tag:blogger.com,1999:blog-66752008913112117032023-11-16T17:42:10.825+07:00MATEMATIKA CERIABlog ini berisi tentang masalah matematika yang sederhana, tetapi sangat mendasar. Isinya antara lain pembuktian rumus-rumus matematika yang jarang diketahui para siswa atau bahkan guru sekalipun. Selain itu juga ditampilkan matematika yang unik, seperti angka berulang, hitungan unik, dan lain-lainomsonhttp://www.blogger.com/profile/17839095311693084453noreply@blogger.comBlogger59125tag:blogger.com,1999:blog-6675200891311211703.post-33048036502922977962015-10-16T19:06:00.000+07:002015-12-05T13:38:15.574+07:00Dari Minus YaSeseorang mengendarai sepeda motor, ....mampir ke pom bensin, tentunya untuk beli bensin<br />
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Ketika akan mengisi bensin, petugas seperti biasa mengatakan "Dari nol ya mass ..."<br />
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Pembeli langsung ngomong ...."Saya penginnya dari minus ... bisa engga ?"<br />
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Petugas : " ....???"<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhPska4ZNCu3QotQI23m_BcHGNjfnv8AwGVybigcbS5fjS6pUfeD2SpsvzGMt0pMX8F5J6EjtNjwdg8BdV7qzUldXitwAGtih_eGOzz5SmbDJE4pyRrqgHSM-n68G-LUpwZpPTRpAlgH3g/s1600/pom+bensin.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhPska4ZNCu3QotQI23m_BcHGNjfnv8AwGVybigcbS5fjS6pUfeD2SpsvzGMt0pMX8F5J6EjtNjwdg8BdV7qzUldXitwAGtih_eGOzz5SmbDJE4pyRrqgHSM-n68G-LUpwZpPTRpAlgH3g/s1600/pom+bensin.jpg" /></a></div>
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Coba bayangkan, kalau kita beli bensin mulai dari minus, tentu sangat menguntungkan.<br />
Misalnya mulai dari minus 1 liter, dan setelah diisi sampai 2 liter ...kita bisa dapat 3 liter kan ??<br />
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Silkan kunjugi <a href="http://supermatematika.com/rumus-cepat-matematika" target="_blank">Rumus Cepat Matematika</a>. Anda akan menemukan matematika yang menarik.omsonhttp://www.blogger.com/profile/17839095311693084453noreply@blogger.com0tag:blogger.com,1999:blog-6675200891311211703.post-64252386134254656902015-10-01T11:38:00.001+07:002015-12-10T11:14:21.559+07:00Spesial Matematika IndonesiaBahasa Indonesia memiliki keunikan dalam memberi nama Angka<br />
1 = Satu<br />
2 = Dua<br />
3 = Tiga<br />
4 = Empat<br />
5 = Lima<br />
6 = Enam<br />
7 = Tujuh<br />
8 = Delapan<br />
9 = Sembilan<br />
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Perhatikan bahwa angka-angka yang memiliki huruf depan sama jika dijumlahkan hasilnya pasti 10<br />
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<b><span style="font-size: large;"><span style="color: red;">S</span>atu + <span style="color: red;">S</span>embilan = 10</span></b><br />
<b><span style="font-size: large;"><span style="color: red;">D</span>ua + <span style="color: red;">D</span>elapan = 10</span></b><br />
<b><span style="font-size: large;"><span style="color: red;">T</span>iga + <span style="color: red;">T</span>ujuh = 10</span></b><br />
<b><span style="font-size: large;"><span style="color: red;">E</span>mpat + <span style="color: red;">E</span>nam = 10</span></b><br />
<b><span style="font-size: large;"><span style="color: red;">L</span>ima + <span style="color: red;">L</span>ima = 10</span></b><br />
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Asyik kan. Sebagai orang Indonesia kita perlu bangga<br />
Ini tidak terjadi di bahasa manapun selain bahasa Indonesia<br />
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Kalau bahasa malaysia gimana ?<br />
Kalau bahasa Malaysia .....> 8 = Lapan<br />
Jadi ga bakalan cocok bro<br />
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Jangan lupa kunjungi website <a href="http://supermatematika.com/akar-akar-saling-berlawanan" target="_blank">persamaan kuadrat</a>, wb matematika yang paling lengkap<br />
<br />omsonhttp://www.blogger.com/profile/17839095311693084453noreply@blogger.com0tag:blogger.com,1999:blog-6675200891311211703.post-9651758044135065552015-08-04T20:22:00.001+07:002015-08-04T20:22:12.348+07:00Teka-Teki Menarik Dalam KotakTeka-Teki Menarik Dalam Kotak<br />
Perhatikan kotak berikut ini<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhHvsjGHF_auTQsGFT660jvj55j7EmBMUFn5mAa4RCX65d86zNIofi994lrQTiRq23m29Qtq_OSjZVyc3oGnjZzdCnpQte5OM8sr51UyL4Ay8umHJmgkGGJMDIqUlqrRW4dmX5kCLBOZ6I/s1600/Teka+Teki+Kotak.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="323" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhHvsjGHF_auTQsGFT660jvj55j7EmBMUFn5mAa4RCX65d86zNIofi994lrQTiRq23m29Qtq_OSjZVyc3oGnjZzdCnpQte5OM8sr51UyL4Ay8umHJmgkGGJMDIqUlqrRW4dmX5kCLBOZ6I/s400/Teka+Teki+Kotak.jpg" width="400" /></a></div>
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Coba, lakukan hal-hal berikut ini :<br />
1. Hubungkan huruf-huruf yang sama dengan menggunakan garis<br />
2. Garis penghubung tidak boleh keluar dari kotak besar.<br />
3. Garis penghubung tidak boleh memotong ataupun berimpit dengan garis tebal<br />
4. Antara garis penghubung yang satu dengan yang lain tidak boleh berpotongan<br />
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Coba lakukan jika kamu memang mampu<br />
<br />omsonhttp://www.blogger.com/profile/17839095311693084453noreply@blogger.com2tag:blogger.com,1999:blog-6675200891311211703.post-17848016069024903682014-02-20T19:16:00.000+07:002014-02-20T19:18:27.395+07:00Salah Konsep PembagianBerapakah hasilnya jika 25 dibagi 5 ? Tentu saja jawabannya adalah 5<br />
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Pada video berikut ada 2 orang yang saling memperdebatkan masalah pembagian, yaitu 25 dibagi 5. Orang pertama menjelaskan hasilnya 5, sedangkan orang kedua menjelaskan hasilnya adalah 14. Anehnya yang menang adalah yang menjawab 14.<br />
Aneh memang ......<br />
Pada awalnya orang pertama menjelaskan pembagian 25 dibagi 5. Tentu saja jika bentuk ini dijelaskan dengan pembagian kurung agak membingungkan, sehingga orang pertama akhirnya hanya memberikan jawaban akhir saja. Tentu saja orang kedua tidak puas, sehingga dia menunjukkan caranya sendiri. Walaupun caranya salah, tetapi dia bisa menjelaskan prosesnya...... Bagaimana caranya , lihat saja di :<br />
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<iframe allowfullscreen="" frameborder="0" height="270" src="//www.youtube.com/embed/Jq6_u0cyLj4?feature=player_detailpage" width="480"></iframe>
<br />omsonhttp://www.blogger.com/profile/17839095311693084453noreply@blogger.com1tag:blogger.com,1999:blog-6675200891311211703.post-63356709668165429882014-01-01T14:07:00.001+07:002014-01-01T14:07:18.314+07:00Grafik CintaSekarang grafik tanda cinta bisa dibuat secara intan, caranya ketik saja di google tulisan ini :<br />
(sqrt(cos(x))*cos(200*x)+sqrt(abs(x))-0.7)*(4-x*x)^0.01, sqrt(9-x^2), -sqrt(9-x^2)<br />
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Setelah itu akan muncul gambar sebagai berikut<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhGWSncbdDW0-4p25l3iG20N6RuOER8Y75qGtb6GgjaEpG1YPyJhP1Z7EZuW1x5Aeho4TZB74owD1ur3uCv5viBkoX4wJ2B8cXek6RRiTmq2SRUOag3kytNBe0iOwon4wk2zb7lLMUfVRI/s1600/love.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="268" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhGWSncbdDW0-4p25l3iG20N6RuOER8Y75qGtb6GgjaEpG1YPyJhP1Z7EZuW1x5Aeho4TZB74owD1ur3uCv5viBkoX4wJ2B8cXek6RRiTmq2SRUOag3kytNBe0iOwon4wk2zb7lLMUfVRI/s400/love.JPG" width="400" /></a></div>
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Oke, selamat mencoba<br />
Jangan lupa .....kirimkan ini kepada orang yang kamu cintai<br />
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omsonhttp://www.blogger.com/profile/17839095311693084453noreply@blogger.com0tag:blogger.com,1999:blog-6675200891311211703.post-53156689833678081672013-12-31T14:58:00.001+07:002013-12-31T15:09:37.700+07:00Penjumlahan unikBanyak sekali penjumlahan-penjumlahan yang unik ada di matematika. salah satunya adalah penjumlahan bilangan-bilangan asli sebagai berikut<br />
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1 + 2 = 3<br />
4 + 5 + 6 = 7 + 8<br />
9 + 10 + 11 + 12 = 13 + 14 + 15<br />
16 + 17 + 18 + 19 + 20 = 21 + 22 + 23 + 24<br />
25 + 26 + 27 + 28 + 29 + 30 = 31 + 32 + 33 + 34 + 35<br />
36 + 37 + 38 + 39 + 40 + 41 + 42 = 43 + 44 + 45 + 46 + 47 + 48<br />
49 + 50 + 51 + 42 + 53 + 54 + 55 + 56 = 57 + 58 + 59 + 60 + 61 + 62 + 63<br />
dan seterusnya ...<br />
anda sudah pasti bisa melanjutkannya .....okey ?omsonhttp://www.blogger.com/profile/17839095311693084453noreply@blogger.com0tag:blogger.com,1999:blog-6675200891311211703.post-20252468776795028312012-08-07T02:35:00.000+07:002012-08-07T02:35:56.297+07:00Ilmuwan Matematika MuslimBanyak sekali tokoh-tokoh ilmuwan muslim yang banyak memberikan sumbangan pada perkembangan matematika. AKan tetapi ilmuwan-ilmuwan ini agak jarang disebut. Berikut ini adalah tokoh-tokohnya :<br />
1.Al-Khawarizm<br />
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2. Al-Kindi<br />
3.Al-Karaji<br />
4. Al-Batani<br />
5. Al-Biruni<br />
6. Umar Khayam<br />
7. Ibnu Sinaomsonhttp://www.blogger.com/profile/17839095311693084453noreply@blogger.com0tag:blogger.com,1999:blog-6675200891311211703.post-2557117386814092122012-08-01T07:29:00.003+07:002012-08-01T07:33:25.777+07:00Segitiga Tak Mungkin<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgIJEk_emLBHzYOrw5HrfxiXTYHHnkrK4MILIv_XcHELCkncqmCLhgHRp2bJAIspLKE0QrEMIADv3sXDfEsvjlE7gZpiW5zn0wgf37RKbKL38fFUwt2jSYzqqbf45k4FW-YZyq_tHeNDYY/s1600/Segitiga.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" height="166" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgIJEk_emLBHzYOrw5HrfxiXTYHHnkrK4MILIv_XcHELCkncqmCLhgHRp2bJAIspLKE0QrEMIADv3sXDfEsvjlE7gZpiW5zn0wgf37RKbKL38fFUwt2jSYzqqbf45k4FW-YZyq_tHeNDYY/s200/Segitiga.jpg" width="200" /></a></div>
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Dari namanya, mungkin terkesan sangat aneh, ya segitiga tak mungkin. Apanya yang tak mungkin ? Segitiga ini tidak mungkin dibuat dalam dunia nyata, makanya disebut segitiga tak mungkin.</div>
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Segitiga seperti ini pertama kali dibuat pada tahun 1934 oleh Oscar Reutersvärd, seniman Swedia. Seringkali segitiga ini disebut Segitiga Penrose, atau disebut juga dengan nama tribar. Objek seperti ini memang mustahil. Akhirnya gambar ini semakin populer setelah dikenalkan oleh seorang matematikawan Roger Penrose. Dengan mempopulerkan segitiga ini pada tahun 1950, maka namanya seringkali dipakai untuk memberi nama segitiga ini, yaitu segitiga penrose. </div>
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Penrose membuat deskripsi "kemustahilan dalam bentuk termurni". Pada bentuk ini, penampilan sengaja dibuat menyolok oleh seniman grafis yang bernama M.C. Escher. Karya-karya M.C. Escher sebagian mendapat inspirasi dari objek yang mustahil semacam ini. <br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg0gySNFUsNLht0AvUxLYGeDuIuV5hS1Tun-ljDRH9aK0AJ8Rg0o-yBtOgLjqycOzQO8NM23phU2CVqMm9Add71pox4PFx192gnWMUtQ_p49UK2fdfa5ai6JHowaxfpZtDu5EvA0E6O8HA/s1600/segitiga+tak+mungkin.JPG" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="132" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg0gySNFUsNLht0AvUxLYGeDuIuV5hS1Tun-ljDRH9aK0AJ8Rg0o-yBtOgLjqycOzQO8NM23phU2CVqMm9Add71pox4PFx192gnWMUtQ_p49UK2fdfa5ai6JHowaxfpZtDu5EvA0E6O8HA/s200/segitiga+tak+mungkin.JPG" width="200" /></a></div>
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Pada gambar tersebut, terlihat bahwa sebuah bangun ruang yang aneh. Bangun ruang ini, terbuat dari balok lurus sebanyak 3 buah, dengan penampang silang yang berbentuk segi empat. Dengan mempertemukan ujung-ujung balok satu sama lain, maka terbentuk sebuah segitiga. Sebenarnya pola-pola seperti ini tidak mungkin dan tidak akan pernah menjadi sebuah segitiga dalam 3 dimensi. Pola-pola ini hanya bisa dibuat dalam gambar saja. Akan tetapi walaupun begitu, kita bisa membuat bentuk imensi 3 nya asalakan kita porong pada bagian tertentu.</div>omsonhttp://www.blogger.com/profile/17839095311693084453noreply@blogger.com0tag:blogger.com,1999:blog-6675200891311211703.post-47222857495653359492012-03-15T13:57:00.002+07:002012-03-15T14:01:49.035+07:00KehidupanTahukah anda bahwa kehidupan bisa dihitung dengan matematika? Ada persamaan sebagai berikut :<br /> hidup + cinta = bahagia<br /> hidup - cinta = sedih<br />Jika kedua ruas dijumlahkan maka diperoleh<br />2 hidup = bahagia + sedih<br />Jadi<br /> hidup = <!--[if gte mso 9]><xml> <w:worddocument> <w:view>Normal</w:View> <w:zoom>0</w:Zoom> <w:trackmoves/> <w:trackformatting/> <w:punctuationkerning/> <w:validateagainstschemas/> <w:saveifxmlinvalid>false</w:SaveIfXMLInvalid> <w:ignoremixedcontent>false</w:IgnoreMixedContent> <w:alwaysshowplaceholdertext>false</w:AlwaysShowPlaceholderText> <w:donotpromoteqf/> <w:lidthemeother>IN</w:LidThemeOther> <w:lidthemeasian>X-NONE</w:LidThemeAsian> <w:lidthemecomplexscript>X-NONE</w:LidThemeComplexScript> <w:compatibility> <w:breakwrappedtables/> <w:snaptogridincell/> <w:wraptextwithpunct/> <w:useasianbreakrules/> <w:dontgrowautofit/> <w:splitpgbreakandparamark/> <w:dontvertaligncellwithsp/> <w:dontbreakconstrainedforcedtables/> <w:dontvertalignintxbx/> <w:word11kerningpairs/> <w:cachedcolbalance/> 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/></span>omsonhttp://www.blogger.com/profile/17839095311693084453noreply@blogger.com0tag:blogger.com,1999:blog-6675200891311211703.post-76240099749983760532012-03-14T18:07:00.003+07:002012-03-14T18:15:25.682+07:00Habis dibagi 2Apa ciri-cirinya sebuah bilangan dikatakan genap ?<br />Jawabannya tentu saja habis dibagi 2.<br />Misalkan ada bilangan 135.734, pertanyaannya, bilangan tersebut ganjil atau genap ? tentu saja bilangan tersebut adalah genap. Anehnya tanpa membagi dengan 2, kita langsung tahu bahwa bilangan tersebut adalah genap. Dari mana tahunya?<br />Tentu saja dari angka terakhir. Karena angka terakhirnya genap, maka bilangan tersebut pasti genap.<br />walaupun angka-angka sebelumnya ganjil, jika angka terakhir genap maka bilangan tersebut pastilah genapomsonhttp://www.blogger.com/profile/17839095311693084453noreply@blogger.com0tag:blogger.com,1999:blog-6675200891311211703.post-59264800171953048592012-02-08T12:36:00.005+07:002012-02-08T13:08:57.705+07:00Bilangan UnikSeringkali beberapa bilangan memiliki keunikan sendiri-sendiri. Kita bisa menyebutnya bilangan unik. Misalnya bilangan tersebut adalah<div><br /></div><br />1<sup>3</sup>+5<sup>3</sup>+3<sup>3</sup> = 153<br />3<sup>3</sup>+7<sup>3</sup>+0<sup>3</sup> = 370<br />3<sup>3</sup>+7<sup>3</sup>+1<sup>3</sup> = 371<br />4<sup>3</sup>+0<sup>3</sup>+7<sup>3</sup> = 407<br />1<sup>4</sup>+6<sup>4</sup>+3<sup>4</sup>+ 4<sup>4</sup> = 1634<br />8<sup>4</sup>+2<sup>4</sup>+0<sup>4</sup>+ 8<sup>4</sup> = 8208<br />9<sup>4</sup>+4<sup>4</sup>+7<sup>4</sup>+ 4<sup>4</sup> = 9474<br />4<sup>5</sup>+1<sup>5</sup>+5<sup>5</sup>+ 0<sup>5</sup> = 4150<br />4<sup>5</sup>+1<sup>5</sup>+5<sup>5</sup>+ 1<sup>5</sup> = 4151<br />5<sup>5</sup>+4<sup>5</sup>+7<sup>5</sup>+ 4<sup>5</sup> + 8<sup>5</sup> = 54748<br />9<sup>5</sup>+2<sup>5</sup>+7<sup>5</sup>+ 2<sup>5</sup> + 7<sup>5</sup> = 92727<br />9<sup>5</sup>+3<sup>5</sup>+0<sup>5</sup>+ 8<sup>5</sup> + 4<sup>5</sup> = 93084omsonhttp://www.blogger.com/profile/17839095311693084453noreply@blogger.com0tag:blogger.com,1999:blog-6675200891311211703.post-2843708495098374092012-01-29T15:11:00.003+07:002012-01-29T16:18:13.100+07:00Perkalian MinusAda yang menanyakan "min kali min apa?" .... mendadak ada yang menjawab "lompat". Jawaban ini tentu saja membuat orang terheran-heran. <div>...... maksud nya ada orang yang panggilannya "Min" .... ketika di depan Min ada kali (parit) ... dia diberi peringatan "Min, kali Min" ..... lansung Min melompat ...</div><div>(Sory ya, bercanda dulu .... biar ga tegang .... sekarang kita mulai serius)</div><div>Perhatikan garis bilangan berikut, </div><div><img style="float:left; margin:0 10px 10px 0;cursor:pointer; cursor:hand;width: 400px; height: 83px;" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgcgu_EF52bbiCrI9aZg-smRmzWcEn4CNVRJg-_YskBr_ZM4rGzyz2Dcw58IXis_OB3iCwRHrLKBTFQj2zmqSB8SHi5vhMPQ6pjLhiNTg77jVskWDBSvxX2DnjL7tSb-cJQgY_aLMlxZUg/s400/Perkalian+minus.jpg" border="0" alt="" id="BLOGGER_PHOTO_ID_5702980465744272770" /></div><div><br /></div><div><br /></div><div><br /></div><div><br /></div><div><br /></div><div><br /></div><div>Bilangan positif bisadikatakan mengarah ke kanan, sedangkan bilangan negatif mengarah ke kiri. </div><div>Pada saat pertama kali kita belajar perkalian maka tentunya kita hanya mengalikan dua bilangan positif, sehingga bisa dikatakan </div><div> </div><div> positif x positif = positif</div><div><br /></div><div>Contoh, misalnya 3x2, artinya ada bilangan 2 sebanyak 3 buah</div><div>jadi, </div><div> 3x2 = 2 + 2 + 2 = 6</div><div>Jika diperhatikan dari jaraknya ke angka nol maka angka 2 setelah dikali dengan 3 maka akan menjauh dari 0 sebanyak 3 kali</div><div><br /></div><div>Jika kita mengalikan 2x3 artinya ada bilangan 3 sebanyak 2 buah</div><div>jadi,</div><div> 2x3 = 3 + 3 = 6</div><div><div>Jika diperhatikan dari jaraknya ke angka nol maka angka 3 setelah dikali dengan 2 maka akan menjauh dari 0 sejauh 2 kali</div><br class="Apple-interchange-newline"></div><div>Dari segi arti memang berbeda, tapi dari segi hasil adalah sama.</div><div>karena hasilnya sama maka</div><div> 3x2 = 2x3</div><div>bentuk terakhir ini sering disebut sifat komutatif, artinya dibolak-balik sama.</div><div>Jadi perkalian bersifat komutatif.</div><div><br /></div><div><b>Kesimpulan</b></div><div><b>1. axb artinya ada b sebanyak a</b></div><div><b>2. axb = bxa </b></div><div><b>3. dari bentuk axb bisa dikatakan bilangan b setelah dikali dengan a, akan menjauh dari nol sejauh a kali.</b></div><div><br /></div><div>Sekarang kita mulai perkalian bilangan dengan bilangan negatif</div><div>misalnya 5x(-2)</div><div>artinya anad -2 sebanyak 5 (dari kesimpulan 1)</div><div>jadi, </div><div> 5 x (-2) =(-2) + (-2)+(-2)+(-2)+(-2) = -10</div><div>dari kesimpulan 3 bisa dikatakan, bilangan -2 setelah dikali dengan 5 akan menjauh dari nol sejauh 5 kali, sehingga diperoleh -10</div><div>Karena -2 ada di sebelah kiri nol, maka menjauh 5 kali berarti makin ke kiri sejauh 5 langkah (satu langkah bernilai 2).</div><div><br /></div><div> 5 x(-2) = -10</div><div> positif xnegatif = negatif</div><div><br /></div><div>Sekarang bagaimana dengan negatif kali positif ?</div><div>misalnya (-2)x5</div><div>ingat dari kesimpulan nomor 2, perkalian bersifat komutatif</div><div> (-2)x5 = 5 x(-2) = -10</div><div>Jadi,</div><div> (-2)x5 = -10</div><div> negatif x positif = negatif</div><div><br /></div><div>Perhatikan bahwa (-2)x5 = -10 artinya bilangan 5 dijauhkan terhadap nol sebanyak 2 kali dengan arah berlawanan dengan 5. (lima menngarah ke kanan, sementara -10 mengarah ke kiri)</div><div>Jadi perkalian dengan bilangan negatif menyebabkan arahnya berlawanan.</div><div><br /></div><div>Kesimpulan 4 : (jika a bil positi, .... atau bisa dikatakan -a bilangan negatif)</div><div><b>-a x b berarti b menjauh sebanyak a kali lipat dengan arah berlawanan dengan arah b.</b></div><div><br /></div><div>sekarang bagaimana dengan (-4)x(-3)</div><div>berdasarkan kesimpulan 4, arahnya harus melawan arah -3. Karena (-3) mengarah ke kiri, hasilnya harus mengarah ke kanan, dan ini berarti hasilnya positif.</div><div>Jadi, </div><div>(-4)x(-3) = 12 </div><div>negatif x negatif = positif</div><div><br /></div>omsonhttp://www.blogger.com/profile/17839095311693084453noreply@blogger.com4tag:blogger.com,1999:blog-6675200891311211703.post-65040712743590727832012-01-01T08:32:00.002+07:002012-01-01T08:43:43.514+07:00Sistem Persamaan LinearSistem persamaan linear merupakan kumpulan dari beberapa persamaan linear. Banyaknya persamaan tergantung pada berapa banyaknya variabel. Misalnya ada 2 variabel, yaitu x dan y, maka akan dibutuhkan 2 persamaan, misalnya<br /><br />ax + by = e<br />cd + dy = f<br /><br />Dengan adanya 2 persamaan ini maka <span style="font-weight:bold;">diharapkan</span> sistem persamaan ini akan memiliki tepat 1 penyelesaian. Jika ada 1 persamaan saja maka persamaan akan memiliki banyak penyelesaian. <br /><br />Kenapa tadi hanya dikatakan diharapkan ?<br />Karena bisa jadi dengan 2 persamaan di atas tidak memiliki penyelesaian. Hal ini terjadi jika kedua persamaan merupakan garis yang sejajar.<br />atau bisa dikatakan, sistem persamaan linear tersebut mempunyai sifat ad-bc = 0<br /><br />Terkadang ada sistem persamaan linear yang sifatnya hanya main-main<br /><span style="font-weight:bold;">Misalnya </span><br /><br />a + b + c + d = 6<br />b + c + d + e = 16<br />c + d + e + f = 21<br />d + e + f + g = 3<br />e + f + g + h = 11<br />Tentukan n ilai dari<br />a + b + c + d + e + f + g + h<br /><br />Jawab <br />Kalau kita lihat bentuk di atas memiliki 8 variabel, sementara hanya ada 5 persamaan. Tentunya sistem persamaan tersebut memiliki banyak penyelesaian. Jadi, kita akan kesulitan menntukan nilai masing-masing variabel, karena banyak kemungkinan.<br />AKan tetapai perhatikan bahwa soal tidak menanyakan nilai masing-masing variabel. Soal hanya menanyakan jumlah semua variabel.<br />Jadi kita tinggal menjumlahkan persamaan pertama dan terakhir<br /><br />Jadi, <br />a + b + c + d = 6<br />e + f + g + h = 11<br />kalau kedua persamaan dijumlahkan maka diperoleh <br />a + b + c + d + e + f + g + h = 17omsonhttp://www.blogger.com/profile/17839095311693084453noreply@blogger.com1tag:blogger.com,1999:blog-6675200891311211703.post-32498244986973545842011-11-20T20:05:00.003+07:002012-01-01T09:42:40.586+07:00sin dan cosKita sudah sering mengetahi bahwa<br />sin<sup>2</sup>x + cos<sup>2</sup>x = 1<br />Dari manakah asal rumus ini ?<br /><br /><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhRmDemG3Tlb1IFKkxjPVK26zuKeQBNIFCAhDx6JPEHT-iqdSZw1Ze1UTwCXx1oPcuAVvApVS_O9aaBNKacaOiKAlja1s9g2TbgbIl2C7Zj28J0ua7mtofk81xTeExdo-BAvU3iIbIwjVY/s1600/sin+dan+cos.jpg"><img style="float: left; margin: 0pt 10px 10px 0pt; cursor: pointer; width: 200px; height: 136px;" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhRmDemG3Tlb1IFKkxjPVK26zuKeQBNIFCAhDx6JPEHT-iqdSZw1Ze1UTwCXx1oPcuAVvApVS_O9aaBNKacaOiKAlja1s9g2TbgbIl2C7Zj28J0ua7mtofk81xTeExdo-BAvU3iIbIwjVY/s200/sin+dan+cos.jpg" alt="" id="BLOGGER_PHOTO_ID_5692488410576199186" border="0" /></a><br /><br /><br /><br /><br /><br /><br /><br /><br />Untuk mengetahuinya maka gunakan pythagoras<br /><br />a<sup>2</sup> + b<sup>2</sup> = r<sup>2</sup><br />(r sin x)<sup>2</sup> + (r cos x)<sup>2</sup> = r<sup>2</sup><br />r<sup>2</sup> sin <sup>2</sup> x + r<sup>2</sup> cos <sup>2</sup> x = r <sup>2</sup><br />Jika dibagi dengan r<sup>2</sup> maka diperoleh<br />sin<sup>2</sup>x + cos<sup>2</sup>x = 1<br /><br />Rumus ini seringkali dibalik sehingga<br />sin<sup>2</sup>x = 1 - cos<sup>2</sup>x<br />atau<br />cos<sup>2</sup>x = 1 - sin<sup>2</sup>x<br /><br />Jika<br />sin<sup>2</sup>x + cos<sup>2</sup>x = 1<br />dibagi dengan cos<sup>2</sup>x maka diperoleh<br />tan<sup>2</sup> x + 1 = sec<sup>2</sup> x<br /><br />Jika<br />sin<sup>2</sup>x + cos<sup>2</sup>x = 1<br />dibagi dengan sin<sup>2</sup>x maka diperoleh<br />1 + cot<sup>2</sup> x = csc<sup>2</sup> xomsonhttp://www.blogger.com/profile/17839095311693084453noreply@blogger.com1tag:blogger.com,1999:blog-6675200891311211703.post-83576062850996359882011-10-28T12:20:00.007+07:002012-01-02T20:12:30.796+07:00sudut-sudut bangun datarSegitiga memiliki 3 sudut. Jumlah ketiga sudutnya alaha 180<sup>o</sup>. Kenapa 180<sup>o</sup>? 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<!--[endif]-->* satu putaran penuh adalah 360<sup>o</sup>.<br /></w:lsdexception>* setengah putaran atau satu garis lurus adalah 180<sup>o</sup>.<br /><br />Perhatikan segitiga ABC berikut<br /><xml><w:latentstyles deflockedstate="false" defunhidewhenused="true" defsemihidden="true" defqformat="false" defpriority="99" latentstylecount="267"></w:latentstyles><w:lsdexception locked="false" priority="70" semihidden="false" unhidewhenused="false" name="Dark List Accent 1"><br 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Sudut yang berseberangan besarnya pasti sama. 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Grid Accent 5"><w:lsdexception locked="false" priority="63" semihidden="false" unhidewhenused="false" name="Medium Shading 1 Accent 5"><w:lsdexception locked="false" priority="64" semihidden="false" unhidewhenused="false" name="Medium Shading 2 Accent 5"><w:lsdexception locked="false" priority="65" semihidden="false" unhidewhenused="false" name="Medium List 1 Accent 5"><w:lsdexception locked="false" priority="66" semihidden="false" unhidewhenused="false" name="Medium List 2 Accent 5"><w:lsdexception locked="false" priority="67" semihidden="false" unhidewhenused="false" name="Medium Grid 1 Accent 5"><w:lsdexception locked="false" priority="68" semihidden="false" unhidewhenused="false" name="Medium Grid 2 Accent 5"><w:lsdexception locked="false" priority="69" semihidden="false" unhidewhenused="false" name="Medium Grid 3 Accent 5"><w:lsdexception locked="false" priority="70" semihidden="false" unhidewhenused="false" name="Dark List Accent 5"><w:lsdexception locked="false" priority="71" 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Jadi :<br />sudut A + sudut B + sudut C = 1</w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:latentstyles></xml></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:lsdexception></w:latentstyles></xml></w:worddocument></xml>80<sup>o</sup><br 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MatematikaBelajar matematika, sebenarnya tidak jauh berbeda dengan bermain game. Pada dasarnya matematika menyenangkan bagi semua orang. Hanya saja matematika ini memiliki banyak sekali level. Jika kita mendapati matematika tidak sesuai dengan level kita, maka kita tidak akan menyukainya.<br /><br />Sama persis dengan main game. Jika seseorang bermain game dengan level yang tidak sesuai (terlalu sulit) maka orang tersebut tidak akan menyukainya. Untuk menyukainya seseorang harus bermain dari level awal, setelah menguasai maka dia bisa menaikkan levelnya, menuju ke level yang lebih sulit.<br /><br />Nah, matematika memiliki banyak sekali level. Untuk mudahnya, anggap saja level ini sama dengan kelas yang ada di sekolah. Mulai dati kelas 1, kelas 2, kelas 3, .... dan seterusnya samapai kelas 12.... belum lagi tingkatan yang ada di perguruan tinggi. Jika seseorang kurang menguasai matematika di kelas tertentu, maka untuk kelas lanjutannya dia tidak akan bisa. Ini tentunya membuat orang tersebut semakin tidak suka dengan matematika. Semakin dia tidak suka, maka potensi yang digunakan untuk belajar matematika semakin kecil. Begitulah efek ini terus berlanjut dan saling mempengaruhi.<br /><br />Jika seseorang tidak menguasai trigonometri yang ada di kelas 10, tentunya di kelas 11 dia semakin tidak bisa dengan trigonometri. semakin dia tidak bisa trigonemetri kelas 11 maka dia makin tidak menyukainya, dan efek berikutnya dia tidak mau mempelajarinya. Efek lanjutannya bisa ditebak, dia makain tidak menyukai trigonometri.<br /><br />Nah bagi untuk mencegah supaya kita tidak benci matematika, mulai sekarang perbanyaklah latihan soal-soal matematika. Mulailah dari soal-soal yang mudah, dan saya sarankan 70% soal yang kita kerjakan adalah soal-soal yang mudah, supaya kita selalu merasa bisa. Anehnya kalau kita banyak latihan soal-soal yang mudah, maka soal yang sulit akan terasa mudah, karena pada dasarnya soal yang sulit adalah kumpulan dari berbagai soal-soal yang mudah.<br /><br />Jika anda masih membenci matematika, luangkalah selama 90 hari untuk mengerjakan soal-soal matematika yang mudah, setiap hari 2 jam, jika tidak bisa 2 jam, 1 jam saja, jiika 1 jam tidak bisa, 1/2 jam juga boleh, yang penting anda lakukan setiap hari. Setelah 90 hari maka anda akan makin suka dengan matematika. Dengan semakin sukanya anda dengan matematika, maka anda makin ingin meluangkan waktu lagi untuk mempelajari matematika. Akibatnya kemampuan anda semakin meningkat. Setelah kemampuan anda meningkat, anda akan semakin suka dengan matematika.<br /><br />Seringkali terjadi hal-hal yang sangat ironis. Ketik seorang siswa disuruh mengerjakan soal matemamatika, dia mengatakan, "ah, ini sudah bisa". Akibatnya dia tidak mau mengerjakannya. Ketika gurunya memberikan soal yang lebih sulit, dia memengatakan,"wah kalau ini saya tidak bisa." Karena tidak bisa, dia juga tidak mau mengerjakannya. Jadi maunya apa? soal mudah tidak mau mengerjakan, soal sulit juga tidak mau ... Orang yang seperti itu pada intinya memang malas, dan inilah yang menyebabkan dia tidak bisa dan tidak suka matematika.<br /><br />Untuk itu kerjakanlah soal yang sekiranya kamu bisa. Jika tidak bisa, pikirkanlah minimal 5 menit, atau mungkin 10 menit. Jika masih tidak bisa, carilah soal-soal lain yang lebih mudah, jangan lansgsung menyerah. Luangkan ini setiap hari 2 jam, maka dalam 90 hari anda akan menuai hasil, tanpa sadar nilai matematika kamu di sekolah akan semakin naik, kamu jadi suka dengan matematika, dan akhirnya kamua akan meluangkan waktu lebih banyak, dan semakin lama akan semakin bisa. Ini tentunya akan mempermudah cita-citamu untuk melanjutkan ke perguruan tinggi yang kamu inginkan, 99% tes perguruan tinggi pasti ada matematikanya. Ok, selamat mencobaomsonhttp://www.blogger.com/profile/17839095311693084453noreply@blogger.com0tag:blogger.com,1999:blog-6675200891311211703.post-66248001095943581632011-09-11T03:51:00.006+07:002011-09-11T04:35:48.749+07:00SegitigaSegitiga adalah bangun datar yang memiliki 3 sisi. Berdasarkan jenisnya maka ada 5 macam segitiga<br />1. Segitiga siku-siku : yaiyu segitiga yang salah satu sudutnya 90<sup>o</sup><br />2. Segitiga tumpul : segitiga yang salah satu sudutnya lebih besar dari 90<sup>o</sup><br />3. Segitiga lancip : segitiga yang setiap sudutnya lebih kecil dari 90<sup>o</sup><br />4. Segitiga sama kaki : segitiga yang memiliki 2 sisi yang sama bisa juga disebut segitiga yang memiliki 2 sudut yang sama<br />5. Segitiga sama sisi : segitiga yang memiliki 3 sisi yang sama, bisa juga disebut segitiga yang memiliki 3 sudut yang sama<br /><br />Pada setiap segitiga, jumlah ketiga sudutnya adalah 180<sup>o</sup>. Jadi, jika ada segitiga ABC maka<br /><br />A + B + C = 180<sup>o</sup><br /><br />Mengapa jumlah ketiga sudut pada segitiga 180<sup>o</sup>?<br />sebelum menjawab ini kita lihat aturan perhitungan sudut sebagai berikut :<br />* satu putaran penuh = 360<sup>o</sup><br />* Setengah putaran = 180<sup>o</sup><br />* Seperempat putaran = 90<sup>o</sup><br />dan sebagainya<br /><br />Jadi maksud dari pertanyaan di atas adalah buktikan bahwa jumlah ketiga sudut segitiga adalah setengah putaran.<br />Nah sekarang perhatikan gambar berikut :<br /><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj8hJdrJ3mhNdNOpy0pihKu7j9O9q3MRBJMWJhU4I4UEzk0ou5CzXqT6dZejg3VV2Kw2cTnJn6IXq3hde4PeF8o5IIT9g4Fh5dWxYin_kb7MnN-1gPZJMrSk-lXzsGOVl4GaqFo3zkdego/s1600/segitiga.jpg"><img style="display: block; margin: 0px auto 10px; text-align: center; cursor: pointer; width: 320px; height: 173px;" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj8hJdrJ3mhNdNOpy0pihKu7j9O9q3MRBJMWJhU4I4UEzk0ou5CzXqT6dZejg3VV2Kw2cTnJn6IXq3hde4PeF8o5IIT9g4Fh5dWxYin_kb7MnN-1gPZJMrSk-lXzsGOVl4GaqFo3zkdego/s200/segitiga.jpg" alt="" id="BLOGGER_PHOTO_ID_5650843417758097090" border="0" /></a><br />Jadi<br /><div style="text-align: center;"><a href="http://www.codecogs.com/eqnedit.php?latex=%5Csmall%20%5Cdpi%7B100%7D%20%5Cfn_jvn%20%5Calpha%20@plus;%5Cbeta%20@plus;%5Cgamma%20=180%5E%7Bo%7D" target="_blank"><img src="http://latex.codecogs.com/gif.latex?%5Csmall%20%5Cdpi%7B100%7D%20%5Cfn_jvn%20%5Calpha%20+%5Cbeta%20+%5Cgamma%20=180%5E%7Bo%7D" title="\small \dpi{100} \fn_jvn \alpha +\beta +\gamma =180^{o}" /></a><br /></div>Jadi terbukti bahwa jumlah ketiga sudut pada segitiga adalah 180<sup>o</sup><br /><br />Luas segitiga<br />Untuk menghitung luas segitiga, ada beberapa rumus yang kita kenal,<br />1. 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style="line-height: 115%;font-family:";font-size:11pt;" > a = alas<br /> t = tinggi<br /></span> Rumus ini adalah rumus yang dikenal oleh anak-anak sejak masih SD<br /><br />2. Jika diketahui dua sisi dan 1 sudut<br /><div style="text-align: center;"> L =<span style="line-height: 115%;font-family:";font-size:11pt;" >½</span> <span style="line-height: 115%;font-family:";font-size:11pt;" >ab sin C </span><br /> L =<span style="line-height: 115%;font-family:";font-size:11pt;" >½ ac sin B</span><br /> L = <span style="line-height: 115%;font-family:";font-size:11pt;" >½ bc sin A</span><br /></div>Rumus ini biasanya dikenal anak-anak ketika mempelajari Trigonometri<br />3. Jika diketahui ketiga sisinya<br /><a href="http://www.codecogs.com/eqnedit.php?latex=%5Csmall%20%5Cdpi%7B100%7D%20%5Cfn_jvn%20L=%5Csqrt%7Bs%28s-a%29%28s-b%29%28s-c%29%7D" target="_blank"><img src="http://latex.codecogs.com/gif.latex?%5Csmall%20%5Cdpi%7B100%7D%20%5Cfn_jvn%20L=%5Csqrt%7Bs%28s-a%29%28s-b%29%28s-c%29%7D" title="\small \dpi{100} \fn_jvn L=\sqrt{s(s-a)(s-b)(s-c)}" /></a><br />Dengan S = <span style="line-height: 115%;font-family:";font-size:11pt;" >½</span> (a+b+c)omsonhttp://www.blogger.com/profile/17839095311693084453noreply@blogger.com0tag:blogger.com,1999:blog-6675200891311211703.post-14290757760127418412011-09-07T08:47:00.000+07:002011-09-07T09:12:29.056+07:00operasi aljabar pada bentuk akarSeringkali kita mengoperasikan aljabar dalam bentuk pangkat, akan tetapi setelah mempelajari ini, siswa masih kesulitan untuk memodifikasi dalam bentuk akar. Contohnya kita memiliki selisih 2 kuadrat sebagai berikut :<br />
<br />
a<sup>2</sup> - b<sup>2</sup> = (a - b)(a + b)<br />
maka bisa kita modifikasi sebagai berikut :<br />
misal
<a href="http://www.codecogs.com/eqnedit.php?latex=a=%5Csqrt%7Bp%7D" target="_blank"><img src="http://latex.codecogs.com/gif.latex?a=%5Csqrt%7Bp%7D" title="a=\sqrt{p}" /></a>
dan
<a href="http://www.codecogs.com/eqnedit.php?latex=b=%5Csqrt%7Bq%7D" target="_blank"><img src="http://latex.codecogs.com/gif.latex?b=%5Csqrt%7Bq%7D" title="b=\sqrt{q}" /></a>
<br />
maka bentuk di atas bisa diubah menjadi<br />
<a href="http://www.codecogs.com/eqnedit.php?latex=%5Cleft%20%28%20%5Csqrt%7Bp%7D%20%5Cright%20%29%5E%7B2%7D%20-%20%28%5Csqrt%7Bq%7D%29%5E%7B2%7D=%28%5Csqrt%7Bp%7D-%5Csqrt%7Bq%7D%29%28%5Csqrt%7Bp%7D@plus;%5Csqrt%7Bq%7D%29" target="_blank"><img src="http://latex.codecogs.com/gif.latex?%5Cleft%20%28%20%5Csqrt%7Bp%7D%20%5Cright%20%29%5E%7B2%7D%20-%20%28%5Csqrt%7Bq%7D%29%5E%7B2%7D=%28%5Csqrt%7Bp%7D-%5Csqrt%7Bq%7D%29%28%5Csqrt%7Bp%7D+%5Csqrt%7Bq%7D%29" title="\left ( \sqrt{p} \right )^{2} - (\sqrt{q})^{2}=(\sqrt{p}-\sqrt{q})(\sqrt{p}+\sqrt{q})" /></a><br />
atau bisa ditulis menjadi<br />
<a href="http://www.codecogs.com/eqnedit.php?latex=%5Clarge%20%7B%5Ccolor%7BRed%7D%20%5Cmathbf%7Bp-q=%28%5Csqrt%7Bp%7D-%5Csqrt%7Bq%7D%29%28%5Csqrt%7Bp%7D@plus;%7B%5Ccolor%7BRed%7D%20%7D%5Csqrt%7Bq%7D%29%7D%7D" target="_blank"><img src="http://latex.codecogs.com/gif.latex?%5Clarge%20%7B%5Ccolor%7BRed%7D%20%5Cmathbf%7Bp-q=%28%5Csqrt%7Bp%7D-%5Csqrt%7Bq%7D%29%28%5Csqrt%7Bp%7D+%7B%5Ccolor%7BRed%7D%20%7D%5Csqrt%7Bq%7D%29%7D%7D" title="\large {\color{Red} \mathbf{p-q=(\sqrt{p}-\sqrt{q})(\sqrt{p}+{\color{Red} }\sqrt{q})}}" /></a><br />
dalam bentuk selisih pangkat tiga dan jumlah pangkat tiga, kita memiliki rumus sebagai berikut<br />
a<sup>3</sup> - b<sup>3</sup> = (a - b)(a<sup>2</sup> + ab +b<sup>2</sup> )<br />
dan<br />
a<sup>3</sup> + b<sup>3</sup> = (a + b)(a<sup>2</sup> - ab +b<sup>2</sup> )<br />
dengan mengganti<br />
<a href="http://www.codecogs.com/eqnedit.php?latex=a=%5Csqrt[3]%7Bp%7D" target="_blank"><img src="http://latex.codecogs.com/gif.latex?a=%5Csqrt[3]%7Bp%7D" title="a=\sqrt[3]{p}" /></a>
dan
<a href="http://www.codecogs.com/eqnedit.php?latex=b=%5Csqrt[3]%7Bq%7D" target="_blank"><img src="http://latex.codecogs.com/gif.latex?b=%5Csqrt[3]%7Bq%7D" title="b=\sqrt[3]{q}" /></a><br />
maka bentuk selisih pangkat tiga menjadi<br />
<a href="http://www.codecogs.com/eqnedit.php?latex=%5Csmall%20%28%5Csqrt[3]%7Bp%7D%29%5E%7B3%7D-%28%5Csqrt[3]%7Bq%7D%29%5E%7B3%7D=%28%5Csqrt[3]%7Bp%7D-%5Csqrt[3]%7Bq%7D%29%28%28%5Csqrt[3]%7Bp%7D%29%5E%7B2%7D@plus;%5Csqrt[3]%7Bp%7D%5Csqrt[3]%7Bq%7D@plus;%28%5Csqrt[3]%7Bq%7D%29%5E%7B2%7D%29" target="_blank"><img src="http://latex.codecogs.com/gif.latex?%5Csmall%20%28%5Csqrt[3]%7Bp%7D%29%5E%7B3%7D-%28%5Csqrt[3]%7Bq%7D%29%5E%7B3%7D=%28%5Csqrt[3]%7Bp%7D-%5Csqrt[3]%7Bq%7D%29%28%28%5Csqrt[3]%7Bp%7D%29%5E%7B2%7D+%5Csqrt[3]%7Bp%7D%5Csqrt[3]%7Bq%7D+%28%5Csqrt[3]%7Bq%7D%29%5E%7B2%7D%29" title="\small (\sqrt[3]{p})^{3}-(\sqrt[3]{q})^{3}=(\sqrt[3]{p}-\sqrt[3]{q})((\sqrt[3]{p})^{2}+\sqrt[3]{p}\sqrt[3]{q}+(\sqrt[3]{q})^{2})" /></a><br />
atau bisa ditulis menjadi :<br />
<a href="http://www.codecogs.com/eqnedit.php?latex=%5Clarge%20%5Cboldsymbol%7B%5Cmathbf%7B%7B%5Ccolor%7BRed%7D%20%28p-q%29=%28%5Csqrt[3]%7Bp%7D-%5Csqrt[3]%7Bq%7D%29%28%5Csqrt[3]%7Bp%5E%7B2%7D%7D@plus;%5Csqrt[3]%7Bpq%7D@plus;%5Csqrt[3]%7Bq%5E%7B2%7D%7D%29%7D%7D%7D" target="_blank"><img src="http://latex.codecogs.com/gif.latex?%5Clarge%20%5Cboldsymbol%7B%5Cmathbf%7B%7B%5Ccolor%7BRed%7D%20%28p-q%29=%28%5Csqrt[3]%7Bp%7D-%5Csqrt[3]%7Bq%7D%29%28%5Csqrt[3]%7Bp%5E%7B2%7D%7D+%5Csqrt[3]%7Bpq%7D+%5Csqrt[3]%7Bq%5E%7B2%7D%7D%29%7D%7D%7D" title="\large \boldsymbol{\mathbf{{\color{Red} (p-q)=(\sqrt[3]{p}-\sqrt[3]{q})(\sqrt[3]{p^{2}}+\sqrt[3]{pq}+\sqrt[3]{q^{2}})}}}" /></a><br />
untuk jumlah pangkat tiga bisa ditulis menjadi :
<a href="http://www.codecogs.com/eqnedit.php?latex=%5Clarge%20%5Cboldsymbol%7B%5Cmathbf%7B%7B%5Ccolor%7BRed%7D%20%28p@plus;q%29=%28%5Csqrt[3]%7Bp%7D@plus;%5Csqrt[3]%7Bq%7D%29%28%5Csqrt[3]%7Bp%5E%7B2%7D%7D-%5Csqrt[3]%7Bpq%7D@plus;%5Csqrt[3]%7Bq%5E%7B2%7D%7D%29%7D%7D%7D" target="_blank"><img src="http://latex.codecogs.com/gif.latex?%5Clarge%20%5Cboldsymbol%7B%5Cmathbf%7B%7B%5Ccolor%7BRed%7D%20%28p+q%29=%28%5Csqrt[3]%7Bp%7D+%5Csqrt[3]%7Bq%7D%29%28%5Csqrt[3]%7Bp%5E%7B2%7D%7D-%5Csqrt[3]%7Bpq%7D+%5Csqrt[3]%7Bq%5E%7B2%7D%7D%29%7D%7D%7D" title="\large \boldsymbol{\mathbf{{\color{Red} (p+q)=(\sqrt[3]{p}+\sqrt[3]{q})(\sqrt[3]{p^{2}}-\sqrt[3]{pq}+\sqrt[3]{q^{2}})}}}" /></a>
omsonhttp://www.blogger.com/profile/17839095311693084453noreply@blogger.com1tag:blogger.com,1999:blog-6675200891311211703.post-55561343809774846812011-08-04T18:40:00.002+07:002011-08-04T18:48:36.859+07:00Salah hitungSuatu hari Rina dan Rini bertemu. Rina bertanya kepada Rini<br />Rina :"Saya beli buku tulis, harganya Rp 7000,-. Uang yang saya bawa Rp 12.000,-. Berapa kembalian yang saya terima?"<br />Rini :"Pasti Rp 5000,-"<br />Rina :"Salah"<br />Rini :"Ah engak ah"<br />Rina : "Pokoknya salah"<br />Rini :"Kan 12.000 - 7.000 = 5.000"<br />Rina : "Ya, itu benar hitunganmu, tapi prakteknya ga kaya gitu"<br />Rini :"Terus kamu dapat kembalian berapa?"<br />Rina :"Rp 3.000,-"<br />Rini :"Ih rugi dunk, kok kamu mau?"<br />Rina :"Ya mau, wong hitungannya benar kok"<br />Rini:"Kok bisa benar gimana?"<br />Rina :"Saya kan punya Rp 12.000,-. Berhubung harganya Rp 7000,- saya kan tidak perlu serahin semua uang, saya cukup serahin Rp 10.000,-. Jadi kembaliannya Rp 3.000,-"<br />Rini :"Benar juga ya"omsonhttp://www.blogger.com/profile/17839095311693084453noreply@blogger.com0tag:blogger.com,1999:blog-6675200891311211703.post-10710651623421205182011-06-26T15:35:00.007+07:002011-06-27T14:02:01.028+07:00Cara Lain Menyelesaikan Persamaan KuadratPada dasarnya ada 3 cara dalam menyelesaikan persamaan kuadrat, yaitu memfaktorkan, melengkapkan kuadrat sempurna dan berikutnya dengan rumus ABC. Pernahkah terpikir oleh anda untuk menggunakan cara lain dalam menyelesaikan persamaan kuadrat.<br /><br />Cara lain yang pernah saya buat adalah metoda substitusi<br />Jika bentuk umum persamaan kuadrat adalah<br />ax<sup>2</sup> + bx + c = 0<br />maka kita tinggal mensubstitusikan<br /><a href="http://www.codecogs.com/eqnedit.php?latex=x=y-%5Cfrac%7Bb%7D%7B2a%7D" target="_blank"><img src="http://latex.codecogs.com/gif.latex?x=y-%5Cfrac%7Bb%7D%7B2a%7D" title="x=y-\frac{b}{2a}" /></a><br />Contoh :<br />x<sup>2</sup>+2x - 8 = 0<br />misalkan<br /><a href="http://www.codecogs.com/eqnedit.php?latex=x=y-%5Cfrac%7Bb%7D%7B2a%7D=y-%5Cfrac%7B2%7D%7B2.1%7D=y-1" target="_blank"><img src="http://latex.codecogs.com/gif.latex?x=y-%5Cfrac%7Bb%7D%7B2a%7D=y-%5Cfrac%7B2%7D%7B2.1%7D=y-1" title="x=y-\frac{b}{2a}=y-\frac{2}{2.1}=y-1" /></a><br />persamaan kuadrat menjadi :<br />x<sup>2</sup>+2x - 8 = 0<br />(y-1)<sup>2</sup>+2(y-1) - 8 = 0<br />y<sup>2</sup> - 2y + 1 + 2y - 2 - 8 = 0<br />y<sup>2</sup> = 9<br /><a href="http://www.codecogs.com/eqnedit.php?latex=y=%5Cpm%20%5Csqrt%7B9%7D=%5Cpm%203" target="_blank"><img src="http://latex.codecogs.com/gif.latex?y=%5Cpm%20%5Csqrt%7B9%7D=%5Cpm%203" title="y=\pm \sqrt{9}=\pm 3" /></a><br /><br />x = y - 1 maka y = x + 1 sehingga persamaan menjadi :<br /><a href="http://www.codecogs.com/eqnedit.php?latex=x@plus;1=\pm 3" target="_blank"><img src="http://latex.codecogs.com/gif.latex?x+1=\pm 3" title="x+1=\pm 3" /></a><br /><a href="http://www.codecogs.com/eqnedit.php?latex=x=-1\pm 3" target="_blank"><img src="http://latex.codecogs.com/gif.latex?x=-1\pm 3" title="x=-1\pm 3" /></a><br />x = -1 + 3 atau x = -1 - 3<br />x = 2 atau x = -4omsonhttp://www.blogger.com/profile/17839095311693084453noreply@blogger.com2tag:blogger.com,1999:blog-6675200891311211703.post-1897773166056268932011-05-19T23:36:00.003+07:002011-05-19T23:44:31.084+07:00Tahukah Anda?1<sup>3</sup>+2<sup>3</sup>=(1+2)<sup>2</sup><br />1<sup>3</sup>+2<sup>3</sup>+3<sup>3</sup>=(1+2+3)<sup>2</sup><br />1<sup>3</sup>+2<sup>3</sup>+3<sup>3</sup>+4<sup>3</sup>=(1+2+3+4)<sup>2</sup><br />1<sup>3</sup>+2<sup>3</sup>+3<sup>3</sup>+4<sup>3</sup>+5<sup>3</sup>=(1+2+3+4+5)<sup>2</sup><br />1<sup>3</sup>+2<sup>3</sup>+3<sup>3</sup>+4<sup>3</sup>+5<sup>3</sup>+6<sup>3</sup>=(1+2+3+4+5+6)<sup>2</sup><br />1<sup>3</sup>+2<sup>3</sup>+3<sup>3</sup>+4<sup>3</sup>+5<sup>3</sup>+6<sup>3</sup>+7<sup>3</sup>=(1+2+3+4+5+6+7)<sup>2</sup><br />1<sup>3</sup>+2<sup>3</sup>+3<sup>3</sup>+4<sup>3</sup>+5<sup>3</sup>+6<sup>3</sup>+7<sup>3</sup>+8<sup>3</sup>=(1+2+3+4+5+6+7+8)<sup>2</sup><br /><br />dan seterusnyaomsonhttp://www.blogger.com/profile/17839095311693084453noreply@blogger.com0tag:blogger.com,1999:blog-6675200891311211703.post-53916061848916143802011-03-16T00:06:00.004+07:002011-03-16T04:05:49.634+07:00Triple pythagorasJika x, y, dan r merupakan sisi-sisi segitiga dan memenuhi persamaan x<sup>2</sup> + y<sup>2</sup> = r<sup>2</sup> maka segitiga tersebut pastilah siku-siku, dan dikatakan x, y, dan z adalah tripel pythagoras.<br /><br />contoh-contoh bilangan yang mengandung triple pythagoras adalah sbb :<br />3, 4, 5<br />6, 8, 10<br />5, 12, 13<br />dan sebagainya<br /><br />Untuk mencari triple pythagoras yang lain maka kita bisa menggunakan rumus-rumus sebagai berikut :<br />x = a<sup>2</sup>-b<sup>2</sup><br />y = 2ab<br />r = x<sup>2</sup> + y<sup>2</sup><br />dengan ketentuan a > b<br /><br />Kita bisa memakai rumus tersebut sbb :<br /><br /><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgY30Koe6nTCnTn__Q4cJ0VXA-7io5u3MHk5ZzG3hWVfSrLjLbOmSaUz-dePcbwBw9D51LgVo5G2Sed_BYNxAzw_7CujlqyUDColXOjTNYY_s8SLMEbZU7xwgWbPP3ko0qreY-9LLBypHw/s1600/triple+pythagoras.jpg"><img style="float: left; margin: 0pt 10px 10px 0pt; cursor: pointer; width: 402px; height: 256px;" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgY30Koe6nTCnTn__Q4cJ0VXA-7io5u3MHk5ZzG3hWVfSrLjLbOmSaUz-dePcbwBw9D51LgVo5G2Sed_BYNxAzw_7CujlqyUDColXOjTNYY_s8SLMEbZU7xwgWbPP3ko0qreY-9LLBypHw/s200/triple+pythagoras.jpg" alt="" id="BLOGGER_PHOTO_ID_5584415684149931810" border="0" /></a>omsonhttp://www.blogger.com/profile/17839095311693084453noreply@blogger.com6tag:blogger.com,1999:blog-6675200891311211703.post-2806204913845049532010-09-08T09:27:00.002+07:002010-09-08T09:37:20.199+07:00Dikali nolMengapa setiap bilangan yang dikali nolsama dengan nol?<br /><br />Pengertian dikali:<br />Misalnya anda ingin memahami mengama 5 dikali 100 sama dengan 500?<br /><br />Anggap saya punya tas yang penuh berisi koin 100 an. Jika anda saya izinkan mengambil uang dari tas saya, dan setiappengambilan hanya boleh mengambil 1 koin, maka banyaknya uang yang dioeroleh tergantung dari berapa kali anda mengambilokoin. Misalnya jika anda mengambil 5 kali maka uang yang anda cdapat adalah 500.<br /><br />Jika mengambil 10 kali,uang yang didapat sama dengan 10 x 100 =1000<br />Jika mengambil 5 kali,uang yang didapat sama dengan 5 x 100 = 500<br />mengambil 2 kali,uang yang didapat sama dengan 2 x 100 = 200<br />mengambil 1 kali,uang yang didapat sama dengan 1 x 100 = 1000<br /><br />Jika anda mengambil 0 kali (tidak mengambil) otomastis anda tidak dapat uang, atau uang anda = 0<br /><br />jadi 0 x 100 = 0<br /><br />Dengan cara yang sama anda bisa mengganti 100 dengan bilangan lain, dan hasilnya pasti nol.omsonhttp://www.blogger.com/profile/17839095311693084453noreply@blogger.com3tag:blogger.com,1999:blog-6675200891311211703.post-27076153621805863752010-08-12T06:38:00.001+07:002010-08-12T06:40:37.678+07:00Ke Bulan Pakai KertasSaat itu ada orang bermimpi ke bulan dengan ,memakai kertas karton. Caranya ? Jika kertas karton itu tebalnya 1 mm maka anda hanya perlu melipat kertas tersebut 43 kali. Dengan cara itu maka lipatan anda akan sampai ke bulan. Kok bisa?<br /><br />Penjelasannya adalah sebagai berikut :<br />Yang harus kita sepakati adalah setiap kertas yang dilipat, ketebalannya akan menjadi dua kali lipatnya. OK? Kalau tidak percaya coba saja. Karton yang tebalnya 1 mm jika dilipat maka ketebalannya menjadio 2 mm. Dengan demikian kita akan mendapati karton-karton tersebut akan menjadi sebagai berikut. <br />Karton semula = 1 mm<br />Dilipat sekali menjadi 2 mm<br />Dilipat 2 kali menjadi 4 mm<br />Dilipat 3 kali menjadi 8 mm<br />Dilipat 4 kali menjadi 16 mm<br />Dilipat 5 kali menjadi 32 mm<br />Dilipat 6 kali menjadi 64 mm<br />Dilipat 7 kali menjadi 128 mm<br />Dilipat 8 kali menjadi 256 mm<br />Dilipat 9 kali menjadi 512 mm<br />…….<br />Dan seterusnya.<br />Perhatikan bahwa bilangan-bilangan tersebut memenuhi 2<sup>n</sup><br />Dilipat 7 kali menjadi 128 mm = 2<sup>7</sup><br />Dilipat 8 kali menjadi 256 mm = 2<sup>8</sup><br />Dilipat 9 kali menjadi 512 mm = 2<sup>9</sup><br />Dilipat 10 kali menjadi 1024 mm = 2<sup>10</sup><br />….<br />Dan seterusnya.<br />Jika dilipat 43 kali akan menjadi <br />2<sup>43</sup> = 8796093022208<br /><br />Woow nilai 87.96.093.022.208 mm tentunya setara dengan 87.96.093 km. Tentu saja lebih besar dibandingkan jarak bumi dan bulan.omsonhttp://www.blogger.com/profile/17839095311693084453noreply@blogger.com1tag:blogger.com,1999:blog-6675200891311211703.post-76482512181629759132010-07-31T06:45:00.001+07:002010-07-31T07:06:42.651+07:00Cerita Papan Catur<meta equiv="Content-Type" content="text/html; charset=utf-8"><meta name="ProgId" content="Word.Document"><meta name="Generator" content="Microsoft Word 12"><meta name="Originator" content="Microsoft Word 12"><link rel="File-List" href="file:///C:%5CUsers%5Cuser%5CAppData%5CLocal%5CTemp%5Cmsohtmlclip1%5C01%5Cclip_filelist.xml"><link rel="themeData" href="file:///C:%5CUsers%5Cuser%5CAppData%5CLocal%5CTemp%5Cmsohtmlclip1%5C01%5Cclip_themedata.thmx"><link rel="colorSchemeMapping" href="file:///C:%5CUsers%5Cuser%5CAppData%5CLocal%5CTemp%5Cmsohtmlclip1%5C01%5Cclip_colorschememapping.xml"><!--[if gte mso 9]><xml> <w:worddocument> <w:view>Normal</w:View> <w:zoom>0</w:Zoom> <w:trackmoves/> <w:trackformatting/> 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mso-ascii-theme-font:minor-latin; mso-fareast-font-family:"Times New Roman"; mso-fareast-theme-font:minor-fareast; mso-hansi-font-family:Calibri; mso-hansi-theme-font:minor-latin; mso-bidi-font-family:"Times New Roman"; mso-bidi-theme-font:minor-bidi;} </style> <![endif]--> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><o:p> </o:p></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;">Ada suatu cerita. Ketika itu ada seorang pemuda yang cukup bijaksana. Kebetulan dia telah berjasa kepada rajanya. Sang raja akhirnya berkata. Wahai pemuda, aku ingin memberikan hadiah kepadamu. Hadiah apa yang kamu inginkan. Sang pemuda menjawab ,”Wahai rajaku, aku hanya menginginkan butiran jagung. Saya mohon engkau sendiri yang meletakkan butiran jagung itu pada papan catur.<span style=""> </span>Papan catur ini ada 64 kotak. Pada kotak pertama ditaruh satu butir jagung, pada kotak kedua sebanyak 2 kali kotak pertama, pada kotak ketiga 2 kali kotak kedua, pada kotak keempat 2 kali kotak ketiga begitu seterusnya sampai kotak yang ke 64.”</p><p class="MsoNormal" style="margin-bottom: 0.0001pt;">
<br /></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><o:p> </o:p></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;">Sang raja berkata,”Itu saja yang kamu inginkan?”<span style=""> </span>Sang pemuda menjawab iya. Kata raja,”Kamu ini permintaanmu sederhana sekali, padahal jika kamu minta apapun aku akan memenuhi.” Sang pemuda berkata ,”Ya baginda, saya hanya menginginkan itu saja. Akan tetapi kalau baginda tidak bisa memenuhi maka baginda harus menyerahkan istana ini kepada hamba.”</p><p class="MsoNormal" style="margin-bottom: 0.0001pt;">
<br /></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;">Sang raja berkata, “Baik, kalau itu saya pasti bisa.” Sang pemuda membuat kesepakatan dengan raja. Sang pemuda menghendaki perjanjian tertulis, jika raja tidak bisa memenuhi permintaannya maka isatana akan diberikan.” Sang raja setuju.</p><p class="MsoNormal" style="margin-bottom: 0.0001pt;">
<br /></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><o:p> </o:p></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;">Akhirnya raja mulai mengumpulkan biji jagung</p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;">Pada kotak pertama diisi 1</p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;">Pada kotak ke 2 diisi 2</p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;">Pada kotak ke 3 diisi 4</p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;">Pada kotak ke 4 diisi 8</p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;">Pada kotak ke 5 diisi 16</p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;">Pada kotak ke 6 diisi 32</p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;">Pada kotak ke 7 diisi 64</p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;">Pada kotak ke 8 diisi 128</p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;">Pada kotak ke 9 diisi 256</p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;">Pada kotak ke 10 diisi 512</p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;">Pada kotak ke 11 diisi 1024</p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;">Pada kotak ke 12 diisi 2048</p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;">Pada kotak ke 13 diisi 4096</p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;">Pada kotak ke 14 diisi 8192</p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;">…….</p><p class="MsoNormal" style="margin-bottom: 0.0001pt;">
<br /></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;">Raja mulai kelelahan menghitung di sini. Akan tetapi dia harus memenuhi permintaan si pemuda. Raja menghitung terus, sampai kelelahan tidak selesai-selesai.</p><p class="MsoNormal" style="margin-bottom: 0.0001pt;">
<br /></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><o:p> </o:p></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;">Akhirnya sang raja menyerah.<span style=""> </span>Karena memang yang dihitung begitu banyak. Jumlah yang seharusnya dia hitung adalah 18446744073709551615. Jika sang raja menghitung satu butir selama 1 detik maka sang raja harus menghitung butiran tersebut selama 5124095576030431 jam, atau sama dengan 213503982334601 hari <span style=""> </span>atau sama dengan 5849417355 tahun. Woow, waktu yang sangat lama. </p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;">Karena menyerah raja memberikan istananya kepada sang pemuda. </p> omsonhttp://www.blogger.com/profile/17839095311693084453noreply@blogger.com4