Pembuktian √2 merupakan Bilangan Irasional Beserta Bukti Geometrisnya (Materi Analisis Real) February 23, 2017 √2 , analisi real , anril , geometris , irasional , kontradiksi , Matematika , pembuktian , prinsip. dalil , pythagoras , rasional , tips <div dir="ltr" style="text-align: left;" trbidi="on"><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgMsbUYETbrty8ujW9iBnbsQlTDwX26NKlPZgT2sPgSEYCMDPal7KxFCMsJBWHnyCDlDSVAjtGHmFE5lEdWm817yK7qn8iC5UH26oLHkVfwkPPTAD6ASkUM20xi6D-i4aO48ByG2JZO2cU/s1600/pythagoras+unik.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="Pythagoras Muda" border="0" height="275" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgMsbUYETbrty8ujW9iBnbsQlTDwX26NKlPZgT2sPgSEYCMDPal7KxFCMsJBWHnyCDlDSVAjtGHmFE5lEdWm817yK7qn8iC5UH26oLHkVfwkPPTAD6ASkUM20xi6D-i4aO48ByG2JZO2cU/s400/pythagoras+unik.jpg" title="Pythagoras Muda" width="400" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Pythagoras Muda</td></tr></tbody></table><div class="MsoNormal" style="text-align: justify;"><br /></div><div class="MsoNormal" style="text-align: justify;"><br /></div><div class="MsoNormal" style="text-align: justify;">B<span lang="IN">ilangan irasional √2 telah dikenal oleh pythagoras dan para muridnya, sejak abad ke-5 </span>SM<span lang="IN">. Bukti klasik yang mengesahkan statusnya sebagai</span></div><a name='more'></a>bilangan irasional berbunyi sebagai berikut: andaikan √2 rasional, yakni terdapat bilangan asli m dan n, dengan FPB<span lang="IN">(m,n) = 1, sedemikian sehingga m/n = √2. </span><br /><div class="MsoNormal" style="text-align: justify;"><span lang="IN"><br /></span></div><div class="MsoNormal" style="text-align: justify;"><span lang="IN">Dalam hal ini, kita mempunyai m<sup>2</sup> = 2n<sup>2</sup>, yang berarti bahwa m<sub>2</sub> genap. Akibatnya, m juga genap, yakni m = 2k untuk suatu bilangan asli k, dan n mestilah ganjil</span>, <span lang="IN">karena </span>FPB<span lang="IN">(m,n) = 1. Selanjutnya, substitusikan m = 2k ke persamaan m<sup>2</sup> = 2n<sup>2</sup>, kita peroleh 4k<sup>2</sup>= 2n<sup>2</sup> atau 2k</span><sup>2</sup><span lang="IN"> = n<sup>2</sup>. Ini berarti bahwa n<sup>2</sup> genap, dan akibatnya n juga genap. Jadi n ganjil dan sekaligus genap, sesuatu yang mustahil</span> dan tidak mungkin terjadi<span lang="IN">! Karena itu kita simpulkan bahwa √2 tidak mungkin rasional.</span><span lang="IN"> </span>(<span lang="IN">bukti seperti ini dikenal sebagai bukti tidak langsung atau bukti dengan kontradiksi</span>)<span lang="IN">.</span><o:p></o:p></div><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi2ymbSli44UXbIZsfW03Luq9KkQijW5gd0bQmE0HH6_6dfoz8K6xva4E-BFXj1faqrdzpi6xo4sNaG28KUqQ3fuRL4OpcC5HlyHE_CtuRB1CMVNPBWnmnmeKM7Gg3saiYqfgw-kwL8Uhs/s1600/gedung+sesuai+prinsip+pythagoras.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="Prinsip Pythagoras Digunakan dalam Pembangunan Gedung" border="0" height="310" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi2ymbSli44UXbIZsfW03Luq9KkQijW5gd0bQmE0HH6_6dfoz8K6xva4E-BFXj1faqrdzpi6xo4sNaG28KUqQ3fuRL4OpcC5HlyHE_CtuRB1CMVNPBWnmnmeKM7Gg3saiYqfgw-kwL8Uhs/s400/gedung+sesuai+prinsip+pythagoras.jpg" title="Prinsip Pythagoras Digunakan dalam Pembangunan Gedung" width="400" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Prinsip Pythagoras Digunakan dalam Pembangunan Gedung </td></tr></tbody></table><div class="MsoNormal" style="text-align: justify;"><span lang="IN"><br /></span></div><div class="MsoNormal" style="text-align: justify;"><span lang="IN"><br /></span></div><div class="MsoNormal" style="text-align: justify;"><span lang="IN">Dalam </span>ilmu <span lang="IN">matematika, sebuah dalil kadang dapat dibuktikan dengan beberapa cara. </span>Untuk itu<span lang="IN">, selain bukti di atas, terdapat pula bukti geometris yang mengukuhkan irasionalitas √2, sebagai berikut. Andaikan, seperti tadi, terdapat bilangan asli m dan n, dengan FPB(m,n) = 1, sedemikian sehingga m/n = √2. Secara geometris, ini setara dengan eksistensi sebuah segitiga siku-siku sama kaki, ∆ABC, dengan alas dan tinggi sama dengan n dan sisi miring m. Karena FPB(m,n) = 1, segitiga ini merupakan segitiga siku-siku sama kaki <i>terkecil</i> yang ketiga panjang sisinya bilangan asli. Sekarang, tarik busur lingkaran BD dengan titik pusat A, dan tarik garis DE yang menyinggung busur lingkaran di D, seperti pada gambar. Kita peroleh segitiga siku-siku sama kaki ∆CDE, dengan CD = DE = m – n, yang merupakan bilangan asli.</span></div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjP-M6w_J2Hla2HKfq7D4osV5GBASSTyPdE_Y582ltUUgqIdBurpuEgE2YddIzTRQ8onaaZd9WniGjCJ5IVpQoBkHa3geFTBNtoB22FtqFnNFiDHiB6kQydAawJkzj5xfKBoJXalIyeLsA/s1600/geometris+akar+2.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="303" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjP-M6w_J2Hla2HKfq7D4osV5GBASSTyPdE_Y582ltUUgqIdBurpuEgE2YddIzTRQ8onaaZd9WniGjCJ5IVpQoBkHa3geFTBNtoB22FtqFnNFiDHiB6kQydAawJkzj5xfKBoJXalIyeLsA/s320/geometris+akar+2.jpg" width="320" /></a></div><div class="MsoNormal" style="text-align: justify;"><span lang="IN"><br /></span></div><div class="MsoNormal" style="text-align: justify;"><span lang="IN">Selanjutnya, bila kita tarik garis AE (tidak diperlihatkan pada gambar), kita peroleh ∆ABE sama dan sebangun dengan ∆ADE. Akibatnya, BE = DE = m – n dan CE = n – (m – n) = 2n – m, yang juga merupakan bilangan asli. Jadi ∆CDE merupakan segitiga siku-siku sama kaki yang <i>lebih kecil </i>daripada ∆ABC dan juga memiliki panjang alas, tinggi, dan sisi miring bilangan asli. Ini bertentangan dengan fakta bahwa ∆ABC merupakan segitiga terkecil yang bersifat demikian.</span><span lang="IN" style="color: #333333; font-family: "lucida sans unicode" , sans-serif;"> </span><span lang="IN">Jadi, pengandaian bahwa √2 rasional mestilah salah.</span> Setelah mencermati pembuktian diatas sobat allmipa menjadi lebih paham bahwa <span lang="IN">√2</span> adalah irasional.</div></div> Pythagoras Muda B ilangan irasional √2 telah dikenal oleh pythagoras dan para muridnya, sejak abad ke-5 SM . Bukti klasik yang mengesahkan s... Read more »
Teka-Teki Unik Mengenai Pembagian Warisan February 23, 2017 harta , hitung , Matematika , pembagian , teka-teki , tips , warisan <div dir="ltr" style="text-align: left;" trbidi="on"><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiYOFGGlh648pzbDqEEn3TL-Bs5FlpFyERIIZXP64JjKYO5h_OUSwNvhGMQzXtkvySREZuX7xD_xuG_ZwitfepvSSycz_7q7pDAl1DXvOCc7iqvXY4XJbPv7_65CtShGCim82AzYatlVI8/s1600/harta+karun.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="harat warisan" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiYOFGGlh648pzbDqEEn3TL-Bs5FlpFyERIIZXP64JjKYO5h_OUSwNvhGMQzXtkvySREZuX7xD_xuG_ZwitfepvSSycz_7q7pDAl1DXvOCc7iqvXY4XJbPv7_65CtShGCim82AzYatlVI8/s1600/harta+karun.jpg" title="harat warisan" /></a></div><div class="MsoNormal" style="text-align: justify;"><br /></div><div class="MsoNormal" style="text-align: justify;"><br /></div><div class="MsoNormal" style="text-align: justify;">Berbicara mengenai hal warisan terkadang membuat hati kita senang, namun ada pula dikala kita membicarakan warisan terasa jengkel serta penuh emosi. Perkara ini dikarenakan ketidakadilan yang menyelimuti pembagian harta warisan. Warisan bisa merubah</div><a name='more'></a> kawan menjadi lawan bahkan bisa merubah saudara menjadi seperti orang lain yang tidak mempunyai hati. Yang bisa menghindari hal-hal negatif itu semua adalah bersabar dan berserah diri kepada Sang Pencipta. Hehe, sudahlah sobat allmipa itu sedikit prolog mengenai harta gono-gini warisan. Sekarang abaikan sejenak, saya akan bermain angka mengenai masalah dalam warisan yang saya buat teka-teki matematika unik.<o:p></o:p><br /><div class="MsoNormal" style="text-align: justify;"><br /></div><div class="MsoNormal" style="text-align: justify;">Sobat allmipa barang kali<span lang="IN"> pernah mendengar cerita atau teka-teki pembagian warisan </span>seperti <span lang="IN">berikut. Seorang peternak yang baru saja </span>meninggal<span lang="IN"> mewariskan 17 ekor </span>sapi<span lang="IN"> kepada 3 anaknya. Dalam surat wasiatnya tertulis bahwa anak pertama mendapat 1/2 bagian, anak kedua mendapat 1/3 bagian, dan anak ketiga mendapat 1/9 bagian. Ketiga anak bingung bagaimana mereka harus membagi 17 ekor </span>sapi<span lang="IN"> tersebut. Akhirnya mereka bertanya kepada pamannya.</span><o:p></o:p></div><div class="MsoNormal" style="text-align: justify;"><span lang="IN"><br /></span></div><div class="MsoNormal" style="text-align: justify;"><span lang="IN">Sang paman mempunyai solusi cemerlang. Ia meminjamkan seekor </span>sapi<span lang="IN">nya, sehingga jumlah </span>sapi<span lang="IN"> menjadi 18 ekor. Setelah itu pembagian pun dilakukan: anak pertama mendapat 1/2 bagian atau 9 ekor </span>sapi<span lang="IN">, anak kedua mendapat 1/3 bagian atau 6 ekor </span>sapi<span lang="IN">, dan anak ketiga mendapat 1/9 bagian atau 2 ekor </span>sapi<span lang="IN">. Jumlah </span>sapi<span lang="IN">yang dibagikan sama dengan 17 ekor </span>sapi<span lang="IN">, sisanya 1 ekor </span>sapi<span lang="IN"> dikembalikan kepada pamannya. Kok bisa ya?</span></div><div class="MsoNormal" style="text-align: justify;"><span lang="IN"><br /></span></div><div class="MsoNormal" style="text-align: justify;"><span lang="IN">Bila kita hitung: 1/2 + 1/3 + 1/9 = (9 + 6 + 2)/18 = 17/18, ternyata lebih kecil daripada 1. Ada dua hal penting yang perlu diperhatikan dalam hitung-hitungan ini. Pertama, pemilihan bilangan 18 sebagai jumlah </span>sapi<span lang="IN"> yang mudah dibagi (karena mempunyai banyak faktor). Kedua, menguraikan 18 sebagai jumlah dari beberapa faktornya, termasuk 1, yaitu 18 = 9 + 6 + 2 + 1, yang setara dengan 1 = 1/2 + 1/3 + 1/9 + 1/18. Selanjutnya, kita dapat menyusun teka-tekinya dengan ‘menyembunyikan’ suku terakhir yang sama dengan 1/18 itu.</span></div><div class="MsoNormal" style="text-align: justify;"><span lang="IN"><br /></span></div><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhrQJ1OlG5kgHdsDfid_DYVOKiVlhDWWWpndrsTl71IXAXGZmGHutTaZrgZED_I3y_ZC1X5Vfo-AgmMs5tQpglNdVdPoPEFQShGlLP_e4nm1jAYg7u6PLIuVnpNTnuLBSl_rtB8q4sJEMs/s1600/harta+ternak.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="warisan ternak" border="0" height="276" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhrQJ1OlG5kgHdsDfid_DYVOKiVlhDWWWpndrsTl71IXAXGZmGHutTaZrgZED_I3y_ZC1X5Vfo-AgmMs5tQpglNdVdPoPEFQShGlLP_e4nm1jAYg7u6PLIuVnpNTnuLBSl_rtB8q4sJEMs/s640/harta+ternak.jpg" title="warisan ternak" width="640" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">warisan ternak</td></tr></tbody></table><div class="MsoNormal" style="text-align: justify;"><span lang="IN"><br /></span></div><div class="MsoNormal" style="text-align: justify;"><span lang="IN">Nah, dengan memperhatikan dua hal tersebut, kita dapat membuat cerita serupa dengan bilangan berbeda. Misal kita pilih bilangan 20, lalu kita menguraikannya sebagai 20 = 10 + 5 + 4 + 1, yang setara dengan 1 = 1/2 + 1/4 + 1/5 + 1/20. Selanjutnya kita bisa membuat teka-teki berikut: Seorang peternak yang baru saja </span>meninggal<span lang="IN"> mewariskan 19 ekor </span>sapi<span lang="IN"> kepada 3 anaknya. Anak pertama mendapat 1/2 bagian, anak kedua mendapat 1/4 bagian, dan anak ketiga mendapat 1/5 bagian. Bagaimanakah mereka harus membagi habis 19 ekor </span>sapi<span lang="IN"> tersebut, tanpa harus memotong </span>sapi<span lang="IN">nya?</span></div><div class="MsoNormal" style="text-align: justify;"><span lang="IN"><br /></span></div><div class="MsoNormal" style="text-align: justify;"><span lang="IN">Bahkan, jumlah anak yang mendapat warisan bisa lebih banyak. Misal kita pilih bilangan 24 = 12 + 6 + 3 + 2 + 1, yang setara dengan 1 = 1/2 + 1/4 + 1/8 + 1/12 + 1/24. Selanjutnya kita buat teka-teki berikut: Seorang peternak yang baru saja </span>meninggal<span lang="IN"> mewariskan 23 ekor </span>sapi<span lang="IN">kepada 4 anaknya. Anak pertama mendapat 1/2 bagian, anak kedua mendapat 1/4 bagian, anak ketiga mendapat 1/8 bagian, dan anak keempat mendapat 1/12 bagian. Bagaimanakah mereka harus membagi habis 23 ekor </span>sapi<span lang="IN"> tersebut, tanpa harus memotong </span>sapi<span lang="IN">nya?</span></div><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjNpJaRPiQ1H-lyFMK447ploUsox8BEogrhakwq5A5BsmDqc7g0qBMd2T73G9u7MLGf2hwwVgEZv7FoUyjb7uC6M3mwBsvaKpu7l_hrelzr2LIWpzntLmi7v608WUy4bAtisQeaAsPdEkI/s1600/warisan.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="Jangan samapi harta warisan memecah persaudarraan anda" border="0" height="240" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjNpJaRPiQ1H-lyFMK447ploUsox8BEogrhakwq5A5BsmDqc7g0qBMd2T73G9u7MLGf2hwwVgEZv7FoUyjb7uC6M3mwBsvaKpu7l_hrelzr2LIWpzntLmi7v608WUy4bAtisQeaAsPdEkI/s640/warisan.jpg" title="Jangan samapi harta warisan memecah persaudarraan anda" width="640" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Jangan samapi harta warisan memecah persaudarraan anda</td></tr></tbody></table><div class="MsoNormal" style="text-align: justify;"><span lang="IN"><br /></span></div><div class="MsoNormal" style="text-align: justify;"><span lang="IN"><br /></span></div><div class="MsoNormal" style="text-align: justify;">Bagaimana sobat allmipa? Setelah mengutak-atik teka-teki mengaenai warisan yang membuat pusing kepala anda jadi semakin terhibur kan? Hehe kalau memikirkan warisan harta yang tidak jelas hnya membuat tambah beban hidup dan beban pikiran, sebaiknya kita bikin lihai saja agar tidak depresi nantinya.<o:p></o:p></div></div> Berbicara mengenai hal warisan terkadang membuat hati kita senang, namun ada pula dikala kita membicarakan warisan terasa jengkel serta penu... Read more »
Pembuktiaan Ketidakterbatasan Bilangan February 20, 2017 angka , asli , bilangan , ketidakterbatasan. terhingga , ketidakterhinggaan , korespondensi , Matematika , pembuktian , rasional , real , terbatas <div dir="ltr" style="text-align: left;" trbidi="on"><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEguMUvSi31CuIBN4JvH3C_oAEezMi1zwxxyFD2miHC0cw6jjRh8h9rZ2KBhBm7G2Ieqt487ZGiX6kvhXCeCHWVGKpHRnVkokOxkL7uKBTmcHIEQbkfj5q79FlE422hV9eFL1_IjBOe6F10/s1600/kumpulan+angka.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="kumpulan angka" border="0" height="256" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEguMUvSi31CuIBN4JvH3C_oAEezMi1zwxxyFD2miHC0cw6jjRh8h9rZ2KBhBm7G2Ieqt487ZGiX6kvhXCeCHWVGKpHRnVkokOxkL7uKBTmcHIEQbkfj5q79FlE422hV9eFL1_IjBOe6F10/s400/kumpulan+angka.jpg" title="kumpulan angka" width="400" /></a></div><div style="margin-bottom: .0001pt; margin: 0cm; text-align: justify;"><span style="font-family: "Lucida Sans Unicode", sans-serif; font-size: 10pt;"><br /></span></div><div style="margin-bottom: .0001pt; margin: 0cm; text-align: justify;"><span style="font-family: "Lucida Sans Unicode", sans-serif; font-size: 10pt;"><br /></span></div><div style="margin-bottom: .0001pt; margin: 0cm; text-align: justify;"><span style="font-family: "Lucida Sans Unicode", sans-serif; font-size: 10pt;">Pernahkah sobat allmipa berangan maupun berimajinasi tentang Ada berapa sih banyaknya <strong><span style="font-family: "Lucida Sans Unicode","sans-serif";">bilangan asli sebenarnya</span></strong>? </span></div><a name='more'></a>Tak terhingga. Ada berapa banyak <strong><span style="font-family: "Lucida Sans Unicode","sans-serif";">bilangan rasional</span></strong>? Tak terhingga juga. Mana yang lebih banyak: bilangan asli atau bilangan rasional? Sama banyak, dalam arti terdapat <strong><span style="font-family: "Lucida Sans Unicode","sans-serif";">korespondensi satu-ke-satu</span></strong> di antara kedua himpunan bilangan tersebut. Dengan perkataan lain, bilangan rasional ‘dapat dinomori’ satu-per-satu.<o:p></o:p><br /><div style="margin-bottom: .0001pt; margin: 0cm; text-align: justify;"><span style="font-family: "Lucida Sans Unicode", sans-serif; font-size: 10pt;"><br /></span></div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgnecwFWarpKE3wpIx2DQfgcHoF-ExuVDKhNlYkE3yJgwH6uY74PgZDfDNHPGjFxCgNlBS7NlnSfNkhVLVwMCYRyclpJ5sl4sK_mFAYxJctAitV9uVMeNaHtLFFLzXmydWK8h6euKYIrQk/s1600/batasan+bilangan.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="270" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgnecwFWarpKE3wpIx2DQfgcHoF-ExuVDKhNlYkE3yJgwH6uY74PgZDfDNHPGjFxCgNlBS7NlnSfNkhVLVwMCYRyclpJ5sl4sK_mFAYxJctAitV9uVMeNaHtLFFLzXmydWK8h6euKYIrQk/s320/batasan+bilangan.jpg" width="320" /></a></div><div style="margin-bottom: .0001pt; margin: 0cm; text-align: justify;"><span style="font-family: "Lucida Sans Unicode", sans-serif; font-size: 10pt;"><br /></span></div><div style="margin-bottom: .0001pt; margin: 0cm; text-align: justify;"><span style="font-family: "Lucida Sans Unicode", sans-serif; font-size: 10pt;"><!--[if gte vml 1]><v:shapetype id="_x0000_t75" coordsize="21600,21600" o:spt="75" o:preferrelative="t" path="m@4@5l@4@11@9@11@9@5xe" filled="f" stroked="f"> <v:stroke joinstyle="miter"/> <v:formulas> <v:f eqn="if lineDrawn pixelLineWidth 0"/> <v:f eqn="sum @0 1 0"/> <v:f eqn="sum 0 0 @1"/> <v:f eqn="prod @2 1 2"/> <v:f eqn="prod @3 21600 pixelWidth"/> <v:f eqn="prod @3 21600 pixelHeight"/> <v:f eqn="sum @0 0 1"/> <v:f eqn="prod @6 1 2"/> <v:f eqn="prod @7 21600 pixelWidth"/> <v:f eqn="sum @8 21600 0"/> <v:f eqn="prod @7 21600 pixelHeight"/> <v:f eqn="sum @10 21600 0"/> </v:formulas> <v:path o:extrusionok="f" gradientshapeok="t" o:connecttype="rect"/> <o:lock v:ext="edit" aspectratio="t"/></v:shapetype><v:shape id="Picture_x0020_1" o:spid="_x0000_i1025" type="#_x0000_t75" alt="bil-rasional" style='width:205.5pt;height:174pt;visibility:visible; mso-wrap-style:square'> <v:imagedata src="file:///C:\Users\user\AppData\Local\Temp\msohtmlclip1\01\clip_image001.gif" o:title="bil-rasional"/></v:shape><![endif]--><!--[if !vml]--><!--[endif]--></span><span style="font-family: "Lucida Sans Unicode", sans-serif; font-size: 10pt;"><o:p></o:p></span></div><div style="margin-bottom: .0001pt; margin: 0cm; text-align: justify;"><span style="font-family: "Lucida Sans Unicode", sans-serif; font-size: 10pt;">Kemudian ada berapa banyak <strong><span style="font-family: "Lucida Sans Unicode","sans-serif";">bilangan real</span></strong>? Nah, ini seru! Jawabannya tak terhingga juga, namun rupanya tidak sama dengan ketakterhinggaan himpunan bilangan asli. Bilangan real jauh lebih banyak daripada bilangan asli.<o:p></o:p></span></div><div style="margin-bottom: .0001pt; margin: 0cm; text-align: justify;"><br /></div><div style="margin-bottom: .0001pt; margin: 0cm; text-align: justify;"><span style="font-family: "Lucida Sans Unicode", sans-serif; font-size: 10pt;">Dengan menggunakan notasi bilangan desimal, setiap bilangan real <em><span style="font-family: "Lucida Sans Unicode","sans-serif";">x</span></em> dapat dituliskan sebagai <em><span style="font-family: "Lucida Sans Unicode","sans-serif";">x</span></em> := <em><span style="font-family: "Lucida Sans Unicode","sans-serif";">a</span></em>,<em><span style="font-family: "Lucida Sans Unicode","sans-serif";">b<sub>1</sub>b<sub>2</sub>b</span></em><sub>3</sub><em><span style="font-family: "Lucida Sans Unicode","sans-serif";">b</span></em><sub>4</sub><em><span style="font-family: "Lucida Sans Unicode","sans-serif";">b</span></em><sub>5</sub><em><span style="font-family: "Lucida Sans Unicode","sans-serif";">b<sub>6</sub></span></em></span><span style="font-family: "Lucida Sans Unicode", sans-serif; font-size: 7.5pt;"> </span><span style="font-family: "Lucida Sans Unicode", sans-serif; font-size: 10pt;">… dengan <em><span style="font-family: "Lucida Sans Unicode","sans-serif";">a</span></em> bilangan bulat dan <em><span style="font-family: "Lucida Sans Unicode","sans-serif";">b</span></em></span><em><span style="font-family: "Lucida Sans Unicode", sans-serif; font-size: 7.5pt;">i</span></em><span style="font-family: "Lucida Sans Unicode", sans-serif; font-size: 10pt;"> ∊ {0, 1, 2, 3, … , 9}.<o:p></o:p></span></div><div style="margin-bottom: .0001pt; margin: 0cm; text-align: justify;"><span style="font-family: "Lucida Sans Unicode", sans-serif; font-size: 10pt;">Untuk membuktikan bahwa bilangan real jauh lebih banyak daripada bilangan asli, kita gunakan <strong><span style="font-family: "Lucida Sans Unicode","sans-serif";">metode kontradiksi</span></strong>. Andaikan semua bilangan real ‘dapat dinomori’. Dengan perkataan lain, kita dapat mendaftarkannya sebagai {<em><span style="font-family: "Lucida Sans Unicode","sans-serif";">x</span></em></span><span style="font-family: "Lucida Sans Unicode", sans-serif; font-size: 7.5pt;">1</span><span style="font-family: "Lucida Sans Unicode", sans-serif; font-size: 10pt;">, <em><span style="font-family: "Lucida Sans Unicode","sans-serif";">x</span></em></span><span style="font-family: "Lucida Sans Unicode", sans-serif; font-size: 7.5pt;">2</span><span style="font-family: "Lucida Sans Unicode", sans-serif; font-size: 10pt;">, <em><span style="font-family: "Lucida Sans Unicode","sans-serif";">x</span></em></span><span style="font-family: "Lucida Sans Unicode", sans-serif; font-size: 7.5pt;">3</span><span style="font-family: "Lucida Sans Unicode", sans-serif; font-size: 10pt;">, <em><span style="font-family: "Lucida Sans Unicode","sans-serif";">x</span></em></span><span style="font-family: "Lucida Sans Unicode", sans-serif; font-size: 7.5pt;">4</span><span style="font-family: "Lucida Sans Unicode", sans-serif; font-size: 10pt;">, <em><span style="font-family: "Lucida Sans Unicode","sans-serif";">x</span></em></span><span style="font-family: "Lucida Sans Unicode", sans-serif; font-size: 7.5pt;">5</span><span style="font-family: "Lucida Sans Unicode", sans-serif; font-size: 10pt;">, … } dengan:<o:p></o:p></span></div><div style="margin-bottom: .0001pt; margin: 0cm; text-align: justify;"><br /></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; text-align: center;"><em><span style="font-family: Arial, sans-serif; font-size: 10pt;">X</span></em><span style="font-family: Arial, sans-serif; font-size: 10pt;">1 := <em><span style="font-family: "Arial","sans-serif";">a</span></em><sub>1</sub>,<em><b><span style="font-family: "Arial","sans-serif";">b</span></b></em><strong><sub><span style="font-family: "Arial","sans-serif";">11</span></sub></strong><em><span style="font-family: "Arial","sans-serif";">b</span></em><sub>12</sub><em><span style="font-family: "Arial","sans-serif";">b</span></em><sub>13</sub><em><span style="font-family: "Arial","sans-serif";">b</span></em><sub>14</sub><em><span style="font-family: "Arial","sans-serif";">b</span></em><sub>15</sub><em><span style="font-family: "Arial","sans-serif";">b</span></em><sub>16</sub>…<o:p></o:p></span></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; text-align: center;"><em><span style="font-family: Arial, sans-serif; font-size: 10pt;">X</span></em><sub><span style="font-family: Arial, sans-serif; font-size: 10pt;">2</span></sub><span style="font-family: Arial, sans-serif; font-size: 10pt;"> := <em><span style="font-family: "Arial","sans-serif";">a</span></em><sub>2</sub>,<em><span style="font-family: "Arial","sans-serif";">b</span></em><sub>21</sub><em><b><span style="font-family: "Arial","sans-serif";">b</span></b></em><strong><sub><span style="font-family: "Arial","sans-serif";">22</span></sub></strong><em><span style="font-family: "Arial","sans-serif";">b</span></em><sub>23</sub><em><span style="font-family: "Arial","sans-serif";">b</span></em><sub>24</sub><em><span style="font-family: "Arial","sans-serif";">b</span></em><sub>25</sub><em><span style="font-family: "Arial","sans-serif";">b</span></em><sub>26</sub>…<o:p></o:p></span></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; text-align: center;"><em><span style="font-family: Arial, sans-serif; font-size: 10pt;">X</span></em><sub><span style="font-family: Arial, sans-serif; font-size: 10pt;">3</span></sub><span style="font-family: Arial, sans-serif; font-size: 10pt;"> := <em><span style="font-family: "Arial","sans-serif";">a</span></em><sub>3</sub>,<em><span style="font-family: "Arial","sans-serif";">b</span></em><sub>31</sub><em><span style="font-family: "Arial","sans-serif";">b</span></em><sub>32</sub><em><b><span style="font-family: "Arial","sans-serif";">b</span></b></em><strong><sub><span style="font-family: "Arial","sans-serif";">33</span></sub></strong><em><span style="font-family: "Arial","sans-serif";">b</span></em><sub>34</sub><em><span style="font-family: "Arial","sans-serif";">b</span></em><sub>35</sub><em><span style="font-family: "Arial","sans-serif";">b</span></em><sub>36</sub>…<o:p></o:p></span></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; text-align: center;"><em><span style="font-family: Arial, sans-serif; font-size: 10pt;">X</span></em><sub><span style="font-family: Arial, sans-serif; font-size: 10pt;">4</span></sub><span style="font-family: Arial, sans-serif; font-size: 10pt;"> := <em><span style="font-family: "Arial","sans-serif";">a</span></em><sub>4</sub>,<em><span style="font-family: "Arial","sans-serif";">b</span></em><sub>41</sub><em><span style="font-family: "Arial","sans-serif";">b</span></em><sub>42</sub><em><span style="font-family: "Arial","sans-serif";">b</span></em><sub>43</sub><em><b><span style="font-family: "Arial","sans-serif";">b</span></b></em><strong><sub><span style="font-family: "Arial","sans-serif";">44</span></sub></strong><em><span style="font-family: "Arial","sans-serif";">b</span></em><sub>45</sub><em><span style="font-family: "Arial","sans-serif";">b</span></em><sub>46</sub>…<o:p></o:p></span></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; text-align: center;"><em><span style="font-family: Arial, sans-serif; font-size: 10pt;">X</span></em><sub><span style="font-family: Arial, sans-serif; font-size: 10pt;">5</span></sub><span style="font-family: Arial, sans-serif; font-size: 10pt;"> := <em><span style="font-family: "Arial","sans-serif";">a</span></em><sub>5</sub>,<em><span style="font-family: "Arial","sans-serif";">b</span></em><sub>51</sub><em><span style="font-family: "Arial","sans-serif";">b</span></em><sub>52</sub><em><span style="font-family: "Arial","sans-serif";">b</span></em><sub>53</sub><em><span style="font-family: "Arial","sans-serif";">b</span></em><sub>54</sub><em><b><span style="font-family: "Arial","sans-serif";">b</span></b></em><strong><sub><span style="font-family: "Arial","sans-serif";">55</span></sub></strong><em><span style="font-family: "Arial","sans-serif";">b</span></em><sub>56</sub>…<o:p></o:p></span></div><div style="margin-bottom: .0001pt; margin: 0cm; text-align: justify;"><span style="font-family: "Lucida Sans Unicode", sans-serif; font-size: 10pt;">dan seterusnya. Nah, sekarang kita definisikan bilangan <em><span style="font-family: "Lucida Sans Unicode","sans-serif";">y</span></em> := 0,<em><span style="font-family: "Lucida Sans Unicode","sans-serif";">c</span></em></span><sub><span style="font-family: "Lucida Sans Unicode", sans-serif; font-size: 7.5pt;">1</span></sub><em><span style="font-family: "Lucida Sans Unicode", sans-serif; font-size: 10pt;">c</span></em><sub><span style="font-family: "Lucida Sans Unicode", sans-serif; font-size: 7.5pt;">2</span></sub><em><span style="font-family: "Lucida Sans Unicode", sans-serif; font-size: 10pt;">c</span></em><sub><span style="font-family: "Lucida Sans Unicode", sans-serif; font-size: 7.5pt;">3</span></sub><em><span style="font-family: "Lucida Sans Unicode", sans-serif; font-size: 10pt;">c</span></em><sub><span style="font-family: "Lucida Sans Unicode", sans-serif; font-size: 7.5pt;">4</span></sub><em><span style="font-family: "Lucida Sans Unicode", sans-serif; font-size: 10pt;">c</span></em><sub><span style="font-family: "Lucida Sans Unicode", sans-serif; font-size: 7.5pt;">5</span></sub><em><span style="font-family: "Lucida Sans Unicode", sans-serif; font-size: 10pt;">c</span></em><sub><span style="font-family: "Lucida Sans Unicode", sans-serif; font-size: 7.5pt;">6</span></sub><span style="font-family: "Lucida Sans Unicode", sans-serif; font-size: 10pt;">… dengan <em><span style="font-family: "Lucida Sans Unicode","sans-serif";">c</span></em></span><em><span style="font-family: "Lucida Sans Unicode", sans-serif; font-size: 7.5pt;">i</span></em><span style="font-family: "Lucida Sans Unicode", sans-serif; font-size: 10pt;"> := 7 bila 0 ≤ <em><b><span style="font-family: "Lucida Sans Unicode","sans-serif";">b</span></b></em></span><em><b><span style="font-family: "Lucida Sans Unicode", sans-serif; font-size: 7.5pt;">ii</span></b></em><span style="font-family: "Lucida Sans Unicode", sans-serif; font-size: 10pt;"> ≤ 4, dan <em><span style="font-family: "Lucida Sans Unicode","sans-serif";">c</span></em></span><em><span style="font-family: "Lucida Sans Unicode", sans-serif; font-size: 7.5pt;">i</span></em><span style="font-family: "Lucida Sans Unicode", sans-serif; font-size: 10pt;"> := 3 bila 5 ≤ <em><b><span style="font-family: "Lucida Sans Unicode","sans-serif";">b</span></b></em></span><em><b><span style="font-family: "Lucida Sans Unicode", sans-serif; font-size: 7.5pt;">ii</span></b></em><span style="font-family: "Lucida Sans Unicode", sans-serif; font-size: 10pt;"> ≤ 9, untuk setiap <em><span style="font-family: "Lucida Sans Unicode","sans-serif";">i</span></em> = 1, 2, 3, … , maka bilangan ini tidak terdapat dalam daftar di atas, karena <em><span style="font-family: "Lucida Sans Unicode","sans-serif";">c</span></em></span><em><span style="font-family: "Lucida Sans Unicode", sans-serif; font-size: 7.5pt;">i</span></em><span style="font-family: "Lucida Sans Unicode", sans-serif; font-size: 10pt;"> = 3 atau 7 dan pasti berbeda dengan <em><b><span style="font-family: "Lucida Sans Unicode","sans-serif";">b</span></b></em></span><em><b><span style="font-family: "Lucida Sans Unicode", sans-serif; font-size: 7.5pt;">ii</span></b></em><span style="font-family: "Lucida Sans Unicode", sans-serif; font-size: 10pt;"> untuk setiap <em><span style="font-family: "Lucida Sans Unicode","sans-serif";">i</span></em>. Padahal, <em><span style="font-family: "Lucida Sans Unicode","sans-serif";">y</span></em> merupakan bilangan real (di antara 0 dan 1). Jadi, bagaimanapun caranya kita berupaya mendaftarkan bilangan real, akan selalu ada bilangan real yang terlewatkan. Himpunan bilangan asli tidak cukup banyak untuk menomori semua bilangan real.<o:p></o:p></span></div><div style="margin-bottom: .0001pt; margin: 0cm; text-align: justify;"><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhaZa69hnJ4qD1xBQpPkSp8H6fB5N6BkXyPBtoklPT50OeHqhJmxbJoscjJdtPH1xfTMTUYSxsEp_FuLksHAIymGBDiy9yKS0coHqaQ6PFTbMJBkdnsmqSNmqqpI_KQWHhZ1f_Gs2c23A0/s1600/guru.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="metode guru" border="0" height="188" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhaZa69hnJ4qD1xBQpPkSp8H6fB5N6BkXyPBtoklPT50OeHqhJmxbJoscjJdtPH1xfTMTUYSxsEp_FuLksHAIymGBDiy9yKS0coHqaQ6PFTbMJBkdnsmqSNmqqpI_KQWHhZ1f_Gs2c23A0/s400/guru.jpg" title="metode guru" width="400" /></a></div><br /> <br /><div style="margin-bottom: .0001pt; margin: 0cm; text-align: justify;"><span style="font-family: "Lucida Sans Unicode", sans-serif; font-size: 10pt;">Metode pembuktian di atas dikenal sebagai <strong><span style="font-family: "Lucida Sans Unicode","sans-serif";">Metode Diagonalisasi Cantor</span></strong>, yang dicetuskan oleh <strong><span style="font-family: "Lucida Sans Unicode","sans-serif";">George Cantor</span></strong> (1845-1918). Metode ini dapat dipakai untuk membuktikan secara umum bahwa kardinalitas suatu himpunan selalu lebih kecil daripada kardinalitas himpunan kuasanya. Dengan begitu sudah terbukti ketidakterbatasan suatu bilangan baik dari bilangan asli, maupun bilangan real. <o:p></o:p></span></div></div> Pernahkah sobat allmipa berangan maupun berimajinasi tentang Ada berapa sih banyaknya bilangan asli sebenarnya ? Tak terhingga. Ada berapa ... Read more »
Penjabaran Deret Harmonik Berganti Tanda Beserta Rumus February 11, 2017 berganti , deret , divergen , harmonik , konvergen , Matematika , Rumus , tanda <div dir="ltr" style="text-align: left;" trbidi="on"><br /><div class="entry-content" style="background-color: white; box-sizing: border-box; font-family: "lucida grande", "lucida sans unicode", "lucida sans", geneva, verdana, sans-serif; font-size: 13px; width: 620px;"><div style="box-sizing: border-box; margin-bottom: 2.3rem;"><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhL9LdKqD4qx2D6nKteqKOtCjZpgqNsoQkUTq_7LXq4XyHTKDJSVRUojNdI7hOS-dHIUd12ERxog25SGorxLSiOW432DLF2msjOiQNvMO4PZrAvUlb9CNus2Rfx65flKSv0HniwIAynHpk/s1600/deret+harmonik.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="deret harmonik berganti tanda" border="0" height="259" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhL9LdKqD4qx2D6nKteqKOtCjZpgqNsoQkUTq_7LXq4XyHTKDJSVRUojNdI7hOS-dHIUd12ERxog25SGorxLSiOW432DLF2msjOiQNvMO4PZrAvUlb9CNus2Rfx65flKSv0HniwIAynHpk/s640/deret+harmonik.jpg" title="deret harmonik berganti tanda" width="640" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"></div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">Sobat allmipa sudah belajar <a href="http://www.allmipa.com/2017/02/mencermati-deret-harmonik-yang-membuat.html" target="_blank"><b>mengenai deret harmonik</b></a> pada materi sebelumnya. Namun kita cermati deret di bawah ini;</div><a name='more'></a><br /><img alt="deret harmonik berganti tanda" class=" wp-image-588 aligncenter" data-attachment-id="588" data-comments-opened="1" data-image-description="" data-image-meta="{"aperture":"0","credit":"","camera":"","caption":"","created_timestamp":"0","copyright":"","focal_length":"0","iso":"0","shutter_speed":"0","title":"","orientation":"0"}" data-image-title="DERET HARMONIK GANTI TANDA-0" data-large-file="https://bermatematikadotnet.files.wordpress.com/2016/05/deret-harmonik-ganti-tanda-0.png?w=431" data-medium-file="https://bermatematikadotnet.files.wordpress.com/2016/05/deret-harmonik-ganti-tanda-0.png?w=239&h=63" data-orig-file="https://bermatematikadotnet.files.wordpress.com/2016/05/deret-harmonik-ganti-tanda-0.png" data-orig-size="431,113" data-permalink="https://bermatematika.net/2016/05/27/deret-harmonik-berganti-tanda/deret-harmonik-ganti-tanda-0/" height="63" sizes="(max-width: 239px) 100vw, 239px" src="https://bermatematikadotnet.files.wordpress.com/2016/05/deret-harmonik-ganti-tanda-0.png?w=239&h=63" srcset="https://bermatematikadotnet.files.wordpress.com/2016/05/deret-harmonik-ganti-tanda-0.png?w=239&h=63 239w, https://bermatematikadotnet.files.wordpress.com/2016/05/deret-harmonik-ganti-tanda-0.png?w=150&h=39 150w, https://bermatematikadotnet.files.wordpress.com/2016/05/deret-harmonik-ganti-tanda-0.png?w=300&h=79 300w, https://bermatematikadotnet.files.wordpress.com/2016/05/deret-harmonik-ganti-tanda-0.png 431w" style="border: 0px; box-sizing: border-box; display: block; height: auto; margin: 0.5rem auto; max-width: 100%; text-align: justify; vertical-align: middle;" title="deret harmonik berganti tanda" width="239" /></div><div style="box-sizing: border-box; margin-bottom: 2.3rem;"><div style="text-align: justify;">memang deret di atas mirip dengan deret harmonik, tetapi tandanya berganti antara positif dan negatif, berselang-seling. Deret ini dikenal sebagai <strong style="box-sizing: border-box;"><i>deret harmonik berganti tanda</i></strong>.</div></div><div style="box-sizing: border-box; margin-bottom: 2.3rem;"><div style="text-align: justify;">Berbeda dengan deret harmonik yang divergen, kalau deret ini konvergen . Perhatikan bahwa:</div></div><div style="box-sizing: border-box; margin-bottom: 2.3rem;"><img alt="deret harmonik berganti tanda" class=" wp-image-567 aligncenter" data-attachment-id="567" data-comments-opened="1" data-image-description="" data-image-meta="{"aperture":"0","credit":"","camera":"","caption":"","created_timestamp":"0","copyright":"","focal_length":"0","iso":"0","shutter_speed":"0","title":"","orientation":"0"}" data-image-title="DERET HARMONIK GANTI TANDA-2" data-large-file="https://bermatematikadotnet.files.wordpress.com/2016/05/deret-harmonik-ganti-tanda-2.png?w=158&h=208?w=283" data-medium-file="https://bermatematikadotnet.files.wordpress.com/2016/05/deret-harmonik-ganti-tanda-2.png?w=158&h=208?w=229" data-orig-file="https://bermatematikadotnet.files.wordpress.com/2016/05/deret-harmonik-ganti-tanda-2.png?w=158&h=208" data-orig-size="283,371" data-permalink="https://bermatematika.net/2016/05/27/deret-harmonik-berganti-tanda/deret-harmonik-ganti-tanda-2/" height="208" sizes="(max-width: 158px) 100vw, 158px" src="https://bermatematikadotnet.files.wordpress.com/2016/05/deret-harmonik-ganti-tanda-2.png?w=158&h=208" srcset="https://bermatematikadotnet.files.wordpress.com/2016/05/deret-harmonik-ganti-tanda-2.png?w=158&h=208 158w, https://bermatematikadotnet.files.wordpress.com/2016/05/deret-harmonik-ganti-tanda-2.png?w=114&h=150 114w, https://bermatematikadotnet.files.wordpress.com/2016/05/deret-harmonik-ganti-tanda-2.png 283w" style="border: 0px; box-sizing: border-box; display: block; height: auto; margin: 0.5rem auto; max-width: 100%; text-align: justify; vertical-align: middle;" title="deret harmonik berganti tanda" width="158" /></div><div style="box-sizing: border-box; margin-bottom: 2.3rem;"><div style="text-align: justify;">dan seterusnya. Kita peroleh barisan jumlah parsial yang <em style="box-sizing: border-box;">berosilasi</em>. Selanjutnya,</div></div><div style="box-sizing: border-box; margin-bottom: 2.3rem;"><img alt="deret harmonik berganti tanda" class=" wp-image-572 aligncenter" data-attachment-id="572" data-comments-opened="1" data-image-description="" data-image-meta="{"aperture":"0","credit":"","camera":"","caption":"","created_timestamp":"0","copyright":"","focal_length":"0","iso":"0","shutter_speed":"0","title":"","orientation":"0"}" data-image-title="DERET HARMONIK GANTI TANDA-3" data-large-file="https://bermatematikadotnet.files.wordpress.com/2016/05/deret-harmonik-ganti-tanda-3.png?w=237&h=119?w=421" data-medium-file="https://bermatematikadotnet.files.wordpress.com/2016/05/deret-harmonik-ganti-tanda-3.png?w=237&h=119?w=300" data-orig-file="https://bermatematikadotnet.files.wordpress.com/2016/05/deret-harmonik-ganti-tanda-3.png?w=237&h=119" data-orig-size="421,211" data-permalink="https://bermatematika.net/2016/05/27/deret-harmonik-berganti-tanda/deret-harmonik-ganti-tanda-3/" height="119" sizes="(max-width: 237px) 100vw, 237px" src="https://bermatematikadotnet.files.wordpress.com/2016/05/deret-harmonik-ganti-tanda-3.png?w=237&h=119" srcset="https://bermatematikadotnet.files.wordpress.com/2016/05/deret-harmonik-ganti-tanda-3.png?w=237&h=119 237w, https://bermatematikadotnet.files.wordpress.com/2016/05/deret-harmonik-ganti-tanda-3.png?w=150&h=75 150w, https://bermatematikadotnet.files.wordpress.com/2016/05/deret-harmonik-ganti-tanda-3.png?w=300&h=150 300w, https://bermatematikadotnet.files.wordpress.com/2016/05/deret-harmonik-ganti-tanda-3.png 421w" style="border: 0px; box-sizing: border-box; display: block; height: auto; margin: 0.5rem auto; max-width: 100%; text-align: justify; vertical-align: middle;" title="deret harmonik berganti tanda" width="237" /></div><div style="box-sizing: border-box; margin-bottom: 2.3rem;"><div style="text-align: justify;">dan seterusnya. Dengan induksi, dapat dibuktikan bahwa</div></div><div style="box-sizing: border-box; margin-bottom: 2.3rem;"><img alt="deret harmonik berganti tanda" class=" wp-image-578 aligncenter" data-attachment-id="578" data-comments-opened="1" data-image-description="" data-image-meta="{"aperture":"0","credit":"","camera":"","caption":"","created_timestamp":"0","copyright":"","focal_length":"0","iso":"0","shutter_speed":"0","title":"","orientation":"0"}" data-image-title="DERET HARMONIK GANTI TANDA-4" data-large-file="https://bermatematikadotnet.files.wordpress.com/2016/05/deret-harmonik-ganti-tanda-4.png?w=171&h=78?w=296" data-medium-file="https://bermatematikadotnet.files.wordpress.com/2016/05/deret-harmonik-ganti-tanda-4.png?w=171&h=78?w=296" data-orig-file="https://bermatematikadotnet.files.wordpress.com/2016/05/deret-harmonik-ganti-tanda-4.png?w=171&h=78" data-orig-size="296,135" data-permalink="https://bermatematika.net/2016/05/27/deret-harmonik-berganti-tanda/deret-harmonik-ganti-tanda-4/" height="78" sizes="(max-width: 171px) 100vw, 171px" src="https://bermatematikadotnet.files.wordpress.com/2016/05/deret-harmonik-ganti-tanda-4.png?w=171&h=78" srcset="https://bermatematikadotnet.files.wordpress.com/2016/05/deret-harmonik-ganti-tanda-4.png?w=171&h=78 171w, https://bermatematikadotnet.files.wordpress.com/2016/05/deret-harmonik-ganti-tanda-4.png?w=150&h=68 150w, https://bermatematikadotnet.files.wordpress.com/2016/05/deret-harmonik-ganti-tanda-4.png 296w" style="border: 0px; box-sizing: border-box; display: block; height: auto; margin: 0.5rem auto; max-width: 100%; text-align: justify; vertical-align: middle;" title="deret harmonik berganti tanda" width="171" /></div><div style="box-sizing: border-box; margin-bottom: 2.3rem;"><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhQO7Z7pAvOfc7-yeUV4CN-2auaCekbPWnjkqWZCLZRXPyN2MExBohFMu7L4tAuD7knL9b3phLjEMwFu0YDen1Ja1XDE-7bb_jCwT9g34VLp5ub6gNr6Ps26lD9UNiphYZ3Xkobcyidw9w/s1600/bandul+jam.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="deret harmonik berganti tanda" border="0" height="400" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhQO7Z7pAvOfc7-yeUV4CN-2auaCekbPWnjkqWZCLZRXPyN2MExBohFMu7L4tAuD7knL9b3phLjEMwFu0YDen1Ja1XDE-7bb_jCwT9g34VLp5ub6gNr6Ps26lD9UNiphYZ3Xkobcyidw9w/s400/bandul+jam.jpg" title="deret harmonik berganti tanda" width="136" /></a></div><div style="text-align: justify;"><br /></div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">Lebih jauh, kedua sub-barisan ini konvergen ke suatu bilangan yang sama, karena <em style="box-sizing: border-box;">s</em><span style="bottom: -0.25em; box-sizing: border-box; font-size: 9.75px; line-height: 0; position: relative; vertical-align: baseline;">2<em style="box-sizing: border-box;">n</em>+1</span> – <em style="box-sizing: border-box;">s</em><span style="bottom: -0.25em; box-sizing: border-box; font-size: 9.75px; line-height: 0; position: relative; vertical-align: baseline;">2<em style="box-sizing: border-box;">n</em></span> = 1/(2<em style="box-sizing: border-box;">n</em>+1) menuju 0 ketika <em style="box-sizing: border-box;">n</em> menuju tak terhingga. Dengan deret pangkat untuk arctan <em style="box-sizing: border-box;">x</em> , dapat dibuktikan bahwa deret harmonik berganti tanda konvergen ke Ï€/4. Ternyata tidak saling berkaitan satu sama lain antara deret harmonik dan deret harmonik berganti tanda.</div></div></div></div> Sobat allmipa sudah belajar mengenai deret harmonik pada materi sebelumnya. Namun kita cermati deret di bawah ini; memang deret di atas mir... Read more »
Mencermati Deret Harmonik Yang Membuat pusing Kepala February 10, 2017 deret , divergen , geometri , harmonik , induksi , konvergen , Matematika , persegi , Rumus <div dir="ltr" style="text-align: left;" trbidi="on"><div style="background-color: white; box-sizing: border-box; color: #333333; font-family: "Lucida Grande", "Lucida Sans Unicode", "Lucida Sans", Geneva, Verdana, sans-serif; font-size: 13px; margin-bottom: 2.3rem;"><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiKVS4YLgLua_oCZz2m_FDALecEAb0B7u5iAACxv58OIGKxbz0yapTcyGOUQFMk0GHnWHBWzKJqH0WdITIrhxl_JWosNHDCKfWvYtU9U7sP4MGpo79bIpLsVgm1xAQ57JqzjAvWxLtGLn0/s1600/ayunan.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="penerapan deret harmonik dalam kehidupan" border="0" height="205" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiKVS4YLgLua_oCZz2m_FDALecEAb0B7u5iAACxv58OIGKxbz0yapTcyGOUQFMk0GHnWHBWzKJqH0WdITIrhxl_JWosNHDCKfWvYtU9U7sP4MGpo79bIpLsVgm1xAQ57JqzjAvWxLtGLn0/s400/ayunan.jpg" title="penerapan deret harmonik dalam kehidupan" width="400" /></a></div><br />Berbicara tentang materi deret selain deret geometri dan deret persegi, masih banyak deret yang penting dan menarik untuk dipelajari, di antaranya adalah deret seperti di bawah ini:<br /><a name='more'></a></div><div style="background-color: white; box-sizing: border-box; color: #333333; font-family: "Lucida Grande", "Lucida Sans Unicode", "Lucida Sans", Geneva, Verdana, sans-serif; font-size: 13px; margin-bottom: 2.3rem;"><img alt="contoh deret harmonik" class=" wp-image-526 aligncenter" data-attachment-id="526" data-comments-opened="1" data-image-description="" data-image-meta="{"aperture":"0","credit":"","camera":"","caption":"","created_timestamp":"0","copyright":"","focal_length":"0","iso":"0","shutter_speed":"0","title":"","orientation":"0"}" data-image-title="DERET HARMONIK-1" data-large-file="https://bermatematikadotnet.files.wordpress.com/2016/05/deret-harmonik-1.png?w=382" data-medium-file="https://bermatematikadotnet.files.wordpress.com/2016/05/deret-harmonik-1.png?w=190&h=64" data-orig-file="https://bermatematikadotnet.files.wordpress.com/2016/05/deret-harmonik-1.png" data-orig-size="382,128" data-permalink="https://bermatematika.net/2016/05/23/deret-harmonik/deret-harmonik-1/" height="64" sizes="(max-width: 190px) 100vw, 190px" src="https://bermatematikadotnet.files.wordpress.com/2016/05/deret-harmonik-1.png?w=190&h=64" srcset="https://bermatematikadotnet.files.wordpress.com/2016/05/deret-harmonik-1.png?w=190&h=64 190w, https://bermatematikadotnet.files.wordpress.com/2016/05/deret-harmonik-1.png?w=380&h=127 380w, https://bermatematikadotnet.files.wordpress.com/2016/05/deret-harmonik-1.png?w=150&h=50 150w, https://bermatematikadotnet.files.wordpress.com/2016/05/deret-harmonik-1.png?w=300&h=101 300w" style="border: 0px; box-sizing: border-box; display: block; height: auto; margin: 0.5rem auto; max-width: 100%; vertical-align: middle;" title="contoh deret harmonik" width="190" /></div><div style="background-color: white; box-sizing: border-box; color: #333333; font-family: "Lucida Grande", "Lucida Sans Unicode", "Lucida Sans", Geneva, Verdana, sans-serif; font-size: 13px; margin-bottom: 2.3rem;"><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhKR3q1WrfTzkxM_oqMtVT6V3ptklsgZ8-RmN1QfK5ing7WG0HaweZ1yUzR7_tAWWMWdxclVx8-xjUDXfBpSM2_eKyHysoPVr6hcLse7yJerpYdNsZWSUcrepiN7gh-fFsl1anES5IMaig/s1600/geraak+harmonik.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="aplikasi deret harmonik" border="0" height="245" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhKR3q1WrfTzkxM_oqMtVT6V3ptklsgZ8-RmN1QfK5ing7WG0HaweZ1yUzR7_tAWWMWdxclVx8-xjUDXfBpSM2_eKyHysoPVr6hcLse7yJerpYdNsZWSUcrepiN7gh-fFsl1anES5IMaig/s400/geraak+harmonik.jpg" title="aplikasi deret harmonik" width="400" /></a></div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">Untuk lebih dimengerti sobat allmipa, Deret ini dinamai <strong style="box-sizing: border-box;">deret harmonik</strong>, karena suku-sukunya sama dengan panjang gelombang nada-nada tinggi. Namun yang menjadi pertanyaan, Apakah deret ini konvergen? Jawabannya tidak; sekalipun suku-sukunya mengecil dan menuju 0, deret tersebut tetap <em style="box-sizing: border-box;">divergen</em>. Buktinya adalah sebagai berikut. Kita hitung jumlah parsial ke-1, ke-2, ke-4, ke-8, … dari deret tersebut, yaitu:</div></div><div style="background-color: white; box-sizing: border-box; color: #333333; font-family: "Lucida Grande", "Lucida Sans Unicode", "Lucida Sans", Geneva, Verdana, sans-serif; font-size: 13px; margin-bottom: 2.3rem;"><img alt="hitung jumlah parsial dalam deret harmonik" class=" size-medium wp-image-527 aligncenter" data-attachment-id="527" data-comments-opened="1" data-image-description="" data-image-meta="{"aperture":"0","credit":"","camera":"","caption":"","created_timestamp":"0","copyright":"","focal_length":"0","iso":"0","shutter_speed":"0","title":"","orientation":"0"}" data-image-title="DERET HARMONIK-2" data-large-file="https://bermatematikadotnet.files.wordpress.com/2016/05/deret-harmonik-2.png?w=620" data-medium-file="https://bermatematikadotnet.files.wordpress.com/2016/05/deret-harmonik-2.png?w=300&h=148" data-orig-file="https://bermatematikadotnet.files.wordpress.com/2016/05/deret-harmonik-2.png" data-orig-size="630,311" data-permalink="https://bermatematika.net/2016/05/23/deret-harmonik/deret-harmonik-2/" height="148" sizes="(max-width: 300px) 100vw, 300px" src="https://bermatematikadotnet.files.wordpress.com/2016/05/deret-harmonik-2.png?w=300&h=148" srcset="https://bermatematikadotnet.files.wordpress.com/2016/05/deret-harmonik-2.png?w=300&h=148 300w, https://bermatematikadotnet.files.wordpress.com/2016/05/deret-harmonik-2.png?w=600&h=296 600w, https://bermatematikadotnet.files.wordpress.com/2016/05/deret-harmonik-2.png?w=150&h=74 150w" style="border: 0px; box-sizing: border-box; display: block; height: auto; margin: 0.5rem auto; max-width: 100%; vertical-align: middle;" title="hitung jumlah parsial dalam deret harmonik" width="300" /></div><div style="background-color: white; box-sizing: border-box; color: #333333; font-family: "Lucida Grande", "Lucida Sans Unicode", "Lucida Sans", Geneva, Verdana, sans-serif; font-size: 13px; margin-bottom: 2.3rem;">dan seterusnya. Dengan induksi, kita peroleh;</div><div style="background-color: white; box-sizing: border-box; color: #333333; font-family: "Lucida Grande", "Lucida Sans Unicode", "Lucida Sans", Geneva, Verdana, sans-serif; font-size: 13px; margin-bottom: 2.3rem;"><img alt="dengan induksi" class="wp-image-532 aligncenter" data-attachment-id="532" data-comments-opened="1" data-image-description="" data-image-meta="{"aperture":"0","credit":"","camera":"","caption":"","created_timestamp":"0","copyright":"","focal_length":"0","iso":"0","shutter_speed":"0","title":"","orientation":"0"}" data-image-title="DERET HARMONIK -3" data-large-file="https://bermatematikadotnet.files.wordpress.com/2016/05/deret-harmonik-31.png?w=106&h=48?w=197" data-medium-file="https://bermatematikadotnet.files.wordpress.com/2016/05/deret-harmonik-31.png?w=106&h=48?w=197" data-orig-file="https://bermatematikadotnet.files.wordpress.com/2016/05/deret-harmonik-31.png?w=106&h=48" data-orig-size="197,89" data-permalink="https://bermatematika.net/2016/05/23/deret-harmonik/deret-harmonik-3-2/" height="48" sizes="(max-width: 106px) 100vw, 106px" src="https://bermatematikadotnet.files.wordpress.com/2016/05/deret-harmonik-31.png?w=106&h=48" srcset="https://bermatematikadotnet.files.wordpress.com/2016/05/deret-harmonik-31.png?w=106&h=48 106w, https://bermatematikadotnet.files.wordpress.com/2016/05/deret-harmonik-31.png?w=150&h=68 150w, https://bermatematikadotnet.files.wordpress.com/2016/05/deret-harmonik-31.png 197w" style="border: 0px; box-sizing: border-box; display: block; height: auto; margin: 0.5rem auto; max-width: 100%; vertical-align: middle;" title="dengan induksi" width="106" /></div><div style="background-color: white; box-sizing: border-box; color: #333333; font-family: "Lucida Grande", "Lucida Sans Unicode", "Lucida Sans", Geneva, Verdana, sans-serif; font-size: 13px; margin-bottom: 2.3rem;">untuk tiap <em style="box-sizing: border-box;">n</em> = 1, 2, 3, …, yang dapat melampaui bilangan besar berapapun. Karena itu, <em style="box-sizing: border-box;">s<span style="bottom: -0.25em; box-sizing: border-box; font-size: 9.75px; line-height: 0; position: relative; vertical-align: baseline;">N </span></em>menuju tak terhingga ketika <em style="box-sizing: border-box;">N</em> menuju tak terhingga. Ini membuktikan bahwa deret harmonik di atas divergen. Setelah melihat kesimpulan di atas kita menjadi lebih tahu mengenai deret harmonik yang sempat membuat pusing kepala.</div></div> Berbicara tentang materi deret selain deret geometri dan deret persegi, masih banyak deret yang penting dan menarik untuk dipelajari, di ant... Read more »
Penjelasan Bilangan Forbenius dan Contoh Penerapannya Dalam Kehidupan February 09, 2017 angka , bilangan , bulat , contoh , Ferdinand George Frobenius , frobenius , Himpunan , Matematika , penerapan , Rumus <div dir="ltr" style="text-align: left;" trbidi="on"><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhQShYA8Dpagp8Vcen_h-1fCRsenycIHds6XAadR6qkWviUHxu6tKj9DrPy4DRxaD9LrLGvjXBiL9QziNiE5JYzQVz442GfYuz5HfVLW_7wxh7qfRFaIK0HWUdVywT0BUQDIxGhZxNMW_Y/s1600/ferdinand+g.+frobenius.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="Ferdinand George Frobenius" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhQShYA8Dpagp8Vcen_h-1fCRsenycIHds6XAadR6qkWviUHxu6tKj9DrPy4DRxaD9LrLGvjXBiL9QziNiE5JYzQVz442GfYuz5HfVLW_7wxh7qfRFaIK0HWUdVywT0BUQDIxGhZxNMW_Y/s1600/ferdinand+g.+frobenius.jpg" title="Ferdinand George Frobenius" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;"><i style="font-family: Arial, sans-serif; font-size: 14.6667px; text-align: justify;">Ferdinand George Frobenius</i></td></tr></tbody></table><div style="margin-bottom: .0001pt; margin: 0cm; text-align: justify;"><span style="font-family: Arial, sans-serif; font-size: 11pt;">Bilangan Frobenius digunakan Untuk himpunan bilangan bulat positif, bilangan bulat terbesar yang tidak dapat dinyatakan sebagai </span></div><a name='more'></a>kombinasi linear (dengan koefisien bilangan bulat positif) dari unsur-unsurnya. Jumlah Frobenius dari 2 dan 5 adalah 3, karena 3 tidak dapat terbentuk sebagai jumlah yang berpasangan dan sementara setiap bilangan bulat yang lebih besar bisa. <span style="font-family: Arial, sans-serif; font-size: 11pt;">Setelah </span><i style="font-family: Arial, sans-serif; font-size: 11pt;">Ferdinand George Frobenius</i><span style="font-family: Arial, sans-serif; font-size: 11pt;">. Definisi ini kurang memiliki etimologi atau memiliki etimologi tidak lengkap..</span><br /> <div style="margin-bottom: .0001pt; margin: 0cm; text-align: justify;"><br /></div><div style="margin-bottom: .0001pt; margin: 0cm; text-align: justify;"><span style="font-family: Arial, sans-serif; font-size: 11pt;">Perlu diketahui sobat allmipa, Perangko dan Bea Pos</span><span style="font-family: Arial, sans-serif; font-size: 11pt;"> terkait erat dengan </span><strong style="box-sizing: border-box; font-family: Arial, sans-serif; font-size: 11pt;"><span style="font-family: "Arial","sans-serif";">problem Frobenius</span></strong><span style="font-family: Arial, sans-serif; font-size: 11pt;">, yaitu menentukan </span><em style="box-sizing: border-box; font-family: Arial, sans-serif; font-size: 11pt;"><span style="font-family: "Arial","sans-serif";">nilai terbesar</span></em><span style="font-family: Arial, sans-serif; font-size: 11pt;"> yang tidak dapat dinyatakan sebagai jumlah dari bilangan-bilangan bulat positif tertentu. Sebagai contoh, bila sebuah restoran cepat saji menjual sayap ayam goreng hanya dalam kemasan berisi 5 potong, 8 potong, dan 17 potong, maka kita dapat membeli 10 potong atau 13 potong sayap ayam, tetapi tidak dapat membeli 12 potong sayap ayam. Nah, berapa potong sayap ayam terbanyak yang tak dapat dibeli di restoran cepat saji tersebut? Jawabannya adalah 19 potong, ya </span><em style="box-sizing: border-box; font-family: Arial, sans-serif; font-size: 11pt;"><span style="font-family: "Arial","sans-serif";">kan</span></em><span style="font-family: Arial, sans-serif; font-size: 11pt;">? Silahkan simak pembahasannya di bawah ini:.</span></div><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhTtgEk7d54G7UyiQSpubRgRNIUfuI6UjjZIlsM0y4pfwNMxIRx40_qlNvJ-jE2x4Gj-hFx4Wu8f8LeOGowjldOnbbfCL-uIoiytAKUHF0tWlIKD8EtmDpulJK0d0_N36yBIplakcXqL1c/s1600/perangko.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="sampel perangko" border="0" height="223" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhTtgEk7d54G7UyiQSpubRgRNIUfuI6UjjZIlsM0y4pfwNMxIRx40_qlNvJ-jE2x4Gj-hFx4Wu8f8LeOGowjldOnbbfCL-uIoiytAKUHF0tWlIKD8EtmDpulJK0d0_N36yBIplakcXqL1c/s400/perangko.jpg" title="sampel perangko" width="400" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">sampel perangko</td></tr></tbody></table><div style="margin-bottom: .0001pt; margin: 0cm; text-align: justify;"><br /></div><div style="margin-bottom: .0001pt; margin: 0cm; text-align: justify;"><span style="font-family: Arial, sans-serif; font-size: 11pt;">Bilangan 19 dalam cerita di atas dikenal sebagai <strong style="box-sizing: border-box;"><span style="font-family: "Arial","sans-serif";">bilangan Frobenius</span></strong> yang terkait dengan himpunan bilangan 5, 8, dan 17. Secara umum, diberikan sejumlah bilangan bulat positif, <em style="box-sizing: border-box;"><span style="font-family: "Arial","sans-serif";">n</span></em><sub>1</sub>, <em style="box-sizing: border-box;"><span style="font-family: "Arial","sans-serif";">n</span></em><sub>2</sub>, …, <em style="box-sizing: border-box;"><span style="font-family: "Arial","sans-serif";">n<sub>k</sub></span></em>, bilangan Frobenius yang terkait dengan himpunan bilangan ini adalah bilangan bulat terbesar yang tidak dapat dinyatakan sebagai <em style="box-sizing: border-box;"><span style="font-family: "Arial","sans-serif";">a<sub>1</sub>n</span></em><sub>1</sub> + <em style="box-sizing: border-box;"><span style="font-family: "Arial","sans-serif";">a</span></em><sub>2</sub><em><span style="font-family: "Arial","sans-serif";">n</span></em><sub>2</sub> + … + <em style="box-sizing: border-box;"><span style="font-family: "Arial","sans-serif";">a<sub>k</sub>n<sub>k</sub></span></em> dengan <em style="box-sizing: border-box;"><span style="font-family: "Arial","sans-serif";">a</span></em><sub>1</sub>, <em style="box-sizing: border-box;"><span style="font-family: "Arial","sans-serif";">a</span></em><sub>2</sub>, … , <em style="box-sizing: border-box;"><span style="font-family: "Arial","sans-serif";">a<sub>k </sub></span></em>bilangan bulat tak negatif (sebagian di antaranya boleh sama dengan 0). <o:p></o:p></span></div><div style="margin-bottom: .0001pt; margin: 0cm; text-align: justify;"><br /></div><div style="margin-bottom: .0001pt; margin: 0cm; text-align: justify;"><span style="font-family: Arial, sans-serif; font-size: 11pt;">Anda dapat mencoba mencari bilangan Frobenius yang terkait dengan dua bilangan, misalnya 8 dan 17. Semakin banyak bilangan yang terlibat, semakin sulit problemnya, dan tidak ada rumus eksplisit untuk solusinya. Namun diambil sisi positifnya, dengan kita belajar matematika pikiran kita akan terasah terus menerus.<br /><o:p></o:p></span></div></div> Ferdinand George Frobenius Bilangan Frobenius digunakan Untuk himpunan bilangan bulat positif, bilangan bulat terbesar yang tidak dapat diny... Read more »
Pengertian dan Rumus Partisi Bilangan Asli February 08, 2017 angka , bilangan , cara , Definisi , Matematika , partisi , Pengertian , Rumus <div dir="ltr" style="text-align: left;" trbidi="on"><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhhp0T0xLtWDLrKbDbcPw2qQjk7WR6bB6SciIUBF8C5c50YTwfK6wPEws6xoTqqGr9hlXn11-3vdp6w7FcBtBnNxDERAIW3egD_ZcjqTSQPoTVxyrWb7tvwp_IsIPVEimIFeUdgmfp0B3s/s1600/partisi.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="partisi bilangan 5" border="0" height="182" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhhp0T0xLtWDLrKbDbcPw2qQjk7WR6bB6SciIUBF8C5c50YTwfK6wPEws6xoTqqGr9hlXn11-3vdp6w7FcBtBnNxDERAIW3egD_ZcjqTSQPoTVxyrWb7tvwp_IsIPVEimIFeUdgmfp0B3s/s320/partisi.jpg" title="partisi bilangan 5" width="320" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">partisi bilangan 5</td></tr></tbody></table><div style="margin-bottom: .0001pt; margin: 0cm; text-align: justify;"><span style="font-family: "Lucida Sans Unicode", sans-serif; font-size: 10pt;">Sebelum kita membahas lebih jauh mengenai partisi bilangan, kita harus tahu apa yang dimaksud dengan partisi bilangan? Yang dimaksud dengan</span></div><a name='more'></a> partisi bilangan asli <em style="box-sizing: border-box; font-family: "Lucida Sans Unicode", sans-serif; font-size: 10pt;"><span style="font-family: "Lucida Sans Unicode","sans-serif";">n</span></em><span style="font-family: "Lucida Sans Unicode", sans-serif; font-size: 10pt;"> adalah cara menguraikan </span><em style="box-sizing: border-box; font-family: "Lucida Sans Unicode", sans-serif; font-size: 10pt;"><span style="font-family: "Lucida Sans Unicode","sans-serif";">n</span></em><span style="font-family: "Lucida Sans Unicode", sans-serif; font-size: 10pt;"> sebagai jumlah dari beberapa bilangan asli (termasuk </span><em style="box-sizing: border-box; font-family: "Lucida Sans Unicode", sans-serif; font-size: 10pt;"><span style="font-family: "Lucida Sans Unicode","sans-serif";">n</span></em><span style="font-family: "Lucida Sans Unicode", sans-serif; font-size: 10pt;"> sendiri). Sebagai contoh, bilangan 5 dapat diuraikan dalam 7 cara, yaitu:</span><br /> <div align="center" style="margin-bottom: .0001pt; margin: 0cm; text-align: center;"><span style="font-family: "Lucida Sans Unicode", sans-serif; font-size: 10pt;">5,<o:p></o:p></span></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; text-align: center;"><span style="font-family: "Lucida Sans Unicode", sans-serif; font-size: 10pt;">4 + 1,<o:p></o:p></span></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; text-align: center;"><span style="font-family: "Lucida Sans Unicode", sans-serif; font-size: 10pt;">3 + 2,<o:p></o:p></span></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; text-align: center;"><span style="font-family: "Lucida Sans Unicode", sans-serif; font-size: 10pt;">3 + 1 + 1,<o:p></o:p></span></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; text-align: center;"><span style="font-family: "Lucida Sans Unicode", sans-serif; font-size: 10pt;">2 + 2 + 1,<o:p></o:p></span></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; text-align: center;"><span style="font-family: "Lucida Sans Unicode", sans-serif; font-size: 10pt;">2 + 1 + 1 + 1,<o:p></o:p></span></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; text-align: center;"><span style="font-family: "Lucida Sans Unicode", sans-serif; font-size: 10pt;">dan 1 + 1 + 1 + 1 + 1.<o:p></o:p></span></div><div style="margin-bottom: .0001pt; margin: 0cm; text-align: justify;"><br /></div><div style="margin-bottom: .0001pt; margin: 0cm; text-align: justify;"><span style="font-family: "Lucida Sans Unicode", sans-serif; font-size: 10pt;">Kemudian bila <em style="box-sizing: border-box;"><span style="font-family: "Lucida Sans Unicode","sans-serif";">p</span></em>(<em style="box-sizing: border-box;"><span style="font-family: "Lucida Sans Unicode","sans-serif";">n</span></em>) menyatakan banyaknya partisi bilangan <em style="box-sizing: border-box;"><span style="font-family: "Lucida Sans Unicode","sans-serif";">n</span></em>, maka <em style="box-sizing: border-box;"><span style="font-family: "Lucida Sans Unicode","sans-serif";">p</span></em>(5) = 7. (Di sini, 3 + 2 dan 2 + 3 dianggap sebagai satu cara yang sama).<o:p></o:p></span></div><div style="margin-bottom: .0001pt; margin: 0cm; text-align: justify;"><span style="font-family: "Lucida Sans Unicode", sans-serif; font-size: 10pt;">Sobat allmipa mungkin berpikiran, apa yang sulit dengan partisi bilangan asli ini? <em style="box-sizing: border-box;"><span style="font-family: "Lucida Sans Unicode","sans-serif"; font-style: normal; mso-bidi-font-style: italic;">Namun</span></em><i>..</i> coba hitung berapa <em style="box-sizing: border-box;"><span style="font-family: "Lucida Sans Unicode","sans-serif";">p</span></em>(10), <em style="box-sizing: border-box;"><span style="font-family: "Lucida Sans Unicode","sans-serif";">p</span></em>(25), dan … <em style="box-sizing: border-box;"><span style="font-family: "Lucida Sans Unicode","sans-serif";">p</span></em>(200), seperti yang dikerjakan oleh Ramanujan pada waktu itu. <o:p></o:p></span></div><div style="margin-bottom: .0001pt; margin: 0cm; text-align: justify;"><span style="font-family: "Lucida Sans Unicode", sans-serif; font-size: 10pt;">Ada rumus rekursif untuk menghitung <em style="box-sizing: border-box;"><span style="font-family: "Lucida Sans Unicode","sans-serif";">p</span></em>(<em style="box-sizing: border-box;"><span style="font-family: "Lucida Sans Unicode","sans-serif";">n</span></em>). Jika <em style="box-sizing: border-box;"><span style="font-family: "Lucida Sans Unicode","sans-serif";">p</span></em>(<em style="box-sizing: border-box;"><span style="font-family: "Lucida Sans Unicode","sans-serif";">n</span></em>,<em style="box-sizing: border-box;"><span style="font-family: "Lucida Sans Unicode","sans-serif";">m</span></em>) menyatakan banyaknya partisi bilangan <em style="box-sizing: border-box;"><span style="font-family: "Lucida Sans Unicode","sans-serif";">n</span></em> dengan hanya menggunakan bilangan yang lebih kecil daripada atau sama dengan <em style="box-sizing: border-box;"><span style="font-family: "Lucida Sans Unicode","sans-serif";">m, </span></em>maka:<o:p></o:p></span></div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiQ9Y9pScEr51qye5B6qerdINOWegNP7_lO5IQPg8o0Me6SsaOhHzKw6mWESrGALgNQidVXdVlFFwABDTQ_3-fdLKraPIOiNM6woNTp8B6gY6f0C2dY3t6y5Ug0MYjr6jhkLh6WTmgDmyE/s1600/partisi+bilangan+1.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="92" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiQ9Y9pScEr51qye5B6qerdINOWegNP7_lO5IQPg8o0Me6SsaOhHzKw6mWESrGALgNQidVXdVlFFwABDTQ_3-fdLKraPIOiNM6woNTp8B6gY6f0C2dY3t6y5Ug0MYjr6jhkLh6WTmgDmyE/s320/partisi+bilangan+1.jpg" width="320" /></a></div><div style="margin-bottom: .0001pt; margin: 0cm; text-align: justify;"><span style="font-family: "Lucida Sans Unicode", sans-serif; font-size: 10pt;"><br /></span></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; text-align: center;"><span style="font-family: "Lucida Sans Unicode", sans-serif; font-size: 10pt;"><!--[if gte vml 1]><v:shapetype id="_x0000_t75" coordsize="21600,21600" o:spt="75" o:preferrelative="t" path="m@4@5l@4@11@9@11@9@5xe" filled="f" stroked="f"> <v:stroke joinstyle="miter"/> <v:formulas> <v:f eqn="if lineDrawn pixelLineWidth 0"/> <v:f eqn="sum @0 1 0"/> <v:f eqn="sum 0 0 @1"/> <v:f eqn="prod @2 1 2"/> <v:f eqn="prod @3 21600 pixelWidth"/> <v:f eqn="prod @3 21600 pixelHeight"/> <v:f eqn="sum @0 0 1"/> <v:f eqn="prod @6 1 2"/> <v:f eqn="prod @7 21600 pixelWidth"/> <v:f eqn="sum @8 21600 0"/> <v:f eqn="prod @7 21600 pixelHeight"/> <v:f eqn="sum @10 21600 0"/> </v:formulas> <v:path o:extrusionok="f" gradientshapeok="t" o:connecttype="rect"/> <o:lock v:ext="edit" aspectratio="t"/></v:shapetype><v:shape id="Picture_x0020_1" o:spid="_x0000_i1027" type="#_x0000_t75" alt="partition-algorithm" style='width:147pt;height:40.5pt;visibility:visible; mso-wrap-style:square'> <v:imagedata src="file:///C:\Users\user\AppData\Local\Temp\msohtmlclip1\01\clip_image001.gif" o:title="partition-algorithm"/></v:shape><![endif]--><!--[if !vml]--><!--[endif]--></span><span style="font-family: "Lucida Sans Unicode", sans-serif; font-size: 10pt;"><o:p></o:p></span></div><div style="margin-bottom: .0001pt; margin: 0cm; text-align: justify;"><span style="font-family: "Lucida Sans Unicode", sans-serif; font-size: 10pt;">dengan <em style="box-sizing: border-box;"><span style="font-family: "Lucida Sans Unicode","sans-serif";">p</span></em>(<em style="box-sizing: border-box;"><span style="font-family: "Lucida Sans Unicode","sans-serif";">n</span></em>,<em style="box-sizing: border-box;"><span style="font-family: "Lucida Sans Unicode","sans-serif";">m</span></em>) = <em style="box-sizing: border-box;"><span style="font-family: "Lucida Sans Unicode","sans-serif";">p</span></em>(<em style="box-sizing: border-box;"><span style="font-family: "Lucida Sans Unicode","sans-serif";">n</span></em>,<em style="box-sizing: border-box;"><span style="font-family: "Lucida Sans Unicode","sans-serif";">n</span></em>) = <em style="box-sizing: border-box;"><span style="font-family: "Lucida Sans Unicode","sans-serif";">p</span></em>(<em style="box-sizing: border-box;"><span style="font-family: "Lucida Sans Unicode","sans-serif";">n</span></em>) untuk <em style="box-sizing: border-box;"><span style="font-family: "Lucida Sans Unicode","sans-serif";">m</span></em> > <em style="box-sizing: border-box;"><span style="font-family: "Lucida Sans Unicode","sans-serif";">n</span></em> (dan, untuk melengkapi rumus, kita definisikan <em style="box-sizing: border-box;"><span style="font-family: "Lucida Sans Unicode","sans-serif";">p</span></em>(0,<em style="box-sizing: border-box;"><span style="font-family: "Lucida Sans Unicode","sans-serif";">n</span></em>) = <em style="box-sizing: border-box;"><span style="font-family: "Lucida Sans Unicode","sans-serif";">p</span></em>(0) = 1). Sebagai contoh:<o:p></o:p></span></div><div style="margin-bottom: .0001pt; margin: 0cm; text-align: justify;"><em style="box-sizing: border-box;"><span style="font-family: "Lucida Sans Unicode", sans-serif; font-size: 10pt;">p</span></em><span style="font-family: "Lucida Sans Unicode", sans-serif; font-size: 10pt;">(5,4) = <em style="box-sizing: border-box;"><span style="font-family: "Lucida Sans Unicode","sans-serif";">p</span></em>(4,1) + <em style="box-sizing: border-box;"><span style="font-family: "Lucida Sans Unicode","sans-serif";">p</span></em>(3,2) + <em style="box-sizing: border-box;"><span style="font-family: "Lucida Sans Unicode","sans-serif";">p</span></em>(2,3) + <em style="box-sizing: border-box;"><span style="font-family: "Lucida Sans Unicode","sans-serif";">p</span></em>(1,4) = 1 + 2 + 2 + 1 = 6.<o:p></o:p></span></div><div style="margin-bottom: .0001pt; margin: 0cm; text-align: justify;"><span style="font-family: "Lucida Sans Unicode", sans-serif; font-size: 10pt;">Untuk <em style="box-sizing: border-box;"><span style="font-family: "Lucida Sans Unicode","sans-serif";">n</span></em> = 0, 1, 2, …, 7 dan <em style="box-sizing: border-box;"><span style="font-family: "Lucida Sans Unicode","sans-serif";">m</span></em> = 1, 2, 3, …, 7, nilai <em style="box-sizing: border-box;"><span style="font-family: "Lucida Sans Unicode","sans-serif";">p</span></em>(<em style="box-sizing: border-box;"><span style="font-family: "Lucida Sans Unicode","sans-serif";">n</span></em>,<em style="box-sizing: border-box;"><span style="font-family: "Lucida Sans Unicode","sans-serif";">m</span></em>) diberikan dalam tabel di bawah ini:<o:p></o:p></span></div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiyNC8gt9X9mIt7j0L_IlcjZGIMe4h6shvimQ5jeG5KnLaKjiSfRAnK4qbL2HYQwyfNZsGlypzpbIDq-BhYj3WkKSMIXevp7039AaKUtD-HIJsgUY2VP_NEY_QB1rNE5XAAgW8xcz4XwF0/s1600/partisi+bilangan+2.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="233" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiyNC8gt9X9mIt7j0L_IlcjZGIMe4h6shvimQ5jeG5KnLaKjiSfRAnK4qbL2HYQwyfNZsGlypzpbIDq-BhYj3WkKSMIXevp7039AaKUtD-HIJsgUY2VP_NEY_QB1rNE5XAAgW8xcz4XwF0/s320/partisi+bilangan+2.jpg" width="320" /></a></div><div style="margin-bottom: .0001pt; margin: 0cm; text-align: justify;"><span style="font-family: "Lucida Sans Unicode", sans-serif; font-size: 10pt;"><br /></span></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; text-align: center;"><span style="font-family: "Lucida Sans Unicode", sans-serif; font-size: 10pt;"><!--[if gte vml 1]><v:shape id="Picture_x0020_2" o:spid="_x0000_i1026" type="#_x0000_t75" alt="partition-table" style='width:225pt;height:163.5pt;visibility:visible;mso-wrap-style:square'> <v:imagedata src="file:///C:\Users\user\AppData\Local\Temp\msohtmlclip1\01\clip_image002.gif" o:title="partition-table"/></v:shape><![endif]--><!--[if !vml]--><!--[endif]--></span><span style="font-family: "Lucida Sans Unicode", sans-serif; font-size: 10pt;"><o:p></o:p></span></div><div style="margin-bottom: .0001pt; margin: 0cm; text-align: justify;"><i><span style="font-family: "Lucida Sans Unicode", sans-serif; font-size: 10pt;">Tabel Nilai <em style="box-sizing: border-box;"><span style="font-family: "Lucida Sans Unicode","sans-serif";">p</span></em>(<em style="box-sizing: border-box;"><span style="font-family: "Lucida Sans Unicode","sans-serif";">n</span></em>,<em style="box-sizing: border-box;"><span style="font-family: "Lucida Sans Unicode","sans-serif";">m</span></em>) untuk <em style="box-sizing: border-box;"><span style="font-family: "Lucida Sans Unicode","sans-serif";">n</span></em>, <em style="box-sizing: border-box;"><span style="font-family: "Lucida Sans Unicode","sans-serif";">m</span></em> = 1,…,7<o:p></o:p></span></i></div><div style="margin-bottom: .0001pt; margin: 0cm; text-align: justify;"><span style="font-family: "Lucida Sans Unicode", sans-serif; font-size: 10pt;">Tabel ini dapat Anda lanjutkan untuk <em style="box-sizing: border-box;"><span style="font-family: "Lucida Sans Unicode","sans-serif";">n</span></em> dan <em style="box-sizing: border-box;"><span style="font-family: "Lucida Sans Unicode","sans-serif";">m</span></em> lainnya.<o:p></o:p></span></div><div style="margin-bottom: .0001pt; margin: 0cm; text-align: justify;"><span style="font-family: "Lucida Sans Unicode", sans-serif; font-size: 10pt;">Ramanujan dan Hardy menemukan bahwa <em style="box-sizing: border-box;"><span style="font-family: "Lucida Sans Unicode","sans-serif";">p</span></em>(<em style="box-sizing: border-box;"><span style="font-family: "Lucida Sans Unicode","sans-serif";">n</span></em>) membesar hampir secara eksponensial seiring dengan membesarnya <em style="box-sizing: border-box;"><span style="font-family: "Lucida Sans Unicode","sans-serif";">n</span></em>. Persisnya, Ramanujan dan Hardy memperoleh taksiran:<o:p></o:p></span></div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgKIissYMZcuN4yrO-Aobb6G1SLETg2dMm2PURTTBoGUWQaFPXJnD5rpjM0_D2DnaJOG2lqU_qwBefy73ggzTG6EsYzogqWrJFyySklKj9Bz1K-laiJ22DkQWPwSNv_AEF4nLbp0g4D1xY/s1600/partisi+bilangan+3.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="64" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgKIissYMZcuN4yrO-Aobb6G1SLETg2dMm2PURTTBoGUWQaFPXJnD5rpjM0_D2DnaJOG2lqU_qwBefy73ggzTG6EsYzogqWrJFyySklKj9Bz1K-laiJ22DkQWPwSNv_AEF4nLbp0g4D1xY/s320/partisi+bilangan+3.jpg" width="320" /></a></div><div style="margin-bottom: .0001pt; margin: 0cm; text-align: justify;"><span style="font-family: "Lucida Sans Unicode", sans-serif; font-size: 10pt;"><br /></span></div><div align="center" style="margin-bottom: .0001pt; margin: 0cm; text-align: center;"><span style="font-family: "Lucida Sans Unicode", sans-serif; font-size: 10pt;"><!--[if gte vml 1]><v:shape id="Picture_x0020_3" o:spid="_x0000_i1025" type="#_x0000_t75" alt="partition-estimate" style='width:184.5pt;height:34.5pt;visibility:visible;mso-wrap-style:square'> <v:imagedata src="file:///C:\Users\user\AppData\Local\Temp\msohtmlclip1\01\clip_image003.gif" o:title="partition-estimate"/></v:shape><![endif]--><!--[if !vml]--><!--[endif]--></span><span style="font-family: "Lucida Sans Unicode", sans-serif; font-size: 10pt;"><o:p></o:p></span></div><div style="margin-bottom: .0001pt; margin: 0cm; text-align: justify;"><span style="font-family: "Lucida Sans Unicode", sans-serif; font-size: 10pt;">Dengan rumus ini, Anda dapat menaksir bahwa nilai <em style="box-sizing: border-box;"><span style="font-family: "Lucida Sans Unicode","sans-serif";">p</span></em>(200) kira-kira sama dengan 4 trilyun. (Nilai <em style="box-sizing: border-box;"><span style="font-family: "Lucida Sans Unicode","sans-serif";">p</span></em>(200) sebenarnya sama dengan 3.972.999.029.388.) </span></div><div style="margin-bottom: .0001pt; margin: 0cm; text-align: justify;"><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgB4d7cEsE2Z2EXsXripObdNUO5KrTzey-0LIqJcOreFsLqGKpGOX9RW9AVWuhDlfARUa_emiJxcGOFR9NOdNWpu1Pl7OmQCFYRntEx-dfITSVxbZ7Huz9ccy1lU6BI7akxr_UNYPluFRk/s1600/belajar+mat.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="psikomotor matematika" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgB4d7cEsE2Z2EXsXripObdNUO5KrTzey-0LIqJcOreFsLqGKpGOX9RW9AVWuhDlfARUa_emiJxcGOFR9NOdNWpu1Pl7OmQCFYRntEx-dfITSVxbZ7Huz9ccy1lU6BI7akxr_UNYPluFRk/s1600/belajar+mat.jpg" title="psikomotor matematika" /></a></div><div style="margin-bottom: .0001pt; margin: 0cm; text-align: justify;">Bagaimana sobat allmipa? Ternyata partisi bilangan asli tidak semudah yang kita kira bukan? Kalau masih angka satuan atau puluhan begitu masih terlihat mudah namun apabila sudah menginjak ratusan atau ribuan pasti membuat kita berpikir ekstra alias pusing sejuta keliling. Hehe <span style="font-family: Wingdings;">J</span></div><div class="MsoNormal" style="text-align: justify;"><o:p></o:p></div></div> partisi bilangan 5 Sebelum kita membahas lebih jauh mengenai partisi bilangan, kita harus tahu apa yang dimaksud dengan partisi bilangan? Ya... Read more »
Asal - Usul Rumus Hasil Kali Perkalian Skalar February 06, 2017 asal-usul , hasil , kali , Matematika , penerapan , proyeksi , Rumus , skalar , Vektor <div dir="ltr" style="text-align: left;" trbidi="on"><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi5H-uFxvUKHwttMy1aItzymXiTyrBcEeXuw0Nvtmp2AwE7eHlKbeuAMw4YULry86YIMpd2j3C8gC6pMuc4XgdlfP4_7EZLh2zdjf2RVc2uepprPgoM7uVC349bVnW5ta0gl0UeJITnPRM/s1600/tarik+tambang.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="contoh perkalian skalar dalam kehidupan" border="0" height="316" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi5H-uFxvUKHwttMy1aItzymXiTyrBcEeXuw0Nvtmp2AwE7eHlKbeuAMw4YULry86YIMpd2j3C8gC6pMuc4XgdlfP4_7EZLh2zdjf2RVc2uepprPgoM7uVC349bVnW5ta0gl0UeJITnPRM/s640/tarik+tambang.jpg" title="contoh perkalian skalar dalam kehidupan" width="640" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">contoh perkalian skalar dalam kehidupan</td></tr></tbody></table><div style="margin-bottom: .0001pt; margin: 0cm; text-align: justify;"><span style="font-family: Arial, sans-serif;">Sering terjadi dalam buku-buku yang membahas ruang vektor, biasanya ada definisi </span><strong style="box-sizing: border-box; font-family: Arial, sans-serif;"><span style="font-family: "Arial","sans-serif";">hasil kali titik</span></strong><span style="font-family: Arial, sans-serif;"> (</span><em style="box-sizing: border-box; font-family: Arial, sans-serif;"><span style="font-family: "Arial","sans-serif";">dot product</span></em><span style="font-family: Arial, sans-serif;">) atau </span><strong style="box-sizing: border-box; font-family: Arial, sans-serif;"><span style="font-family: "Arial","sans-serif";">hasil kali skalar</span></strong><span style="font-family: Arial, sans-serif;"> (</span><em style="box-sizing: border-box; font-family: Arial, sans-serif;"><span style="font-family: "Arial","sans-serif";">scalar product</span></em><span style="font-family: Arial, sans-serif;">). Di </span><strong style="box-sizing: border-box; font-family: Arial, sans-serif;"><span style="font-family: "Arial","sans-serif";">R</span></strong><em style="box-sizing: border-box; font-family: Arial, sans-serif;"><sub><span style="font-family: "Arial","sans-serif";">n</span></sub></em><span style="font-family: Arial, sans-serif;">, hasil kali skalar antara dua vektor </span><em style="box-sizing: border-box; font-family: Arial, sans-serif;"><span style="font-family: "Arial","sans-serif";">x</span></em><span style="font-family: Arial, sans-serif;"> = (</span><em style="box-sizing: border-box; font-family: Arial, sans-serif;"><span style="font-family: "Arial","sans-serif";">x</span></em><sub style="font-family: Arial, sans-serif;">1</sub><span style="font-family: Arial, sans-serif;">, …, </span><em style="box-sizing: border-box; font-family: Arial, sans-serif;"><span style="font-family: "Arial","sans-serif";">x<sub>n</sub></span></em><span style="font-family: Arial, sans-serif;">) dan </span><em style="box-sizing: border-box; font-family: Arial, sans-serif;"><span style="font-family: "Arial","sans-serif";">y</span></em><span style="font-family: Arial, sans-serif;"> = (</span><em style="box-sizing: border-box; font-family: Arial, sans-serif;"><span style="font-family: "Arial","sans-serif";">y</span></em><sub style="font-family: Arial, sans-serif;">1</sub><span style="font-family: Arial, sans-serif;">, … , </span><em style="box-sizing: border-box; font-family: Arial, sans-serif;"><span style="font-family: "Arial","sans-serif";">y<sub>n</sub></span></em><span style="font-family: Arial, sans-serif;">) didefinisikan dengan rumus</span></div><a name='more'></a><br /> <div style="margin-bottom: .0001pt; margin: 0cm; text-align: justify;"><br /></div><div style="margin: 0cm 0cm 0.0001pt; text-align: center;"><em style="box-sizing: border-box;"><span style="font-family: Arial, sans-serif;"><b>x</b></span></em><span style="font-family: Arial, sans-serif;"><b> ∙ <em style="box-sizing: border-box;"><span style="font-family: "Arial","sans-serif";">y</span></em> = <em style="box-sizing: border-box;"><span style="font-family: "Arial","sans-serif";">xy</span></em><sub>1</sub> + … + <em style="box-sizing: border-box;"><span style="font-family: "Arial","sans-serif";">x<sub>n</sub> y<sub>n</sub></span></em></b><o:p></o:p></span></div><div style="margin-bottom: .0001pt; margin: 0cm; text-align: justify;"><br /></div><div style="margin-bottom: .0001pt; margin: 0cm; text-align: justify;"><span style="font-family: Arial, sans-serif;">Tetapi sering kali si pengarang tidak memberikan penjelasan dari mana rumus tersebut diperoleh atau mengapa rumusnya harus seperti itu. Untuk itu www.allmipa.com akan menjelaskan mengapa hasil kali titik di <strong style="box-sizing: border-box;"><span style="font-family: "Arial","sans-serif";">R</span></strong><em style="box-sizing: border-box;"><sub><span style="font-family: "Arial","sans-serif";">n</span></sub></em> mempunyai rumus seperti itu.<o:p></o:p></span></div><div style="margin-bottom: .0001pt; margin: 0cm; text-align: justify;"><br /></div><div style="margin-bottom: .0001pt; margin: 0cm; text-align: justify;"><span style="font-family: Arial, sans-serif;">Misalkan kita mempunyai sebuah garis <em style="box-sizing: border-box;"><span style="font-family: "Arial","sans-serif";">l</span></em> di <strong style="box-sizing: border-box;"><span style="font-family: "Arial","sans-serif";">R</span></strong><em style="box-sizing: border-box;"><sub><span style="font-family: "Arial","sans-serif";">n</span></sub></em>, yang katakanlah (untuk mempermudah pembahasan) melalui titik asal O(0, …, 0) dan <em style="box-sizing: border-box;"><span style="font-family: "Arial","sans-serif";">A</span></em>(<em style="box-sizing: border-box;"><span style="font-family: "Arial","sans-serif";">a</span></em><sub>1</sub>, …, <em style="box-sizing: border-box;"><span style="font-family: "Arial","sans-serif";">a<sub>n</sub></span></em>). Jadi koordinat sebarang titik di <em style="box-sizing: border-box;"><span style="font-family: "Arial","sans-serif";">l</span></em> adalah <em style="box-sizing: border-box;"><span style="font-family: "Arial","sans-serif";">k</span></em>(<em style="box-sizing: border-box;"><span style="font-family: "Arial","sans-serif";">a</span></em><sub>1</sub>, …, <em style="box-sizing: border-box;"><span style="font-family: "Arial","sans-serif";">a<sub>n</sub></span></em>) untuk suatu bilangan real <em style="box-sizing: border-box;"><span style="font-family: "Arial","sans-serif";">k</span></em>. Misalkan kita juga mempunyai sebuah titik <em style="box-sizing: border-box;"><span style="font-family: "Arial","sans-serif";">Q</span></em>(<em style="box-sizing: border-box;"><span style="font-family: "Arial","sans-serif";">q</span></em><sub>1</sub>, …, <em style="box-sizing: border-box;"><span style="font-family: "Arial","sans-serif";">q<sub>n</sub></span></em>) di luar garis <em style="box-sizing: border-box;"><span style="font-family: "Arial","sans-serif";">l</span></em>, dan kita ingin mencari titik <em style="box-sizing: border-box;"><span style="font-family: "Arial","sans-serif";">P</span></em>(<em style="box-sizing: border-box;"><span style="font-family: "Arial","sans-serif";">p</span></em><sub>1</sub>, …, <em style="box-sizing: border-box;"><span style="font-family: "Arial","sans-serif";">p<sub>n</sub></span></em>) pada garis <em style="box-sizing: border-box;"><span style="font-family: "Arial","sans-serif";">l</span></em> yang paling dekat ke <em style="box-sizing: border-box;"><span style="font-family: "Arial","sans-serif";">Q</span></em>.<o:p></o:p></span></div><div style="margin-bottom: .0001pt; margin: 0cm; text-align: justify;"><br /></div><div style="margin-bottom: .0001pt; margin: 0cm; text-align: justify;"><strong style="box-sizing: border-box;"><span style="font-family: Arial, sans-serif;">Jarak</span></strong><span style="font-family: Arial, sans-serif;"> antara titik <em style="box-sizing: border-box;"><span style="font-family: "Arial","sans-serif";">P</span></em>(<em style="box-sizing: border-box;"><span style="font-family: "Arial","sans-serif";">p</span></em><sub>1</sub>, …, <em style="box-sizing: border-box;"><span style="font-family: "Arial","sans-serif";">p<sub>n</sub></span></em>) dan <em style="box-sizing: border-box;"><span style="font-family: "Arial","sans-serif";">Q</span></em>(<em style="box-sizing: border-box;"><span style="font-family: "Arial","sans-serif";">q</span></em><sub>1</sub>, …, <em style="box-sizing: border-box;"><span style="font-family: "Arial","sans-serif";">q<sub>n</sub></span></em>) di <strong style="box-sizing: border-box;"><span style="font-family: "Arial","sans-serif";">R</span></strong><em style="box-sizing: border-box;"><sub><span style="font-family: "Arial","sans-serif";">n</span></sub></em> dalam hal ini diberikan oleh rumus<o:p></o:p></span></div><div style="margin-bottom: .0001pt; margin: 0cm; text-align: justify;"><span style="font-family: Arial, sans-serif;"><!--[if gte vml 1]><v:shapetype id="_x0000_t75" coordsize="21600,21600" o:spt="75" o:preferrelative="t" path="m@4@5l@4@11@9@11@9@5xe" filled="f" stroked="f"> <v:stroke joinstyle="miter"/> <v:formulas> <v:f eqn="if lineDrawn pixelLineWidth 0"/> <v:f eqn="sum @0 1 0"/> <v:f eqn="sum 0 0 @1"/> <v:f eqn="prod @2 1 2"/> <v:f eqn="prod @3 21600 pixelWidth"/> <v:f eqn="prod @3 21600 pixelHeight"/> <v:f eqn="sum @0 0 1"/> <v:f eqn="prod @6 1 2"/> <v:f eqn="prod @7 21600 pixelWidth"/> <v:f eqn="sum @8 21600 0"/> <v:f eqn="prod @7 21600 pixelHeight"/> <v:f eqn="sum @10 21600 0"/> </v:formulas> <v:path o:extrusionok="f" gradientshapeok="t" o:connecttype="rect"/> <o:lock v:ext="edit" aspectratio="t"/></v:shapetype><v:shape id="Picture_x0020_1" o:spid="_x0000_i1028" type="#_x0000_t75" alt="hasil-kali-skalar_1" style='width:269.25pt;height:39.75pt;visibility:visible; mso-wrap-style:square'> <v:imagedata src="file:///C:\Users\user\AppData\Local\Temp\msohtmlclip1\01\clip_image001.gif" o:title="hasil-kali-skalar_1"/></v:shape><![endif]--><!--[if !vml]--><!--[endif]--></span><span style="font-family: Arial, sans-serif;"><o:p></o:p></span></div><div style="margin-bottom: .0001pt; margin: 0cm; text-align: justify;"><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEipGOsU7MpLKSGnj2lR05CmVGwrxLp4o8CpyWbd8qXPjGxp4siK2bfU4dZD0uH6G3OQafPwhI9uI3KF7kOZQn08RQarrLIt7oNSWXPhV85NWv1jqNYmMqwKF23lFDwFv9ElxH4YDT3Uiss/s1600/hasil+kali+skalar+1.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="50" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEipGOsU7MpLKSGnj2lR05CmVGwrxLp4o8CpyWbd8qXPjGxp4siK2bfU4dZD0uH6G3OQafPwhI9uI3KF7kOZQn08RQarrLIt7oNSWXPhV85NWv1jqNYmMqwKF23lFDwFv9ElxH4YDT3Uiss/s320/hasil+kali+skalar+1.jpg" width="320" /></a></div><br /><div style="margin-bottom: .0001pt; margin: 0cm; text-align: justify;"><span style="font-family: Arial, sans-serif;"><br /></span></div><div style="margin-bottom: .0001pt; margin: 0cm; text-align: justify;"><span style="font-family: Arial, sans-serif;">Rumus jarak ini merupakan perumusan dari <strong style="box-sizing: border-box;"><span style="font-family: "Arial","sans-serif";">Dalil Pythagoras</span></strong> (terkait segitiga siku-siku).<o:p></o:p></span></div><div style="margin-bottom: .0001pt; margin: 0cm; text-align: justify;"><span style="font-family: Arial, sans-serif;">Nah, mengingat <em style="box-sizing: border-box;"><span style="font-family: "Arial","sans-serif";">P</span></em> = <em style="box-sizing: border-box;"><span style="font-family: "Arial","sans-serif";">kA</span></em> untuk suatu bilangan real <em style="box-sizing: border-box;"><span style="font-family: "Arial","sans-serif";">k</span></em>, persoalan di atas sama saja dengan mencari <em style="box-sizing: border-box;"><span style="font-family: "Arial","sans-serif";">k</span></em> sedemikian sehingga |<em style="box-sizing: border-box;"><span style="font-family: "Arial","sans-serif";">kA</span></em> – <em style="box-sizing: border-box;"><span style="font-family: "Arial","sans-serif";">Q</span></em>|<sup>2</sup> minimum. </span></div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgKApBhfLIO-TS_DpEqYH6BjjbVuTeo0_ukwOiwMR7U-1UxWIsVh7FKm8MZiE5Utr9Ym3YmSJOzwfI2IAZ0qrX7nDs2jIZ6GmNfc5cXgHMaulBMtBGQ9nFLq1Nx3ow4skBvpeMCbVJ4ZwM/s1600/hasil+kali+skalar+2.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="66" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgKApBhfLIO-TS_DpEqYH6BjjbVuTeo0_ukwOiwMR7U-1UxWIsVh7FKm8MZiE5Utr9Ym3YmSJOzwfI2IAZ0qrX7nDs2jIZ6GmNfc5cXgHMaulBMtBGQ9nFLq1Nx3ow4skBvpeMCbVJ4ZwM/s400/hasil+kali+skalar+2.jpg" width="400" /></a></div><div style="margin-bottom: .0001pt; margin: 0cm; text-align: justify;"><span style="font-family: Arial, sans-serif;"><br /></span></div><div style="margin-bottom: .0001pt; margin: 0cm; text-align: justify;"><span style="font-family: Arial, sans-serif;"><br /></span></div><div style="margin-bottom: .0001pt; margin: 0cm; text-align: justify;"><span style="font-family: Arial, sans-serif;">Tetapi </span><span style="font-family: Arial, sans-serif;">yang merupakan fungsi kuadrat dalam </span><em style="box-sizing: border-box; font-family: Arial, sans-serif;"><span style="font-family: "Arial","sans-serif";">k</span></em><span style="font-family: Arial, sans-serif;">, dan grafiknya berupa suatu parabola yang terbuka ke atas. Dari pengetahuan tentang fungsi kuadrat, kita tahu bahwa |</span><em style="box-sizing: border-box; font-family: Arial, sans-serif;"><span style="font-family: "Arial","sans-serif";">kA</span></em><span style="font-family: Arial, sans-serif;"> – </span><em style="box-sizing: border-box; font-family: Arial, sans-serif;"><span style="font-family: "Arial","sans-serif";">Q</span></em><span style="font-family: Arial, sans-serif;">|</span><sup style="font-family: Arial, sans-serif;">2</sup><span style="font-family: Arial, sans-serif;"> akan mencapai minimum apabila</span></div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhnWKfzcrpWIWH-bcpWwRJhFusYaOKVcx8o3RqlB_ar5e4hXkvMJxH3NBO5psE4RUtvcAPlDFc5ma06Yn_gG41ROw3B57zqE2A6_2oIJXY1pDkQHU5hJlCPs1ajopCoMf9YJ-jiC9Uv5qk/s1600/hasil+kali+skalar+3.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhnWKfzcrpWIWH-bcpWwRJhFusYaOKVcx8o3RqlB_ar5e4hXkvMJxH3NBO5psE4RUtvcAPlDFc5ma06Yn_gG41ROw3B57zqE2A6_2oIJXY1pDkQHU5hJlCPs1ajopCoMf9YJ-jiC9Uv5qk/s1600/hasil+kali+skalar+3.jpg" /></a></div><div style="margin-bottom: .0001pt; margin: 0cm; text-align: justify;"><span style="font-family: Arial, sans-serif;"><br /></span></div><div style="margin-bottom: .0001pt; margin: 0cm; text-align: justify;"><span style="font-family: Arial, sans-serif;"><!--[if gte vml 1]><v:shape id="Picture_x0020_3" o:spid="_x0000_i1026" type="#_x0000_t75" alt="hasil-kali-skalar_3" style='width:99.75pt;height:51.75pt; visibility:visible;mso-wrap-style:square'> <v:imagedata src="file:///C:\Users\user\AppData\Local\Temp\msohtmlclip1\01\clip_image003.gif" o:title="hasil-kali-skalar_3"/></v:shape><![endif]--><!--[if !vml]--><!--[endif]--></span><span style="font-family: Arial, sans-serif;"><o:p></o:p></span></div><div style="margin-bottom: .0001pt; margin: 0cm; text-align: justify;"><span style="font-family: Arial, sans-serif;">Perhatikan juga bahwa dengan nilai <em style="box-sizing: border-box;"><span style="font-family: "Arial","sans-serif";">k</span></em> di atas, kita mempunyai <o:p></o:p></span></div><div style="margin-bottom: .0001pt; margin: 0cm; text-align: justify;"><span style="font-family: Arial, sans-serif;">|<em style="box-sizing: border-box;"><span style="font-family: "Arial","sans-serif";">kA</span></em> – O|<sup>2</sup> + |<em style="box-sizing: border-box;"><span style="font-family: "Arial","sans-serif";">kA</span></em> – <em style="box-sizing: border-box;"><span style="font-family: "Arial","sans-serif";">Q</span></em>|<sup>2</sup> = |<em style="box-sizing: border-box;"><span style="font-family: "Arial","sans-serif";">Q</span></em> – O|<sup>2</sup>.<o:p></o:p></span></div><div style="margin-bottom: .0001pt; margin: 0cm; text-align: justify;"><br /></div><div style="margin-bottom: .0001pt; margin: 0cm; text-align: justify;"><span style="font-family: Arial, sans-serif;">Dalam hal ini, vektor <em style="box-sizing: border-box;"><span style="font-family: "Arial","sans-serif";">OB</span></em> = <em style="box-sizing: border-box;"><span style="font-family: "Arial","sans-serif";">OQ</span></em> – <em style="box-sizing: border-box;"><span style="font-family: "Arial","sans-serif";">k</span></em>.<em style="box-sizing: border-box;"><span style="font-family: "Arial","sans-serif";">OA</span></em> tegak lurus terhadap vektor <em style="box-sizing: border-box;"><span style="font-family: "Arial","sans-serif";">k</span></em>.<em style="box-sizing: border-box;"><span style="font-family: "Arial","sans-serif";">OA</span></em> (dan akibatnya <em style="box-sizing: border-box;"><span style="font-family: "Arial","sans-serif";">OB</span></em> juga tegak lurus terhadap <em style="box-sizing: border-box;"><span style="font-family: "Arial","sans-serif";">OA</span></em>, termasuk ketika <em style="box-sizing: border-box;"><span style="font-family: "Arial","sans-serif";">k</span></em> = 0), dan vektor <em style="box-sizing: border-box;"><span style="font-family: "Arial","sans-serif";">k</span></em>.<em style="box-sizing: border-box;"><span style="font-family: "Arial","sans-serif";">OA</span></em> dikenal sebagai <strong style="box-sizing: border-box;"><span style="font-family: "Arial","sans-serif";">vektor proyeksi</span></strong> dari vektor <em style="box-sizing: border-box;"><span style="font-family: "Arial","sans-serif";">OQ</span></em> terhadap vektor <em style="box-sizing: border-box;"><span style="font-family: "Arial","sans-serif";">OA</span></em>. Jadi begitulah asal-usul dari perumusan hasil kali skalar.<o:p></o:p></span></div><div style="margin-bottom: .0001pt; margin: 0cm; text-align: justify;"><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh0cT3Se6q3lpwAvbmWBs3rEwuL_HpVouxbmUKtD5uq7xZZZij0Qw0_zJ8Ykk2pXyDna5MMCvX7F1yeLsMditNHRg-snaL2YRgnF0FtL5_8XIiSRk4M1lvU3rrB9FqLTzTQhnLYRXgYMkE/s1600/vektor+proyeksi.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh0cT3Se6q3lpwAvbmWBs3rEwuL_HpVouxbmUKtD5uq7xZZZij0Qw0_zJ8Ykk2pXyDna5MMCvX7F1yeLsMditNHRg-snaL2YRgnF0FtL5_8XIiSRk4M1lvU3rrB9FqLTzTQhnLYRXgYMkE/s1600/vektor+proyeksi.jpg" /></a></div><!--[if gte vml 1]><v:shape id="Picture_x0020_4" o:spid="_x0000_i1025" type="#_x0000_t75" alt="vektor-proyeksi-ortogonal" style='width:148.5pt;height:131.25pt;visibility:visible;mso-wrap-style:square'> <v:imagedata src="file:///C:\Users\user\AppData\Local\Temp\msohtmlclip1\01\clip_image004.gif" o:title="vektor-proyeksi-ortogonal"/></v:shape><![endif]--><!--[if !vml]--><!--[endif]--><o:p></o:p><br /> <div style="margin-bottom: .0001pt; margin: 0cm; text-align: justify;"><br /></div><div style="margin-bottom: .0001pt; margin: 0cm; text-align: justify;"><span style="font-family: Arial, sans-serif;">Nah, karena bentuk penjumlahan hasil kali <em style="box-sizing: border-box;"><span style="font-family: "Arial","sans-serif";">a<sub>1</sub>q</span></em><sub>1</sub> + … <em style="box-sizing: border-box;"><span style="font-family: "Arial","sans-serif";">a<sub>n</sub>q<sub>n</sub></span></em> muncul dalam solusi persoalan yang kita anggap penting, maka kita definisikan hasil kali skalar <em style="box-sizing: border-box;"><b><span style="font-family: "Arial","sans-serif";">a<sub>1</sub>q</span></b></em><strong><sub><span style="font-family: "Arial","sans-serif";">1</span></sub></strong><strong><span style="font-family: "Arial","sans-serif";"> + … </span></strong><em style="box-sizing: border-box;"><b><span style="font-family: "Arial","sans-serif";">a<sub>n</sub>q<sub>n</sub></span></b></em><strong><span style="font-family: "Arial","sans-serif";"> = </span></strong><em style="box-sizing: border-box;"><b><span style="font-family: "Arial","sans-serif";">a</span></b></em><strong><span style="font-family: "Arial","sans-serif";">∙</span></strong><em style="box-sizing: border-box;"><b><span style="font-family: "Arial","sans-serif";">q</span></b></em>. Jadi rumus ini bukan turun dari langit, tetapi diperoleh dari persoalan mencari titik terdekat.<o:p></o:p></span></div><div style="margin-bottom: .0001pt; margin: 0cm; text-align: justify;"><span style="font-family: Arial, sans-serif;">Dengan rumus hasil kali skalar ini, dua vektor <em style="box-sizing: border-box;"><span style="font-family: "Arial","sans-serif";">OA</span></em> dan <em style="box-sizing: border-box;"><span style="font-family: "Arial","sans-serif";">OB</span></em> <em style="box-sizing: border-box;"><span style="font-family: "Arial","sans-serif";">saling tegak lurus</span></em> jika dan hanya jika <em style="box-sizing: border-box;"><span style="font-family: "Arial","sans-serif";">a</span></em>∙<em style="box-sizing: border-box;"><span style="font-family: "Arial","sans-serif";">b</span></em> = <em style="box-sizing: border-box;"><span style="font-family: "Arial","sans-serif";">a<sub>1</sub>b<sub>1</sub></span></em> + … + <em style="box-sizing: border-box;"><span style="font-family: "Arial","sans-serif";">a<sub>n</sub>b<sub>n</sub></span></em> = 0. <o:p></o:p></span></div></div> contoh perkalian skalar dalam kehidupan Sering terjadi dalam buku-buku yang membahas ruang vektor, biasanya ada definisi hasil kali titik ... Read more »