><br />
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<span style="font-family: "verdana" , sans-serif; font-size: large;">Hona hemen aukeraketa egiteko lagungarria den </span><span style="font-family: "verdana" , sans-serif; font-size: large;">arau mnemotekniko bat</span><span style="font-family: "verdana" , sans-serif; font-size: large;">:</span><br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhdJo8wQKubwfe0rAxmdtSHl36VThdFuRbtT77c5YmsWHYxuH05RW8IGk9e9DUfpMmjjrhrnhQanLjoZ0LQYKnVrPAmnC9Y92svItqejjZUfYmEvgjG2tJ1tppdKzaQItpaPXGrG4TL-M_h/s1600/ZATIKAKO+METODOA.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1230" data-original-width="1600" height="488" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhdJo8wQKubwfe0rAxmdtSHl36VThdFuRbtT77c5YmsWHYxuH05RW8IGk9e9DUfpMmjjrhrnhQanLjoZ0LQYKnVrPAmnC9Y92svItqejjZUfYmEvgjG2tJ1tppdKzaQItpaPXGrG4TL-M_h/s640/ZATIKAKO+METODOA.JPG" width="640" /></a></div>
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<span style="font-family: "verdana" , sans-serif; font-size: large;">Irudian ALPES hitzaren hizkien esanahia dugu. Hizki bakoitza funtzio mota bat da, eta honek pista bat ematen digu <b>u</b> eta <b>dv</b> aukeratzerakoan. Aurretik dagoen hizkiak <b>u </b>bezala zein funtzio aukeratu behar den esango digu eta bigarren hizkiak, ordez, <b>dv</b>.</span><br />
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<span style="font-family: "verdana" , sans-serif; font-size: large;">Adibide bat ikusi baino lehen, gogora dezagun zertan datzan zatikako metodoa.</span><br />
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<span style="font-family: "verdana" , sans-serif; font-size: large;">🔻🔻🔻</span></div>
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<span style="color: #0b5394; font-family: "verdana" , sans-serif; font-size: large;">Demagun x-ren menpeko bi funtzio deribagarri: f(x) eta g(x).</span><br />
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<span style="font-family: "verdana" , sans-serif; font-size: large;">Bi funtzio hauen f·g biderkadura deribatuko dugu Lagrangeren notazioa erabiliz,</span></span><br />
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<span style="color: #0b5394; font-family: "verdana" , sans-serif; font-size: large;">(f·g)' = f'·g + f·g' =g·f' + f·g'</span></div>
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<span style="color: #0b5394; font-family: "verdana" , sans-serif; font-size: large;">edo berdina dena baina Leibnizen diferentzialen notazioaz (f' = df/dx),</span></div>
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<span style="color: #0b5394; font-family: "verdana" , sans-serif; font-size: large;">d(f·g)/dx= g·df/dx + f·dg/dx</span></div>
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<span style="color: #0b5394; font-family: "verdana" , sans-serif; font-size: large;">d(f·g) = g·df + f·dg</span></div>
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<span style="color: #0b5394; font-family: "verdana" , sans-serif; font-size: large;">Bi aldeak integratuz, </span></div>
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<span style="font-family: "verdana" , sans-serif; font-size: large;">∫d(f·g) = ∫</span><span style="font-family: "verdana" , sans-serif; font-size: large;">g·df + ∫f·dg</span></span><br />
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<span style="font-family: "verdana" , sans-serif; font-size: large;">f·g = ∫</span><span style="font-family: "verdana" , sans-serif; font-size: large;">g·df + ∫f·dg</span></span><br />
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<span style="color: #0b5394; font-family: "verdana" , sans-serif; font-size: large;"> ∫f·dg isolatuz, zatikako metodoaren oinarrizko formula izango dugu,</span><br />
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<span style="color: #0b5394; font-family: "verdana" , sans-serif; font-size: large;">∫f·dg = </span><span style="color: #0b5394; font-family: "verdana" , sans-serif; font-size: large;">f·g - ∫g·df</span><span style="color: #0b5394;"><span style="color: #666666;"><span style="font-family: "verdana" , sans-serif; font-size: large;"><br /></span>
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<span style="color: #0b5394; font-family: "verdana" , sans-serif; font-size: large;">f = u eta g = v eginez, ezagunagoa den honako adierazpenera heltzen gara,</span><br />
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<span style="color: #0b5394; font-family: "verdana" , sans-serif; font-size: large;"><b style="background-color: #f4cccc; font-size: xx-large;">∫udv = uv - ∫vdu</b></span></div>
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<span style="font-family: "verdana" , sans-serif; font-size: large;">🔺🔺🔺</span></div>
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<span style="font-family: "verdana" , sans-serif; font-size: large;">➤Zatikako metodoa zein integral motatan komeni da erabiltzea?</span><br />
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<span style="font-family: "verdana" , sans-serif; font-size: large;">Integrala berehalakoa ez bada eta integrakizunean mota desberdineko funtzioen biderkadura (funtzio polinomikoa bider funtzio trigonometrikoa, funtzio polinomikoa bider esponentziala, trigonometrikoa bider esponentziala, polinomikoa bider arco funtzioa, arcsinx·dx, lnx·dx,...) agertzen denean zatikako metodoa aplikatzea efektiboa izan daiteke.</span><br />
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<span style="font-family: "verdana" , sans-serif; font-size: large;">Metodo honetan, hasierako integrala bi zatitan banatzen da, <b>u</b> zatia eta <b>dv</b> (bigarren hau dx-rekin batera hartzen da eta integratzeko erraza edo berehalakoa behar du izan); formula aplikatu ondoren, bigarren integrala hasierakoa baino errazagoa izatea espero da.</span><br />
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<span style="font-family: "verdana" , sans-serif; font-size: large;"><b>u </b>eta <b>dv</b> aukeratzeko ikusitako arau mnemoteknikoaren adibide bat doakizue bideo honetan:</span></div>
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<span style="font-family: "verdana" , sans-serif; font-size: large;">Zatikako metodoaren erabileraren beste adibide bat:</span><br />
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<span style="font-family: "verdana" , sans-serif; font-size: large;">➤Beharrezkoa al da zatikako metodoaren formula? Ba al dago integral hauek ebazterik formula erabili gabe?</span></div>
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<span style="font-family: "verdana" , sans-serif; font-size: large;">Formula lortzeko jarraituriko pausoak aztertuta, berehala konturatzen gara u·v biderkaduraren d</span><span style="font-family: "verdana" , sans-serif; font-size: large;">eribatua klabea dela. Biderkadura hau deribatzerakoan integratu nahi dugun <b>f</b> funtzioa agertzea lortuko bagenu, integrala ia ebatzita izango genuke. Beraz, gakoa hor dago, u.v egoki aukeratzean, honen deribatua kalkulatzerakoan (u'·v+u·v') integratu nahi dugun funtzioa (∫<b>f</b>·dx) batugaietako batean agertu behar da (<b>f</b> = u'·v edo <b>f</b> = u·v').</span></div>
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<span style="font-family: "verdana" , sans-serif;"><b><u><span style="font-size: x-large;">Adibide 1</span></u></b><span style="font-size: large;"> </span></span></div>
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<span style="font-family: "verdana" , sans-serif; font-size: large;">Adibide bera, formula eskuratzeko jarraituriko pausoekin konparatuz:</span></div>
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<b style="font-family: Verdana, sans-serif; font-size: x-large;"><u>Adibide 2</u></b></div>
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<span style="font-family: "verdana" , sans-serif; font-size: large;"><b style="font-family: Verdana, sans-serif; font-size: x-large;"><u>Adibide 3 (bideoa)</u></b></span></div>
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<b style="font-family: Verdana, sans-serif; font-size: x-large;"><u>Adibide 4 (aurkezpena)</u></b><br />
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<iframe allowfullscreen="true" frameborder="0" height="389" mozallowfullscreen="true" src="https://docs.google.com/presentation/d/e/2PACX-1vTeyac6iZyI2EQtOkV1QbFCiE5pvnuVXgx3J9JsU_bPGGa9dfZCk6D_5zA1kAbby7f64lpXyPjID9C1/embed?start=false&loop=false&delayms=5000" webkitallowfullscreen="true" width="480"></iframe>
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<span style="color: #ea9999; font-family: "verdana" , sans-serif; font-size: x-large;"><b>*****</b></span></div>
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Egin Matematikahttp://www.blogger.com/profile/01348970164668368901noreply@blogger.com0tag:blogger.com,1999:blog-4037937100819081512.post-43483756258056778302018-10-11T01:16:00.000+02:002019-03-28T15:52:10.267+01:00MATEMATIKA II ikasgaiaren sitesa<br />
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<span style="font-family: "verdana" , sans-serif; font-size: large;">Arrasate BHI ikastetxeko Batxilergo 2. mailako ikasleek "MATEMATIKA II" ikasgaiaren prestakuntzan laguntza izan dezaten eta ikasgaiaren osagarri gisa, honako sites hau sortu dut: </span></div>
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<tr><td class="tr-caption" style="text-align: center;"><span style="font-family: "trebuchet ms" , sans-serif; font-size: large;"><a href="https://sites.google.com/view/matematikaii-arrasatebhi/matematika-ii" target="_blank">MATEMATIKA II ARRASATE BHI</a></span></td></tr>
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<span style="font-family: "verdana" , sans-serif; font-size: large;">Webgune honetan gelan erabiliko ditugun apunteak, ebazteko ariketak, ariketa ebatziak, problemak, teoriaren azalpen-bideoak, ariketa ebatzien bideoak, hautoprobetako ariketa ebatziak eta matematikarekin zerikusia duten gaiak agertuko dira aldiro.</span></div>
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<span style="font-family: "verdana" , sans-serif; font-size: large;">Zabalik dago, edozeini lagungarri edota interesgarria iruditzen bazaio, erabil eta jeitsi dezake berton argitaratzen den materiala; beti ere hezkuntza, irakaskuntza edota norberaren prestakuntzarako erabilera emanez.</span></div>
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Egin Matematikahttp://www.blogger.com/profile/01348970164668368901noreply@blogger.com0tag:blogger.com,1999:blog-4037937100819081512.post-14846315654770280442018-07-20T16:58:00.001+02:002018-07-21T18:25:38.973+02:00PROBLEMA 5 (OPE-2018-EPE)<br />
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<span style="font-family: "verdana" , sans-serif; font-size: large;"><b><span style="font-size: x-large;">PROBLEMA 5 </span><span style="font-size: large;">(2018ko EPEa)</span></b></span></td>
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<span style="font-family: "verdana" , sans-serif; font-size: x-large;">Hurrengo murrizketak emanda,</span></div>
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<span style="font-family: "verdana" , sans-serif; font-size: x-large; text-align: justify;">Zein da zoriz aukeratutako puntu baten koordenatu biak positiboak izateko probabilitatea? Eta bi koordenatuak zenbaki osoak izateko?</span></div>
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<b style="font-family: verdana, sans-serif; font-size: xx-large;"><u><span style="color: #38761d;">1. Galdera</span></u></b><br />
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<span style="font-family: "verdana" , sans-serif; font-size: x-large;">Emandako murrizketak planoko A eremu bat definitzen dute:</span><br />
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<span style="font-family: "verdana" , sans-serif; font-size: x-large;">Koordenatu biak positiboak diren eremua <b>A+ </b>izendatuko dugu:</span><br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEghDTsQovhawp2xSfq2SiKmrM6KMe72aHQKv47cIhaf_omypzPYlmCoMWavwnO8fuPa77O1vhd52g60FkQ-Uhs_nU4oca9e5ze2-8EJNQ49zs36ZzZDcRtDCKySO-tKOcHQU375DpeZu8oc/s1600/PROBLEMA5-OPE2018%25283%2529.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1108" data-original-width="968" height="640" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEghDTsQovhawp2xSfq2SiKmrM6KMe72aHQKv47cIhaf_omypzPYlmCoMWavwnO8fuPa77O1vhd52g60FkQ-Uhs_nU4oca9e5ze2-8EJNQ49zs36ZzZDcRtDCKySO-tKOcHQU375DpeZu8oc/s640/PROBLEMA5-OPE2018%25283%2529.JPG" width="558" /></a></div>
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<span style="font-family: "verdana" , sans-serif; font-size: x-large; text-align: justify;">Homogeneotasun printzipioan oinarrituz, A eremu barruko Aⱼ azpimultzo bakoitzari bere azalarekiko proportzionala den probabilitatea esleitzen zaio; Hain zuzen ere:</span><br />
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<span style="font-family: "verdana" , sans-serif; font-size: x-large; text-align: justify;">P(Aⱼ)=Azalera </span><span style="font-family: "verdana" , sans-serif; font-size: x-large; text-align: justify;">Aⱼ</span><span style="font-family: "verdana" , sans-serif; font-size: x-large; text-align: justify;">/Azalera A=</span><span style="font-family: "verdana" , sans-serif; font-size: x-large; text-align: justify;">|A|/</span><span style="font-family: "verdana" , sans-serif; font-size: x-large; text-align: justify;">|Aⱼ|</span></div>
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<span style="font-family: "verdana" , sans-serif; font-size: x-large; text-align: justify;">Non A-ren probabilitatea P(A)=1 den. Probabilitatea esleitzeko modu hau banaketa uniforme izenez ezaguna da.</span><br />
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<span style="font-family: "verdana" , sans-serif; font-size: x-large; text-align: justify;">Orduan, zoriz aukeratutako puntu baten koordenatu biak positiboak izateko probabilitatea, bi eremu hauen azaleren arteko erlazioak ematen digu:</span><br />
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<span style="font-family: "verdana" , sans-serif; font-size: x-large; text-align: justify;"><b> P(x,y:+,+)=|A+|/|A|</b></span><br />
<span style="font-family: "verdana" , sans-serif; text-align: justify;"> bi koordenatuak +</span><br />
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<span style="font-family: "verdana" , sans-serif; font-size: x-large; text-align: justify;">Bi azalera hauek kalkulatzeko oinarrizko geometria eta kalkulu integrala erabiliko dugu:</span><br />
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<span style="font-family: "verdana" , sans-serif; font-size: x-large; text-align: justify;">Irudiko eremu triangeluarren azalerak oina·altuera/2:</span><br />
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<span style="font-family: "verdana" , sans-serif; font-size: x-large;">y=4/x funtzioak Xardatzarekin x=1 eta x=2ren artean mugaturiko R barrutiaren azalera:</span></div>
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<span style="font-family: "verdana" , sans-serif; font-size: x-large; text-align: justify;">T₂</span><span style="font-family: "verdana" , sans-serif; font-size: x-large; text-align: justify;"> eremuan "y" ordenatua negatiboa da. </span><span style="font-family: "verdana" , sans-serif; font-size: x-large; text-align: center;">P(x,y:+,+) bi koordenatuak positiboak izateko probabilitatea eta </span><span style="font-family: "verdana" , sans-serif; font-size: x-large; text-align: justify;"> </span><span style="font-family: "verdana" , sans-serif; font-size: x-large; text-align: center;">P(x,y:+,-) "y" koordenatua negatiboa izateko probabilitatea izendatzeko erabiltzen badira, zera idatz daiteke</span><span style="font-family: "verdana" , sans-serif; font-size: x-large; text-align: justify;">:</span></div>
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<span style="font-family: "verdana" , sans-serif; font-size: x-large;">P(x,y:+,+)=</span><b style="font-family: verdana, sans-serif; font-size: xx-large; text-align: justify;">|</b><span style="font-family: "verdana" , sans-serif; font-size: x-large; text-align: justify;">A+</span><b style="font-family: verdana, sans-serif; font-size: xx-large; text-align: justify;">|</b><span style="font-family: "verdana" , sans-serif; font-size: x-large; text-align: justify;">/</span><b style="font-family: verdana, sans-serif; font-size: xx-large; text-align: justify;">|</b><span style="font-family: "verdana" , sans-serif; font-size: x-large; text-align: justify;">A</span><b style="font-family: verdana, sans-serif; font-size: xx-large; text-align: justify;">|</b><br />
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<span style="font-family: "verdana" , sans-serif; font-size: x-large;">P(x,y:+,+)=1⎯P(x,y:+,⎯)=1⎯</span><span style="font-family: "verdana" , sans-serif; font-size: x-large; text-align: justify;">|T₂|</span><span style="font-family: "verdana" , sans-serif; font-size: x-large; text-align: justify;">/</span><b style="font-family: verdana, sans-serif; font-size: xx-large; text-align: justify;">|</b><span style="font-family: "verdana" , sans-serif; font-size: x-large; text-align: justify;">A</span><b style="font-family: verdana, sans-serif; font-size: xx-large; text-align: justify;">|</b><br />
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<span style="font-family: "verdana" , sans-serif; font-size: x-large;">Hau da,</span></div>
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<span style="font-family: "verdana" , sans-serif; font-size: x-large;">Edo zuzenean:</span></div>
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<span style="font-family: "verdana" , sans-serif; font-size: x-large;"><b><u><span style="color: #38761d;">2. Galdera</span></u></b></span></div>
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<span style="font-family: "verdana" , sans-serif; font-size: x-large;">Bi koordenatuak osoak izateko probabilitatea zero da; puntu multzo honen azalera (neurria) nulua delako; ez du ℜ² espazio euklidearreko <b>A</b> multzoaren barnean azalerarik mugatzen.</span></div>
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<span style="font-family: "verdana" , sans-serif; font-size: x-large;"><b><span style="color: #3d85c6;">*****</span></b></span></div>
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<br />Egin Matematikahttp://www.blogger.com/profile/01348970164668368901noreply@blogger.com0tag:blogger.com,1999:blog-4037937100819081512.post-24877361666339791272018-07-17T18:38:00.000+02:002018-07-17T20:24:37.435+02:00PROBLEMA 4 (OPE-2018-EPE)<br />
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<span style="font-family: "verdana" , sans-serif; font-size: large;"><b><span style="font-size: x-large;">PROBLEMA 4 </span><span style="font-size: large;">(2018ko EPEa)</span></b></span></td>
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<span style="color: #444444; font-family: "verdana" , sans-serif; font-size: large;"><i><span style="font-size: x-large;">Edozein </span><b style="font-size: xx-large;">n</b><span style="font-size: x-large;"> zenbaki arruntarentzat </span><b><span style="font-size: x-large;">A</span><span style="font-size: large;">n</span><span style="font-size: x-large;">=2ⁿ+A²ⁿ+A³ⁿ</span></b><span style="font-size: x-large;"> definitzen da.</span></i></span></div>
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<span style="color: #444444; font-family: "verdana" , sans-serif; font-size: large;"><i><b style="font-size: xx-large;"> </b><span style="font-size: x-large;"><b>i. </b>Frogatu n-ren balio guztientzat <b>A</b></span><span style="font-size: large;"><b>n+3</b></span><span style="font-size: x-large;"> kongruente <b>A</b></span><span style="font-size: large;"><b>n</b></span><span style="font-size: x-large;"> ,</span></i></span></div>
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<span style="color: #444444; font-family: "verdana" , sans-serif; font-size: large;"><i><span style="font-size: x-large;"> moduluz <b>7</b> dela.</span></i></span></div>
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<span style="color: #444444; font-family: "verdana" , sans-serif; font-size: large;"><i><span style="font-size: x-large;"> <b>ii.</b> Aurkitu <b>n</b>-ren zein balioarentzat </span></i></span><i style="color: #444444; font-family: verdana, sans-serif;"><b><span style="font-size: x-large;">A</span><span style="font-size: large;">n</span></b><span style="font-size: x-large;"> </span></i><i style="color: #444444; font-family: verdana, sans-serif;"><span style="font-size: x-large;"><b>7</b> zenbakiaz</span></i></div>
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<i style="color: #444444; font-family: verdana, sans-serif;"><span style="font-size: x-large;"> zatigarria den (erabili aurreko emaitza).</span></i></div>
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<span style="font-family: "verdana" , sans-serif; font-size: x-large;">Gogora dezagun nola definitzen den Zenbaki-Teorian m modulodun (m∈𝐙, m≥1) kongruentzia:</span><br />
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<span style="color: #0b5394; font-family: "verdana" , sans-serif; font-size: x-large;"><i><b>a</b> eta <b>b</b> bi zenbaki oso <b>m</b> moduluz kongruenteak dira, <b>a≡b (mod. m)</b>, <b>m </b>modulua <b>a-b</b> diferentziaren zatitzailea bada, <b>a-b=km k∈𝐙</b>. Bestela esanda, <b>m </b>moduluaz zatituz hondar berdina ematen dutenean: <b>a=k₁·m+r</b>, <b>b=k₂·m+r </b>eta <b>0≤r≤m-1</b></i></span></div>
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<b style="font-family: Verdana, sans-serif; font-size: xx-large;"><u><i><span style="color: #38761d;">1.Atala</span></i></u></b><br />
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<span style="font-family: "verdana" , sans-serif; font-size: x-large;">Emandako adierazpena, honela ere adieraz daiteke:</span><br />
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<i style="color: #444444; font-family: verdana, sans-serif;"><span style="font-size: x-large;"><b>A</b></span><span style="font-size: large;"><b>n+3</b></span></i><span style="font-family: "verdana" , sans-serif; font-size: x-large;"> kongruente </span><i style="color: #444444; font-family: verdana, sans-serif;"><span style="font-size: x-large;"><b>A</b></span><span style="font-size: large;"><b>n</b></span></i><span style="font-family: "verdana" , sans-serif; font-size: x-large;">, moduluz 7 da, baldin bien arteko diferentzia 7ren multiploa bada; hau da,</span></div>
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<span style="font-family: "verdana" , sans-serif; font-size: x-large;">Hortaz, <b>edozein n arruntarentzat</b> </span><i style="color: #444444; font-family: verdana, sans-serif;"><span style="font-size: x-large;"><b>A</b></span><span style="font-size: large;"><b>n+3</b></span></i><span style="font-family: "verdana" , sans-serif; font-size: x-large;">≡</span><i style="color: #444444; font-family: verdana, sans-serif;"><span style="font-size: x-large;"><b>A</b></span><span style="font-size: large;"><b>n</b></span></i><span style="font-family: "verdana" , sans-serif; font-size: x-large;"> <b>(mod. 7)</b></span></div>
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<span style="font-family: "verdana" , sans-serif; font-size: x-large;">Beste era honetan ere arrazoitu daiteke:</span><br />
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<span style="font-family: "verdana" , sans-serif; font-size: x-large;">2-ren berreketen kongruentziak (7 moduluz) koadro honetan laburtuz,</span><br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiE38wTrJEea_m1k9es9ExIbyqnCrriJHMErRgGCCziaDAQr5MlDrLwhiXAJX5KcJUbnZg0Yn2Vx0-R_5O09kErlUEflQFyIZM9NcgDPanfkOO08mZspTESztdyuuLHaZH1EathfZs8iJqH/s1600/PROBLEMA4-OPE2018%25283%2529.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="323" data-original-width="1251" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiE38wTrJEea_m1k9es9ExIbyqnCrriJHMErRgGCCziaDAQr5MlDrLwhiXAJX5KcJUbnZg0Yn2Vx0-R_5O09kErlUEflQFyIZM9NcgDPanfkOO08mZspTESztdyuuLHaZH1EathfZs8iJqH/s1600/PROBLEMA4-OPE2018%25283%2529.JPG" /></a></div>
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<span style="font-family: "verdana" , sans-serif; font-size: x-large;">Non kongruentzien propietateak erabili diren,</span><br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjmzFMoPoDLk5QTa1S6ouFdayiE9WG-_6hWQqSMK8LBrYCtKWxQalNUcBL7wLRKoW-kXtLRGd9EX3rzb4IgEaePvxvyYHD1oyJAF60Y_rIDijkD39yh0s0k_IJjhlyF7HGl5gYVtpl-IsX9/s1600/PROBLEMA4-OPE2018%25284%2529.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="572" data-original-width="1231" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjmzFMoPoDLk5QTa1S6ouFdayiE9WG-_6hWQqSMK8LBrYCtKWxQalNUcBL7wLRKoW-kXtLRGd9EX3rzb4IgEaePvxvyYHD1oyJAF60Y_rIDijkD39yh0s0k_IJjhlyF7HGl5gYVtpl-IsX9/s1600/PROBLEMA4-OPE2018%25284%2529.JPG" /></a></div>
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<span style="font-family: "verdana" , sans-serif; font-size: x-large;">Bestetik, gogoratu m moduludun aritmetikan, a zenbaki osoak alderantzizkoa du baldin eta bakarrik soilik zkh(a,m)=1 bada.</span><br />
<span style="font-family: "verdana" , sans-serif; font-size: x-large;"> Orduan, <b>a·b≡1</b> bada, <b>a⁻¹=b</b> eta <b>b⁻¹=a </b></span><br />
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<span style="font-family: "verdana" , sans-serif; font-size: x-large;">Propietate hau honela aplikatu da,</span><br />
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<span style="font-family: "verdana" , sans-serif; font-size: x-large;">2·4=8≡1 (mod. 7) denez, 2ren alderantzizkoa 2⁻¹=4 eta 4ren alderantzizkoa 4</span><span style="font-family: "verdana" , sans-serif; font-size: x-large;">⁻¹=2</span><br />
<span style="font-family: "verdana" , sans-serif; font-size: x-large;"><br /></span><span style="font-family: "verdana" , sans-serif; font-size: x-large;">Azkenik, taulako azkenengo errenkadatik, honako ondoriora iristen gara:</span><br />
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<b><i style="color: #444444; font-family: verdana, sans-serif;"><span style="font-size: x-large;">A</span><span style="font-size: large;">n+3</span></i><span style="font-family: "verdana" , sans-serif; font-size: x-large;">-</span><i style="color: #444444; font-family: verdana, sans-serif;"><span style="font-size: x-large;">A</span><span style="font-size: large;">n</span></i><span style="font-family: "verdana" , sans-serif; font-size: x-large;"> </span><span style="font-family: "verdana" , sans-serif; font-size: x-large;">≡0-0=0 </span></b><span style="font-family: "verdana" , sans-serif; font-size: x-large;"><b>(mod. 7)</b> edo </span><b><i style="color: #444444; font-family: verdana, sans-serif;"><span style="font-size: x-large;">A</span><span style="font-size: large;">n+3</span></i><span style="font-family: "verdana" , sans-serif; font-size: x-large;">-</span><i style="color: #444444; font-family: verdana, sans-serif;"><span style="font-size: x-large;">A</span><span style="font-size: large;">n</span></i><span style="font-family: "verdana" , sans-serif; font-size: x-large;"> </span><span style="font-family: "verdana" , sans-serif; font-size: x-large;">≡3-3=0 </span><span style="font-family: "verdana" , sans-serif; font-size: x-large;">(mod. 7)</span></b><br />
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<b style="font-family: Verdana, sans-serif; font-size: xx-large;"><u><i><span style="color: #38761d;">2.Atala</span></i></u></b><br />
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<span style="font-family: "verdana" , sans-serif; font-size: x-large;">Aurreko ataletik zera ondorioztatu daiteke, </span><i style="color: #444444; font-family: verdana, sans-serif;"><span style="font-size: x-large;">A</span><span style="font-size: large;">n</span></i><span style="font-family: "verdana" , sans-serif; font-size: x-large;"> 7 zenbakiaz zatigarria da baldin n ez bada 3ren multiploa:</span><br />
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<i style="color: #444444; font-family: verdana, sans-serif;"><b><span style="font-size: x-large;"> A</span><span style="font-size: large;">n </span></b></i><b><span style="font-family: "verdana" , sans-serif; font-size: x-large;">7az zatigarria, baldin n≠3k non k∈𝐍</span></b><br />
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<span style="font-family: "verdana" , sans-serif; font-size: x-large;">Edo, kongruentzien hizkuntza erabiliz, gauza bera dena:</span><br />
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<i style="color: #444444; font-family: verdana, sans-serif;"><b><span style="font-size: x-large;"> A</span><span style="font-size: large;">n </span></b></i><b><span style="font-family: "verdana" , sans-serif; font-size: x-large;">7az zatigarria, baldin n≡1 edo </span><span style="font-family: "verdana" , sans-serif; font-size: x-large;">n≡2 (mod. 3)</span></b><br />
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<span style="font-family: "verdana" , sans-serif; font-size: x-large;"><b><span style="color: #0b5394;">*****</span></b></span></div>
Egin Matematikahttp://www.blogger.com/profile/01348970164668368901noreply@blogger.com0tag:blogger.com,1999:blog-4037937100819081512.post-84979824515267346592018-07-12T17:46:00.002+02:002018-07-19T15:31:55.547+02:00PROBLEMA 3 (OPE-2018-EPE)<br />
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<span style="font-family: "verdana" , sans-serif; font-size: large;"><b><span style="font-size: x-large;">PROBLEMA 3 </span><span style="font-size: large;">(2018ko EPEa)</span></b></span></td>
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<span style="color: #444444; font-family: "verdana" , sans-serif; font-size: x-large;"><i><b>ABC</b> triangelu zorrotz baten altuerak <b>H</b> ortozentroan ebakitzen dira. Badakigu <b>AB=CH</b> dela. Zehaztu <b>∡BCA </b>angeluaren balioa.</i></span></div>
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<span style="font-family: "verdana" , sans-serif; font-size: x-large;">Triangelu zorrotza denez, atuerak triangeluaren barnealdean ebakitzen dira H ortozentroan. Honela, triangelua binaka antzekoak diren 6 triangelu zuzenetan banatuta geratzen da.</span><br />
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<span style="font-family: "verdana" , sans-serif; font-size: x-large;"><b>H</b> erpinean, erpinez aurkako angeluak berdinak dituzten triangeluak antzekoak dira:</span><br />
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<span style="color: purple; font-family: "verdana" , sans-serif; font-size: x-large;"><b><u>Metodo 1</u></b></span></div>
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<span style="font-family: "verdana" , sans-serif; font-size: x-large;"><b>CHA'</b> eta </span><span style="font-family: "verdana" , sans-serif; font-size: x-large;"><b>ABC'</b> triangelu </span><span style="font-family: "verdana" , sans-serif; font-size: x-large;">zuzenetan oinarrizko erlazio trigonometrikoak aplikatuz:</span></div>
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<span style="font-family: "verdana" , sans-serif; font-size: x-large;"> <span style="color: #0b5394;">➤</span><b>CHA'</b> triangeluan <b>HA'=c·sin𝝰</b></span><br />
<span style="font-family: "verdana" , sans-serif; font-size: x-large;"> <span style="color: #0b5394;">➤</span><b>ABA'</b> triangeluan <b>BA'=c</b></span><span style="font-family: "verdana" , sans-serif; font-size: x-large;"><b>·sin𝝰</b></span><br />
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<span style="font-family: "verdana" , sans-serif; font-size: x-large;">Hemendik, <b>BHA'</b> triangelu zuzen isoszelea da (eta baita bere antzekoa den <b>AHB'</b> ere). Beraz, triangelu hauen angelu zorrotzak <b>𝞬=45º</b>koak dira.</span><br />
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<span style="font-family: "verdana" , sans-serif; font-size: x-large;"><b>CAA'</b> triangelu zuzenean (edo <b>CBB'</b></span><b style="font-family: verdana, sans-serif; font-size: xx-large;"> </b><span style="font-family: "verdana" , sans-serif; font-size: x-large;">triangeluan)</span><span style="font-family: "verdana" , sans-serif; font-size: x-large;">, angelu zorrotzen batura <b>90º </b></span><span style="font-family: "verdana" , sans-serif; font-size: x-large;">da:</span><br />
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<span style="font-family: "verdana" , sans-serif; font-size: x-large;"><b>(𝞪+𝞫)+𝞬=90º</b></span><br />
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<b><span style="font-family: "verdana" , sans-serif; font-size: x-large;">𝞪+𝞫+45</span><span style="font-family: "verdana" , sans-serif; font-size: x-large;">º</span><span style="font-family: "verdana" , sans-serif; font-size: x-large;">=90º</span></b><br />
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<b><span style="font-family: "verdana" , sans-serif; font-size: x-large;">𝞪+𝞫</span><span style="font-family: "verdana" , sans-serif; font-size: x-large;">=</span><span style="font-family: "verdana" , sans-serif; font-size: x-large;">45</span><span style="font-family: "verdana" , sans-serif; font-size: x-large;">º</span></b><br />
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<b><span style="color: #0b5394;"><u><i style="font-family: verdana, sans-serif; font-size: xx-large;">∡C</i><span style="font-family: "verdana" , sans-serif; font-size: x-large;">=</span><span style="font-family: "verdana" , sans-serif; font-size: x-large;">45</span><span style="font-family: "verdana" , sans-serif; font-size: x-large;">º</span></u></span></b></div>
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<span style="font-family: "verdana" , sans-serif; font-size: x-large;"> <b>CAA'</b> eta <b>CBB'</b></span><b style="font-family: verdana, sans-serif; font-size: xx-large; text-align: start;"> </b><span style="font-family: "verdana" , sans-serif; font-size: x-large;">triangeluak ere zuzen isoszeleak dira.</span><br />
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<span style="font-family: "verdana" , sans-serif; font-size: x-large;">Beste era honetan ere ebatz dezakegu:</span></div>
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<span style="font-family: "verdana" , sans-serif; font-size: x-large;">CHB' eta ABB' triangeluak kongruenteak dira , bi angelu eta bi angeluen arteko aldea berdinak direlako:</span></div>
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<span style="font-family: "verdana" , sans-serif; font-size: x-large;"> ▷CH=AB=c hipotenusak luzera berekoak</span></div>
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<span style="text-align: center;"><span style="font-family: "verdana" , sans-serif; font-size: x-large;"> ▷𝞫 angelu zorrotza berdina </span></span></div>
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<span style="text-align: center;"><span style="font-family: "verdana" , sans-serif; font-size: x-large;"> ▷Triangelu zuzenak direnez, baita beste angelu zorrotza</span></span></div>
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<span style="text-align: center;"><span style="font-family: "verdana" , sans-serif; font-size: x-large;"> ere </span></span><span style="font-family: "verdana" , sans-serif; font-size: x-large; text-align: center;">berdina.</span></div>
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<span style="text-align: center;"><span style="font-family: "verdana" , sans-serif; font-size: x-large;">Ondorioz, BB'=B'C⇒CBB' triangelu zuzen isoszelea:</span></span></div>
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<span style="text-align: center;"><span style="font-family: "verdana" , sans-serif; font-size: x-large;"> tan(</span></span><span style="font-family: "verdana" , sans-serif; font-size: x-large; text-align: center;">𝞪+𝞫)=tan(⦛C)=</span><span style="font-family: "verdana" , sans-serif; font-size: x-large; text-align: center;">BB'/B'C=1⇒</span><span style="color: #0b5394; font-family: "verdana" , sans-serif; font-size: x-large; text-align: center;"><b><u>⦛C=45º</u></b></span></div>
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<span style="color: #0b5394; font-family: "verdana" , sans-serif; font-size: x-large;"><b>*****</b></span></div>
Egin Matematikahttp://www.blogger.com/profile/01348970164668368901noreply@blogger.com0tag:blogger.com,1999:blog-4037937100819081512.post-86862233099587782572018-07-11T12:08:00.000+02:002018-07-17T18:47:38.750+02:00PROBLEMA 2 (OPE-2018-EPE)<br />
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<span style="font-family: "verdana" , sans-serif; font-size: large;"><b><span style="font-size: x-large;">PROBLEMA 2 </span><span style="font-size: large;">(2018ko EPEa)</span></b></span></td>
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<span style="color: #444444; font-family: "verdana" , sans-serif; font-size: x-large;"><i>Izan bedi <b>p</b> zenbaki lehena. Zehaztu <b>k∈𝚭</b> zenbaki oso guztiak:</i></span></div>
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<span style="color: #444444; font-family: "verdana" , sans-serif; font-size: x-large;"><i><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhVeWGr0DBl86vnnBWP3c6LT_qaRMjx3ZXKKPDFB73KLWCOdue2iYFfscYH5aZCRhk37hly6Yah9XlkjfFDvry1pF7TcvDHbxTcYjM29Ntdch_4Zp-DjFY7A7XG_bafOfz-hiitcdAdudfK/s1600/erroa2.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="50" data-original-width="128" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhVeWGr0DBl86vnnBWP3c6LT_qaRMjx3ZXKKPDFB73KLWCOdue2iYFfscYH5aZCRhk37hly6Yah9XlkjfFDvry1pF7TcvDHbxTcYjM29Ntdch_4Zp-DjFY7A7XG_bafOfz-hiitcdAdudfK/s1600/erroa2.JPG" /></a><i style="text-align: left;"></i></i></span></div>
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<span style="color: #444444; font-family: "verdana" , sans-serif; font-size: x-large;"><i><i>zenbaki oso ez negatiboa izateko.</i></i></span></div>
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<span style="font-family: "verdana" , sans-serif; font-size: x-large;">Emandako adiezpena zenbaki oso ez negatiboa izan dadin, <b>k²-kp</b> zenbaki karratua <b>a²</b> izan behar da:</span><br />
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<span style="font-family: "verdana" , sans-serif; font-size: x-large;">k zenbakia osoa izan dadin, beharrezko baldintza p²+4a² karratu perfektua izatea:</span><br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiUz10ZA0kwJ3R6nQm1TylzwYYg95TRCNVEZ3MCFmeJzC28laJY0EfMa0dgqeXAKz3RVd4KJluzYFbOGKbANkW32X7QPkAd-RpjKUAU6K_oN8YxK4suzKryNpPBsjzWVac3RgNDGqL8SWp0/s1600/PROBLEMA2-OPE2018%25284%2529.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="318" data-original-width="549" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiUz10ZA0kwJ3R6nQm1TylzwYYg95TRCNVEZ3MCFmeJzC28laJY0EfMa0dgqeXAKz3RVd4KJluzYFbOGKbANkW32X7QPkAd-RpjKUAU6K_oN8YxK4suzKryNpPBsjzWVac3RgNDGqL8SWp0/s1600/PROBLEMA2-OPE2018%25284%2529.JPG" /></a></div>
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<span style="font-family: "verdana" , sans-serif; font-size: x-large;">Kontuan izanik:</span><br />
<span style="font-family: "verdana" , sans-serif; font-size: x-large;">►b+2a⩾b-2a</span><br />
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<span style="font-family: "verdana" , sans-serif; font-size: x-large;">►p lehena eta p² ren zatitzaileak: {1,p,p²} (zenbaki lehenen karratuek hiru eta bakarrik hiru zatitzaile dituzte)</span></div>
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<span style="font-family: "verdana" , sans-serif; font-size: x-large;">Honako bi kasu hauek ager daitezke:</span></div>
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<span style="font-family: "verdana" , sans-serif; font-size: x-large;">Zenbaki lehen bikoiti bakarra dago p=2. Kasu honetan, p+1=3 bakoitia denez, (p+1)/2=3/2 ez litzateke osoa izango. Ondorioz, azkeneko bi soluzioak baztertu behar dira p=2 kasurako.</span><br />
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<span style="font-family: "verdana" , sans-serif; font-size: x-large;">Laburtuz:</span><br />
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<span style="font-family: "verdana" , sans-serif; font-size: x-large;">Problema PDFan:</span></div>
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<span style="font-family: "verdana" , sans-serif; font-size: x-large;"><b><span style="color: #274e13;">*****</span></b></span></div>
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</span>Egin Matematikahttp://www.blogger.com/profile/01348970164668368901noreply@blogger.com0tag:blogger.com,1999:blog-4037937100819081512.post-50708538455852333952018-07-10T11:52:00.003+02:002018-07-17T18:47:38.727+02:00PROBLEMA 1 (OPE-2018-EPE)<br />
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<span style="font-family: "verdana" , sans-serif; font-size: large;"><b><span style="font-size: x-large;">PROBLEMA 1 </span><span style="font-size: large;">(2018ko EPEa)</span></b></span></td>
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<span style="color: #444444; font-family: "verdana" , sans-serif; font-size: x-large;"><i>Izan bedi koordenatu ardatzen alde positiboek eta </i></span></div>
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<span style="color: #444444;"><span style="font-family: "verdana" , sans-serif; font-size: x-large; font-style: italic;"><b>y = 2 cos x</b> kurbak, </span><span style="font-family: "verdana" , sans-serif; font-size: x-large;"><i style="font-weight: bold;">0 ≤ x ≥ </i><b>𝜋</b><i style="font-weight: bold;">/2</i><i> izanik, zehazten duten R eskualdea. Aurkitu a parametroaren balioa </i><b style="font-style: italic;">y = a sin x</b><i> kurbak R eskualdea azalera bereko bi zatitan banatzeko.</i></span></span></div>
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<span style="font-family: "verdana" , sans-serif; font-size: x-large;">Koordenatu ardatzek eta <i><b>y=2cosx </b></i>funtzioak mugaturiko eskualdearen azalera <b>2</b> unitate karratukoa da,</span><br />
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<span style="font-family: "verdana" , sans-serif; font-size: x-large;"><i><b>y=asinx</b></i> funtzioak aurreko eskualdea azalera bereko bi zati berdinetan banatzen du (zati bakoitzaren azalera <b>1</b> unitate karratukoa da)</span><br />
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<span style="font-family: "verdana" , sans-serif;"><span style="font-size: x-large;"><b><i>y=2cosx</i></b> eta <b><i>y=asinx</i></b> funtzioen arteko ekabi-puntua,</span></span><br />
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<span style="font-family: "verdana" , sans-serif;"><span style="font-size: x-large;">Ordenatuen ardatzak eta bi funtzio hauek <b><i>[0,x₀]</i></b> tartean definituriko eskualdearen azalera <b>1</b> da,</span></span></div>
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<span style="font-family: "verdana" , sans-serif;"><span style="font-size: x-large;">Identitate trigonometrikoak gogoratuz, sinua eta kosinua "<b><i>a</i></b>" parametroaren menpe adieraziko dugu (</span></span><i style="color: #444444; font-family: verdana, sans-serif; font-size: xx-large; font-weight: bold;">0 ≤ x ≥ </i><b style="color: #444444; font-family: verdana, sans-serif; font-size: xx-large;">𝜋</b><i style="color: #444444; font-family: verdana, sans-serif; font-size: xx-large; font-weight: bold;">/2 </i><span style="color: #444444; font-family: verdana, sans-serif; font-size: x-large;">kontuan izanik)</span><span style="font-family: verdana, sans-serif; font-size: x-large;">,</span></div>
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<span style="font-family: "verdana" , sans-serif;"><span style="font-size: x-large;">Azkenik, (<b><span style="color: red;">I</span></b>)an ordezkatuz,</span></span></div>
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Egin Matematikahttp://www.blogger.com/profile/01348970164668368901noreply@blogger.com0tag:blogger.com,1999:blog-4037937100819081512.post-14176662459614862592018-05-23T23:37:00.000+02:002018-07-04T20:14:32.914+02:00Problema matematikoen itsasoan murgilduz<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhBbeYvdQL66Mpx35-ILjougQRyqRV9rsm2lHCr2ulLMysNqmWz_ERjKYkedCgKBKdzTWJ-5O7T2ntZgudbJ-dhsrupGqXkq9k2kQn7CISFw4_3wMMc5P3z2__mUAAc4gAK9oB4dRUAqsfE/s1600/Los+elegantes+triangulos.JPG" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" data-original-height="668" data-original-width="1189" height="177" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhBbeYvdQL66Mpx35-ILjougQRyqRV9rsm2lHCr2ulLMysNqmWz_ERjKYkedCgKBKdzTWJ-5O7T2ntZgudbJ-dhsrupGqXkq9k2kQn7CISFw4_3wMMc5P3z2__mUAAc4gAK9oB4dRUAqsfE/s320/Los+elegantes+triangulos.JPG" width="320" /></a><span style="color: #0b5394;"><span style="font-size: large;"><span style="font-family: "verdana" , sans-serif;">➤Sestaoko XIX. Matematika Jardunaldietan problema matematikoen ebazpena gai hartuta "Problemak ebaztea, esperientzia atsegina!" / "Resolver problemas, una experiencia gratificante" ponentzia aurkeztu nuen otsailaren 20an. Niretzat oso esperientzia atsegina eta aberasgarria izan zenez, hitzaldirako gertatu nuen materiala gogora ekarri nahi dut gaurkoan eta nire blogaren bidez berriz ere partekatu. </span></span></span><br />
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<span style="color: #0b5394; font-family: "verdana" , sans-serif; font-size: large;">➤2015-16 eta 2016-17 ikasturteetan Prest_Gara programaren barruan eman nituen "Problemen Ebazpena Lankidetzan Blog Batekin" ikastaroetan landu genituen problemen bildumak ere uzten ditut hemen. Enuntziatuak eta proposatutako ebazpideak, denera 160 problema maila guztietakoak:</span></div>
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<span style="color: #990000; font-family: "verdana" , sans-serif; font-size: x-large;"><b><a href="https://drive.google.com/file/d/1rJ432fCV7Kw14pFOLtpnHCDf1L8I7-EH/view?usp=sharing" target="_blank">PROBLEMEN EBAZPENA I</a></b></span></div>
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<span style="color: #990000; font-family: "verdana" , sans-serif; font-size: x-large;"><b><a href="https://drive.google.com/file/d/1qeX7dIn5utTyDRs-GSUfaIa3I7Ruly01/view?usp=sharing" target="_blank">PROBLEMEN EBAZPENA II</a></b></span></div>
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<span style="color: #0b5394; font-family: "verdana" , sans-serif; font-size: large;">➤Ondorengo loturan klikatuz, Luis Miguel Iglesias Albarranek otsailaren 19an aurkezturiko "Irakasleen konpetentzia digitala IKTen erabilera Matematikak ikas eta irakasteko" ponentziaren eta nire hitzaldiaren bideoak ikusgai daude:</span></div>
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<span style="color: #0b5394; font-family: "verdana" , sans-serif; font-size: large;"><a href="https://sites.google.com/b03sestao.es/xixmatematikajardunaldiak/1-programa-d-b-h" style="background-color: #fff2cc;" target="_blank">XIX. MATEMATIKA JARDUNALDIEN BIDEOAK</a></span></div>
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<span style="color: #0b5394; font-family: "verdana" , sans-serif; font-size: x-large;"><b>*****</b></span></div>
<span style="color: #0b5394; font-family: "verdana" , sans-serif; font-size: large;"><br /></span>Egin Matematikahttp://www.blogger.com/profile/01348970164668368901noreply@blogger.com0tag:blogger.com,1999:blog-4037937100819081512.post-18666534265826111632018-04-07T00:21:00.000+02:002018-04-08T23:41:26.439+02:00Zenbaki lehenen multzoak ez du maximorik<span style="font-family: "verdana" , sans-serif; font-size: large;"><br /></span>
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<span style="font-family: "verdana" , sans-serif; font-size: large;">Zenbaki lehenen multzoa infinitua al da? Bestela esanda, zenbaki lehenen multzoak maximorik al du?</span><br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg1lv9sjx7gXhy8tl9hyvdzw77f2NhLdHRTZj5VIpbY88SOeuLPXnML-FbQau3gDqH9KwobDYqjDvwJGrvKgRu5pimHottKYUXjFa5OreMZR3eQSrxSaPruXQUHKVRj6oY53xTUUzsucaVt/s1600/Euclid_of_Megara.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="365" data-original-width="260" height="400" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg1lv9sjx7gXhy8tl9hyvdzw77f2NhLdHRTZj5VIpbY88SOeuLPXnML-FbQau3gDqH9KwobDYqjDvwJGrvKgRu5pimHottKYUXjFa5OreMZR3eQSrxSaPruXQUHKVRj6oY53xTUUzsucaVt/s400/Euclid_of_Megara.jpg" width="283" /></a><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiwCDDeSgvmaRyQ08cfk4UoiyDH8ImGJ443-lpppNIOdKfUnnn6y3o1eaXrNWkjY8vtRx7SNkvfFNMKJodRANcau0eQnfvMJJMMNrzk4ArJvv1053GpjjgR2DUGP5lSztPid1qNChVMbG2x/s1600/ZENBAKI+LEHENAK3.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="181" data-original-width="1134" height="100" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiwCDDeSgvmaRyQ08cfk4UoiyDH8ImGJ443-lpppNIOdKfUnnn6y3o1eaXrNWkjY8vtRx7SNkvfFNMKJodRANcau0eQnfvMJJMMNrzk4ArJvv1053GpjjgR2DUGP5lSztPid1qNChVMbG2x/s640/ZENBAKI+LEHENAK3.JPG" width="640" /></a></div>
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<span style="font-family: "verdana" , sans-serif; font-size: large;">Euclides matematikari greziarrak zenbaki lehenen multzoak ez zuela maximorik egiaztatu zuen; hau da, infinitua zela. Eskaini zigun egiaztapena dotorea da oso, harri bitxi bat.</span><br />
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<span style="font-family: "verdana" , sans-serif; font-size: large;">Honako prozedura hau jarraitu zuen:</span><br />
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<span style="color: #0b5394; font-family: "verdana" , sans-serif; font-size: large;">Demagun zenbaki lehen maximoa <b>p </b>badagoela,</span></blockquote>
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<span style="color: #0b5394; font-family: "verdana" , sans-serif; font-size: large; text-align: center;"> <b>2, 3, 5, 7, 11, 13, 17, 19, ..., p</b></span></blockquote>
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<span style="color: #0b5394; font-family: "verdana" , sans-serif; font-size: large;">zenbaki lehenen zerrenda da, <b>p</b> zenbaki lehen guztien artean handiena izanik.</span></blockquote>
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<span style="color: #0b5394; font-family: "verdana" , sans-serif; font-size: large;">Zerrendako zenbaki guztien biderkadurari unitate bat batuz, <b>N</b> zenbakia izango dugu:</span></blockquote>
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<span style="color: #0b5394; font-family: "verdana" , sans-serif; font-size: large; text-align: center;"> <b>N = 2·3·5·7·11· ... ·p + 1</b></span></blockquote>
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<span style="color: #0b5394; font-family: "verdana" , sans-serif; font-size: large;"><b>N</b> zenbakia zati 2 eginez:</span></blockquote>
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<span style="color: #0b5394;"><b><span style="font-family: "verdana" , sans-serif; font-size: large;">(</span><span style="font-family: "verdana" , sans-serif; font-size: large;">2·3·5·7·11· ... ·p + 1)/2 = </span><span style="font-family: "verdana" , sans-serif; font-size: large;">3·5·7·11· ... ·p + 1/2</span></b></span></div>
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<span style="color: #0b5394;"><span style="font-family: "verdana" , sans-serif; font-size: large;">Hau da, <b>N</b> ez da </span><span style="font-family: "verdana" , sans-serif; font-size: large; text-align: center;">2gatik zatigarria.</span></span></blockquote>
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<span style="color: #0b5394; font-family: "verdana" , sans-serif; font-size: large; text-align: center;">Era berean egiazta daiteke <b>N</b> ez dela,</span></blockquote>
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<span style="font-family: "verdana" , sans-serif; font-size: large; text-align: center;"><span style="color: #0b5394;"> <b> 3, 5, 7, 11, 13, 17, 19, ..., p</b></span></span></blockquote>
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<span style="font-family: "verdana" , sans-serif; font-size: large;"><span style="color: #0b5394;">zenbakien multiploa.</span></span></blockquote>
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<span style="font-family: "verdana" , sans-serif; font-size: large;"><span style="color: #0b5394;">Hau dela eta,<b> N</b> zenbakia lehena da edo, lehena ez bada, zerrendan ez dagoen eta zerrendako zenbakiak baino handiagoa den zenbaki lehen bategatik zatigarria da nahitaez. </span></span></blockquote>
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<span style="font-family: "verdana" , sans-serif; font-size: large; text-align: center;"><span style="color: #0b5394;">Kasu batean zein bestean, <b>p</b> baino handiagoa den zenbaki lehen bat aurkitu dugu. Hau hasieran egindako suposizioaren aurka doa. </span></span></blockquote>
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<span style="font-family: "verdana" , sans-serif; font-size: large; text-align: center;"><span style="color: #0b5394;">Ondorioz, zenbaki lehenen multzoak ez du maximorik, infinitua da.</span></span></blockquote>
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<span style="font-family: "verdana" , sans-serif; font-size: large; text-align: center;"><br /></span><span style="font-family: "verdana" , sans-serif; font-size: large; text-align: center;">Évariste Galois matematikari gazteak esaten zuenez, zenbaki lehenen tauletan ez dago ordenarik, ezta auraurik ere.</span><br />
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<span style="font-family: "verdana" , sans-serif; font-size: large; text-align: center;">Honako bideo honetan, zenbaki lehenak menderatu nahian edo, Eratostenesen bahea egiteko forma bitxia aurkezten da:</span><br />
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<span style="color: #0b5394; font-family: "verdana" , sans-serif; font-size: x-large;"><b>*****</b></span></div>
Egin Matematikahttp://www.blogger.com/profile/01348970164668368901noreply@blogger.com0