Wikipedia niawiki https://nia.wikipedia.org/wiki/Wikipedia:Olayama MediaWiki 1.47.0-wmf.24 first-letter Media Spesial Huhuo Sangoguna Huhuo zangoguna Wikipedia Huhuo Wikipedia Berkas Huhuo berkas MediaWiki Huhuo MediaWiki Templat Huhuo templat Fanolo Huhuo wanolo Kategori Huhuo kategori Portal Huhuo portal TimedText TimedText talk Modul Pembicaraan Modul Acara Huhuo Acara 0 3964 29960 28188 2026-10-22T14:32:08Z Ziv 2591 ([[c:GR|GR]]) [[File:Cardano.jpg]] → [[File:Gerolamo Cardano, self-portrait (1501–1576).jpg]] [[c:COM:FR#FR2|Criterion 2]] · More precise and not just a meaningless one word name 29960 wikitext text/x-wiki [[Berkas:Quadratic formula.svg|jmpl|Rumus [[persamaan kuadrat]] mengungkapkan solusi dari persamaan derajat dua <math>ax^2 + bx +c=0</math> dalam koefisien <math>a, b, c</math>, dimana <math>a</math> bukan nol.]] '''Aljabar''' andre (dari [[bahasa Arab]] الجبر ''"al-jabr"'' ma na la fo'eluaha ia lamane, fangowuloi mbagia si no tekiko ''("pengumpulan bagian yang rusak")''<ref name="oed">{{Cite web|url=http://www.oxforddictionaries.com/us/definition/english/algebra|title=algebra|website=Oxford English Dictionary|publisher=Oxford University Press|access-date=2017-02-21|archive-date=2013-12-31|archive-url=https://web.archive.org/web/20131231173558/http://www.oxforddictionaries.com/us/definition/english/algebra|dead-url=yes}}</ref>) ya'ia da'ö sambua mbagia wamahaö moroi ba [[Matematika]], sifao ba ''teori bilangan'', [[Geometri]] ba tola gö ba wo ''Analisis matematis''. No si no oi la'ila niha, aljabar ya'ia da'ö fangi'ila ba ngawalö dandra-dandra ''(simbol-simbol)'' matematika ba goi-goi hewisa wo fangamöi tandra-tandra ni'oguna'ö ba wangerai andrö;<ref>I. N. Herstein, ''Topics in Algebra'', "An algebraic system can be described as a set of objects together with some operations for combining them." p. 1, Ginn and Company, 1964</ref> aljabar no sambua wamahaö samarahu arakhagö fefu mbagia ba matematika.<ref>I. N. Herstein, ''Topics in Algebra'', "...it also serves as the unifying thread which interlaces almost all of mathematics." p. 1, Ginn and Company, 1964</ref> Baero da'ö, aljabar andre, no sambua dane-dane ba wangalui lala hewisa wo fosinöndra ngawalö ni'erai ba matetika simane ''abstraksi grup matematika'', ''Gelanggang matematika'' ba ''Medan matematika''. Idugu oya mbagia dane-dane moroi ba aljabar latötöi faoma ''aljabar elementer'', na aljabar ''abstrak'' latötöi faoma ''aljabar abstrak'' mazui ''aljabar modern''. Aljabar elementer sito'ölönia labali'ö soguna sibai ba wamahaö matematika, ''ilmu pengetahuan'', mazui ''teknik'', faoma ''aplikasi'' sifakhai ba ''kesehatan'' ba simanö göi ba wamahaö ''ekonomi''. Aljabar abstrak sambua duho ba matematika salaŵa ma'ifu ma ''tingkat lanjut'', ni'oguna'ö ya'ira si no ''profesional'' ba ere matematika. Aljabar elementer faehu ma'ifu faoma [[Aritmetika]] ba wangoguna'ö ''abstraksi'', simane mangoguna'ö hurufo ba wo salahini numero-numero si lö mu'ilana mazui tetehege tola ihalö oya zinöndra ''(nilai)''. Simane duma-duma, ba <math>x + 2 = 5</math> hurufo <math>x</math> lö mu'ila, ba hiza huku ''inversi'' tola la'oguna'ö ba wangalui sinöndra ma ''nilai'': <math>x=3</math>. Bakha ba ''(Ekivalensi massa-energi)'', hurufo <math>E</math> faoma <math>m</math> da'ö ni'oguna'ö tobali ''variabel'', ba hurufo <math>c</math> sitobali ''Konstanta (matematika)'', ''kecepatan cahaya'' bakha ba ''vakum''. Aljabar ibe'e lala ''metode'' ba wangalui wamagölösi ba hewisa ba wo faehagö ''rumus'' saoha (khöra ya'ira si no to'ölö mangoguna'ö ''konsep'' da'a) moroi na i'oguna'ö ''metode konvensional'', simane lasura fefu zinöndra faoma hurufo ma ngawua wehede. Ngawua wehede ''aljabar'' andre, oya sibai la'oguna'ö ba ngawalö lala sakhökhö. Ngawalö gangolifa ba dandrösa matematika ba aljabar abstrak da'ö nifotöi "aljabar", fehede da'a la'oguna'ö, simane ba wanötöi [[aljabar linear]] faoma topologi aljabar. Samösa gere matematika si no mamalua famareso ma ba [[li Indonesia]] lamane ''penelitian'' sanandrösa ba aljabar lakaoni faoma fanötöi; '''aljabarwan'''. == Etimologi == Fehede ''aljabar'' tefoli moroi ba Bahasa Arab {{Lang|ar|الجبر}} (''al-jabr'' molo'ö famo'eluaha moroi ba ngawua wehede andrö, eluahania ya'ia da'ö; "fangowuloi ngawalö zi no tekiko") fehede ma fanötöi da'a, tehalö moroi ba zi sambua mbuku nibe'e högö mbuku andrö ''Al Kitaab al muhtasar fii hisaab al jabr wa'l muqabaala''<ref>Ekkehard Kopp "[https://www.openbookpublishers.com/product/1279 Making up Numbers : A History of Invention in Mathematics] {{Webarchive|url=https://web.archive.org/web/20220407031354/https://www.openbookpublishers.com/product/1279 |date=2022-04-07 }}" Open Book Publishers</ref> nisura matematikawan ba astronom [[Persia]], [[Muḥammad bin Mūsā al-Khawārizmī|Al-Khwarizmi]]. Ngawua wehede da'a no möi ia ba bahasa Inggris ba abad si 15, he moroi ba [[Spanyol]], [[Italia]], mazui moroi ba Pertengahan Latin. Aljabar ba zi'oföna latötöi ia ''prosedur operasi pengaturan patah'' ma ''dislokasi tulang''. Eluahania ba matematika si'oföna sibai lasura ba abad 16. == Ngawalö geluaha "aljabar" == Ngawua wehede "aljabar" so lala wangoguna'ö ya'ia bakha ba matematika, sitobali fehede nitandrösaigö ba dandrösa mazui faoma ''kualifikasi''. * Sitobali fehede fanandrösa ba dandrösa ma ''kata tunggal'' si lö mangoguna'ö kata sandang, "aljabar" töi sambua wamahaö ba wamahaö matematika. * Sitobali fehede fanandrösa ba dandrösa ma ''kata tunggal'' mazui ba wanötöi ha wa'oya, "aljabar" ifaehagö wenaeta ma ''struktur'' matematika faoma fa'akhökhö, ba famo'eluaha satulö, te balazigö hewisa gamöta dödö zanura ya'ia. Sito'ölönia fenaeta da'a so khönia wamonönö ''(penambahan, perkalian, ba skalar perkalian)''. Ba ginötö zanura mangoguna'ö fehede "aljabar", lahaogö sambua ''subset'' moroi ba ''asumsi tambahan'' simane: ''asosiatif, komutatif, unital, mazui/faoma ''dimensi berhingga''. Bakha ba aljabar universal, fehede "aljabar" tefaolo ba ''generalisasi'' moroi ba ''konsep'' da'ö moroi yaŵa, tola manö; ''n-ary'' operasi. * Faoma ''kualifikasi'', so wa'afabö'ö sifagölö: ** Lö hadöi khönia ''kata sandang'', eluahania no föfö moroi ba aljabar, simane ''aljabar linier, aljabar elementer'' ma ''simbol-manipulasi'' goi-goi nifaha'ö ba naha wamaha'ö ma ''kursus sd'' matematika sitobali dane-dane moroi ba wamahaö ''pendidikan dasar'' faoma ''menengah'', mazui ''aljabar abstrak''. ** Sangoguna'ö ''kata sandang'', eluahania sambua duma-duma moroi ba ''struktur abstrak'', simane ''aljabar bentang'', ''aljabar asosiatif'', mazui ''aljabar operator verteks''. ** Fa'itaria dombua geluaha da'a la'oguna'ö ba wangerai sifagölö, simane bakha ba ''kalimat'': ''aljabar Komutatif adalah studi tentang gelanggang komutatif, yang merupakan aljabar komutatif atas bilangan bulat''. == Aljabar moroi ba Matematika == Aljabar andre tebörötaigö moroi ba wangerai sifagölö ba [[Aritmetika]], sangoguna'ö hurufo ba wo salahini numero. Da'a tola möi sitobali dane-dane moroi ba zindruhunia heŵa'ae lö lafaogö khönia numero-numero. Simane duma-dumania, ba ''persamaan kuadrat'' :<math>ax^2+bx+c=0,</math> <math>a, b, c</math> tola tobali ''bilangan'' gofu hadia manö(kecuali bahwa <math>a</math> tidak dapat bernilai <math>0</math>), ba rumus kuadrat andre tola la'oguna'ö wa'alio ba aoha ba wanöndra ''nilai-nilai'' moroi ''kuantitas'' <math>x</math> si lö mu'ila ma ba wofönui ''persamaan''. Rumus kuadrat la'oguna'ö ba wamatunö persamaan, ba aefa da'ö ba wanöndra hadia lala moroi ba ''persamaan'' da'ö. Molo'ö waö-waö, ba famahaö ba ginötö iada'a, famareso aljabar tebörötaigö moroi hewisa ba wamosinöndra ''persamaan'' simane ''persamaan kuadrat'' da'ö yaŵa. Aefa da'ö, oya wanofu-wanofu sito'ölö, simane; "hadia ''persamaan'' simane da'ö so lala wamosinöndra ya'ia?", "hawa'oya lala nitörö ba wangasiwai ngawalö wamosinöndra ''persamaan'' da'a?", "hadia zitola lafaehagö moroi ba lala wamosinöndra fanofu andrö?". Fanofu-fanofu simane da'a, möi ba wa'atohare ngawalö mbua wangera-ngera sifakhai ba ''bentuk, struktur'' faoma ''simetri''.<ref>{{Cite book|title=The Common Sense of Teaching Mathematics|last=Gattengo|first=Caleb|publisher=Educational Solutions Inc.|year=2010|isbn=978-0878252206}}</ref> ''Sifat-sifat struktural'' moroi ba ''objek-objek non-numerik'' ma si tenga numero, aefa da'ö awena la ''abstraksi'' ba wanutunö waö-waö ''struktur-struktur'' aljabar simane grup (matematika), ''gelanggang'' (matematika), faoma ''medan (matematika). Fatua lö abad si-16, matematika so ba zidombua ''subbidang'', ya'ia da'ö [[Aritmetika]] faoma [[Geometri]]. Heŵa'ae ba zi ha'uga ngawalö ''metode'', si no oföna mu'ehaogö, te da'ö ni'ila niha sitobali aljabar, ma fa'atohare aljabar faoma lö ara aefa da'ö, ''Kalkulus infinitesimal'' sitobali ''subbidang'' matematika ha ba gotalua abad si-16 ma abad si-17. Moroi ba dalu abad si-19, oya zibohou sifakhai ba matematika, nifaehagö moroi ba aritmetika faoma geometri, ba arakhagö fefu mangoguna'ö aljabar. Ba ginötö iada'a, aljabar no idugu sökhi irege oya wamahaö moroi ba matematika zangoguna'ö simane sitola mufaigi ba [[Klasifikasi Subjek Matematika]]<ref>{{Cite web|url=http://www.ams.org/mathscinet/msc/msc2010.html|title=2010 Mathematics Subject Classification|access-date=2014-10-05|archive-date=2014-06-06|archive-url=https://web.archive.org/web/20140606010248/http://www.ams.org/mathscinet/msc/msc2010.html|dead-url=yes}}</ref> hiza, lö hadöi he ha sambua moroi ba wenaeta si'oföna (dua digit entri) nifotöi ''aljabar''. Iada'a aljabar no irugi 08-sistem-sistem aljabar umum, 12-[[Medan (matematika)|teori medan]] dan [[Polinomial]], 13-[[Aljabar Komutatif]], 15-[[Aljabar Linear]] dan [[Aljabar Multilinear|multilinear]]; [[Matriks (matematika)|teori matriks]], 16-[[Aljabar Asosiatif]], 17-[[Aljabar tak-Asosiatif]], 18-[[Teori Kategori]]; [[Aljabar Homologis]], 19-[[Teori-K]], faoma 20-[[Teori Grup]]. Aljabar andrö la'oguna'ö göi ia molo'ö ''ekstensif'' bakha 11-[[Teori Bilangan]] faoma 14-[[Geometri Aljabar]]. == Waö-waö ''(Sejarah)'' == === Böröta aljabar ''(Sejarah awal aljabar)'' === [[Berkas:Image-Al-Kitāb al-muḫtaṣar fī ḥisāb al-ğabr wa-l-muqābala.jpg|jmpl|Nga'örö moroi ba mbuku ni sura [[Muḥammad bin Mūsā al-Khawārizmī|Al-Khwarizmi]] yang berjudul '''al-Kitab al-muḫtaṣar fī ḥisāb al-ğabr wa-l-muqābala''' (''Buku Ringkasan tentang Perhitungan dengan Pelengkapan dan Penyetimbangan'']] Ŵa'a moroi ba aljabar andre, tola mu'osisi'ö ofeta ba ginötö me Babilonia kuno,<ref>{{Cite book|title=A Concise History of Mathematics|url=https://archive.org/details/concisehistoryof0000stru_m6j1|last=Struik|first=Dirk J.|publisher=Dover Publications|year=1987|isbn=0-486-60255-9|location=New York}}</ref> samazaewe ma sangodödögö fangi'ila sifakhai ba ''sistem aritmetika lanjut'' ni'oguna'ö ba wangerai ''menurut gaya algoritme''. Bangsa Babilonia la odödögö ba wangi'ila ''rumus'' ba wangerai hewisa lala wanema li ma hewisa ba wangalui sinöndra moroi ba ngawalö zedöna la'erai ba ginötö iada'a, hiza ba wangalui sinöndrönania la'oguna'ö lala ''persamaan linear, persamaan kuadrat'', faoma ''persamaan taktentu''. Hiza, tenga fefu, so göi matematika Mesir Kuno ma ni'oguna'ö ira niha [[Mesir]] ba ginötö iada'a faoma Matematika Yunani ba matematika Tiongkok ba milenium 1 SM sito'ölönia lafosinöndra wanofu andrö faoma ''metode geometris'', simane si no lafaehagö bakha ba ''Papirus Matematika Rhind'', ''Elemen Euklides'', faoma ''Sembilan Bab mengenai Seni Matematika''. Karya geometris moroi Yunani, simane si no lasura bakha ''Elemen'', no lahenagö döla-döla dödö ma ''kerangka kerja'' ba wame'e ''rumus'' ma na'i manalu'i moroi ba zinöndra nitötöna irege da'a nigo'ö zato ba wangalui sinöndra, heŵa'ae da'a lö si tehöngö fatua lö si muledo ''Matematika Islam'' ba ''abad pertengahan''.<ref>{{harvnb|Boyer|1991}}</ref> Ba ginötö [[Plato]], matematika Yunani no so wa'atedou. Niha Yunani no la'ila nifotöi aljabar geometri, hiza moroi ba wamahaö andrö tefaehagö awa ''suku-suku matematika'' tetötöi ia moroi ba ''sisi-sisi'' ba ''objek geometri'', sito'ölönia simane famadölö ma ''garis'', sangokhögö hurufo-hurufo sifakhai ba wangerai.<ref name=citeboyer>{{Harv|Boyer|1991|loc="Europe in the Middle Ages" p. 258}} "In the arithmetical theorems in Euclid's ''Elements'' VII-IX, numbers had been represented by line segments to which letters had been attached, and the geometric proofs in al-Khwarizmi's ''Algebra'' made use of lettered diagrams; but all coefficients in the equations used in the ''Algebra'' are specific numbers, whether represented by numerals or written out in words. The idea of generality is implied in al-Khwarizmi's exposition, but he had no scheme for expressing algebraically the general propositions that are so readily available in geometry."</ref> [[Diofantus]] (abad si-3 Masehi) ya'ia da'ö samösa gere matematika ma laŵa'ö ba wehede lamane Matematikawan Yunani moroi ba mbanua ''Iskandariyah'' ba si no manura oya mbuku nifotöi ''[[Arithmetica]]''. Sinura-sinura andre fakhai ia ba wamosinöndra ngawalö wanofu moroi ba ''persamaan aljabar'',<ref>{{Cite book|url=https://books.google.com/?id=gZ2Us3F7dSwC&pg=PA34&dq#v=onepage&q=&f=false|title=A History of Elementary Mathematics – With Hints on Methods of Teaching|last=Cajori|first=Florian|year=2010|isbn=1-4460-2221-8|page=34|author-link=Florian Cajori}}</ref> ba hiza no möi famarou ba si ba wangi'ila nifotöi ''persamaan Diofantin'' bakha ba ''teori bilangan''. Sito'ölö nifalui ba ginötö iada'a na la faedogö moroi ba ni fa'ema ba da'ö moroi yaŵa, no manandrösa ia ba Matematikawan bangsa Persia, Muḥammad ibn Mūsā al-Khwārizmī (kira-kira 780–850). Aefa da'ö isura mbuku nibe'enia högö mbuku andrö: ''Buku Ringkasan tentang Perhitungan dengan Pelengkapan dan Penyetimbangan'', nifangamöi aljabar sitobali famahaö matematika ma ''disiplin'' matematika si lö mombalazigö geometri faoma aritmetika.<ref>{{Cite journal|title=Al Khwarizmi: The Beginnings of Algebra|author=Roshdi Rashed|publisher=Saqi Books|date=November 2009|isbn=0-86356-430-5|ref=harv|postscript= }}</ref> Matematikawan ba ginötö periode Helenistik, sotöi Heron moroi ba Iskandariyah faoma Diofantus,<ref>{{cite web|url=http://library.thinkquest.org/25672/diiophan.htm |title=Diophantus, Father of Algebra |publisher= |accessdate=2014-10-05 |deadurl=yes |archiveurl=https://web.archive.org/web/20130727040815/http://library.thinkquest.org/25672/diiophan.htm |archivedate=2013-07-27 |df= }}</ref> simanö göi Matematikawan India simane Brahmagupta manohugö sito'ölö lafalua ma tradisi-tradisi Mesir faoma Babilonia, heŵa'ae ''Arithmetica''-nia Diophantus faoma ''Brāhmasphuṭasiddhānta''-nia Brahmagupta so ba dadaoma sabölö alaŵa.<ref>{{cite web|url=http://www.algebra.com/algebra/about/history/|title=History of Algebra|publisher=|accessdate=2014-10-05}}</ref> Simane duma-dumania, lala wangalui aritmetika si'oföna ''(termasuk solusi nol dan negatif)'' soguna ba ''persamaan kuadrat'', hulö simane si no ifaehagö Brahmagupta ba mbukunia ''Brahmasphutasiddhanta''. Aefa da'ö, Matematikawan Persia faoma Arab no la'odödögö hewisa ba wondrou'ö metode-metode aljabar ba wogamö wetaro sabölö alaŵa. Heŵa'ae Diofantus faoma bangsa Babilonia asese sibai mangoguna'ö metode ''ad hoc'' nifaehagö ni'oguna'öra ba wangalui sinöndra persamaan-persamaan, bua wangera-ngera Al-Khwarizmi no tobali dane-dane si sökhi. No i'asiwai persamaan linear faoma kuadrat si lö mangoguna'ö ''simbolisme'' aljabar, ''bilangan negatif'', mazui ''nol'', börö da'ö, andrö wa si lötolalö'ö i faehusi ngawalö persamaan.<ref name="Meri2004">{{cite book|author=Josef W. Meri|title=Medieval Islamic Civilization|url=https://books.google.com/books?id=H-k9oc9xsuAC&pg=PA31|accessdate=25 November 2012|year=2004|publisher=Psychology Press|isbn=978-0-415-96690-0|page=31}}</ref> Ba wamahaö aljabar andre, no la'ila niha ya'ia da'ö fakhai ia ba wamagölö ngawalö wangerai ma lamane ''teori persamaan'', Matematikawan Yunani, sotöi Diofantus molo'ö wangi'ila niha sito'ölö, lakaoni ia faoma ere aljabar ma abölö enahöi wanötöi ya'ia, ya'ia da'ö: "bapak aljabar" ba hiza ba wa'anumalö ginötö, oya wabu'a-bu'asa sifakhai khö gere tanöbö'ö simane sotöi ''al-Khwarizmi'', sangahaogö fangi'ila nifotöi: ''al-jabr'', tola göi lakaoni ia faoma ''ama aljabar''.<ref>{{cite book |first=Carl B. |last=Boyer |title=A History of Mathematics |url=https://archive.org/details/historymathemati00boye_328 |edition=Second |location= |publisher=Wiley |year=1991 |pages=[https://archive.org/details/historymathemati00boye_328/page/n197 178], 181 |isbn=0-471-54397-7 }}</ref> Ya'ira sifao khö zöndra Diofantus zanguŵa'ö wa aljabar nisöndra bakha ba ''Al-Jabr'' tehude ma'ifu moroi ba nisöndra ba ''Arithmetica'' ba börö da'ö ''Arithmetica'' no hulö ni'adogo'ö, na ''Al-Jabr'' ma'afefu no hulö ''retoris''.<ref>{{cite book |first=Carl B. |last=Boyer |title=A History of Mathematics |url=https://archive.org/details/historymathemati00boye_328 |edition=Second |location= |publisher=Wiley |year=1991 |page=[https://archive.org/details/historymathemati00boye_328/page/n247 228] |isbn=0-471-54397-7 }}</ref> Na ya'ira sifao ba zöndra Al-Khwarizmi sifakhai ma samaehagö nifotöi metode "[[reduksi (matematika)|reduksi]]" faoma "penyetimbangan" ''(transposisi suku-suku yang diambil ke ruas lain suatu persamaan, yaitu, pencoretan suku-suku yang memiliki [[variabel (matematika)|variabel]] dan [[eksponensiasi|pangkat]] sama pada ruas lain suatu persamaan)'', nituturu ma ni aekhugö moroi ba ''al-jabr'' ba mböröta,<ref name=Boyer-229>{{Harv|Boyer|1991|loc="The Arabic Hegemony" p. 229}} "It is not certain just what the terms ''al-jabr'' and ''muqabalah'' mean, but the usual interpretation is similar to that implied in the translation above. The word ''al-jabr'' presumably meant something like "restoration" or "completion" and seems to refer to the transposition of subtracted terms to the other side of an equation; the word ''muqabalah'' is said to refer to "reduction" or "balancing" – that is, the cancellation of like terms on opposite sides of the equation."</ref> ba no ifaehagö molo'ö wangi'ilania ngawalö wamosinöndra sifakhai ba ''persamaan kuadrat'' ma fangerai ba numeronia samösa,<ref>{{Harv|Boyer|1991|loc="The Arabic Hegemony" p. 230}} "The six cases of equations given above exhaust all possibilities for linear and quadratic equations having positive root. So systematic and exhaustive was al-Khwarizmi's exposition that his readers must have had little difficulty in mastering the solutions."</ref> i'abe'e'ö dödö niha molo'ö ngawalö zi so khönia sitobali dane-dane ''geometris'', sifao ba wangoguna'ö aljabar sitonali sambua dane-dane ma ''disiplin'' nifaehagö molo'ö fa'ato'ölö zangoguna'ö ya'ia.<ref>Gandz and Saloman (1936), ''The sources of al-Khwarizmi's algebra'', Osiris i, p. 263–277: "In a sense, Khwarizmi is more entitled to be called "the father of algebra" than Diophantus because Khwarizmi is the first to teach algebra in an elementary form and for its own sake, Diophantus is primarily concerned with the theory of numbers".</ref> Aljabar andre, lö sa'ae i'angeragö ngawalö wangalui sinöndra ba ngawalö wanofu, hiza sambua ''eksposisi'' ba mböröta sifakhai ba ''suku-suku primitif'' ba ngawalö wangorudugö ba wamagölösi sinöndra, ba tetohugö ia molo'ö ''eksplisit'' ba wangalui sinöndra satulö". Ya'ia göi no i'alui wamagölösi ma ''persamaan'' soguna khönia samösa ba lala wangerai sito'ölö simane nifalua niha bö'ö, ba hiza no iföfögö ba wangalui sambua zinöndra si lö hadöi amozua ma abölö sökhi wanötöi ya'ia ba [[li Indonesia]] lamane: ''tak terbatas''.<ref name=Rashed-Armstrong>{{Cite book | last1=Rashed | first1=R. | last2=Armstrong | first2=Angela | year=1994 | title=The Development of Arabic Mathematics | publisher=[[Springer Science+Business Media|Springer]] | isbn=0-7923-2565-6 | oclc=29181926 | pages=11–2 | ref=harv | postscript= }}</ref> Matematikawan Persia tanöbö'ö, sotöi ''Umar Khayyām'' si no latema ngawalö wangi'ila ba woföfögö dane-dane ''geometri aljabar'' ba ya'ia samösa ere samaehagö ''solusi geometris umum'' ba wangerai famagölösi ''fungsi kubik'' ma ''persamaan kubik''. Buku nisurania nibe'e högö mbuku: ''Risalah tentang Peragaan Soal-Soal Aljabar'' (1070), so honogöigö ngawalö ''prinsip-prinsip aljabar'', si no angolifa moroi ba zifakhai ba Matematika, hiza Matematika Persia andre, sindruhunia la fa'ohe'ö ba Eropa.<ref>[[#refmathmaster|Mathematical Masterpieces: Further Chronicles by the Explorers]], p. 92</ref> Matematikawan Persia tanöbö'ö, simane ''Sharaf al-Din al-Tusi'', sanöndra lala wamosinöndra aljabar faoma numero ba ngawalö zifakhai ba wamagölösi fangerai kubik ma ''persamaan kubik.''<ref>{{MacTutor|id=Al-Tusi_Sharaf|title=Sharaf al-Din al-Muzaffar al-Tusi}}</ref> Dia juga mengembangkan konsep mengenai [[fungsi (matematika)|fungsi]].<ref>{{Cite journal|last=Victor J. Katz|first=Bill Barton|title=Stages in the History of Algebra with Implications for Teaching|journal=Educational Studies in Mathematics|publisher=[[Springer Science+Business Media|Springer Netherlands]]|volume=66|issue=2|date=October 2007|doi=10.1007/s10649-006-9023-7|pages=185–201 [192]|last2=Barton|first2=Bill|ref=harv|postscript= }}</ref> Matematikawan India, sotöi [[Mahavira (matematikawan)|Mahavira]] faoma [[Bhāskara II]], Matematikawan Persia [[Al-Karaji]],<ref name="Boyer al-Karkhi ax2n">{{Harv|Boyer|1991|loc="The Arabic Hegemony" p. 239}} "Abu'l Wefa was a capable algebraist as well as a trigonometer.&nbsp;... His successor al-Karkhi evidently used this translation to become an Arabic disciple of Diophantus – but without Diophantine analysis!&nbsp;... In particular, to al-Karkhi is attributed the first numerical solution of equations of the form ax²ⁿ + bxⁿ = c (only equations with positive roots were considered),"</ref> faoma Matematikawan Tiongkok, [[Zhu Shijie]], mangasiwai ngawalö wangerai sifakhai ba ''persamaan kubik'', ''persamaan kuartik ma kuartik'', ''persamaan kuintik'', faoma famagölösi ''polinomial'' sasese laŵa'ö ''berorde'' abölö alaŵa mangoguna'ö ''metode numerik''. Ba abad ke-13, fangasiwaisi ''persamaan kubik'' nifalua [[Fibonacci]] ya'ia da'ö sosalahini moroi ba mböröta wangi'ila aljabar Eropa. [[Abū al-Ḥasan ibn ʿAlī al-Qalaṣādī]] (1412–1486) manga'i mbosi-bosi si'oföna sinumalö ba wangi'ila ngawalö dandra aljabar, ma ba [[li Indonesia]] lamane: "langkah-langkah pertama menuju perkenalan simbolisme aljabar". Hiza, no göi i'erai ∑''n''², ∑''n''³ ba i'oguna'ö wamahatö (suksesif) ba wotatugöi ''akar kuadrat''.<ref>{{Cite web|url=http://www-history.mcs.st-andrews.ac.uk/Biographies/Al-Qalasadi.html|title=Al-Qalasadi biography|website=www-history.mcs.st-andrews.ac.uk|access-date=2017-10-17}}</ref> Ba ginötö [[Islam]] so ma'ifu we'amöi furi, dunia Eropa hi no wa'amanöi wangi'ila ba matematika. Ba da'ö aljabar andre, tedou wa'amozawili ba zi sagörö ulidanö. === Waö-waö Sibohou Aljabar ''(Sejarah modern aljabar)'' === [[Berkas:Gerolamo Cardano, self-portrait (1501–1576).jpg|jmpl|258x258px|Matematikawan Italia [[Girolamo Cardano]] mamatumbu'ö lala wangalui sinöndra kubik ma ba wehede matematika asese lamane ''fungsi kubik'' faoma ''fungsi kuadrat'' nisurania bakha ba mbuku ba ndröfi si 1545 nibe'e högö mbuku: ''[[Ars Magna (Gerolamo Cardano)|Ars magna]]''.]] Karya [[François Viète]] sifakhai ba aljabar sibohou, ba gafuriata abadi si-16 no sambua bosi soguna sibai sinumalö ba wangoguna'ö ''aljabar modern''. Ba ndröfi si 1637, [[René Descartes]] mamatumbu'ö ''La Géométrie'', sifao ba wanöndra sambua lala nifotöi ''geometri analitis'' ba hiza ifaehagö göi ba niha sato nifotöi ''notasi aljabar modern''. Salua tanöbö'ö zifakhai ba wondrou'ö aljabar ya'ia da'ö ba wangasiwaisi ''aljabar'' si no oi la'ila niha sato ba zi sagörö ulidanö meluo da'ö ba wangalui hewisa wamagölösi nifotöi ''persamaan kubik'' faoma ''kuartik'', nifanöi wangi'ila ba ginötö abad si-16. Bua wangera-ngera sifakhai ba ''determinan'' nifanöi moroi khö matematikawan Jepang sotöi [[Seki Kōwa]] ba abad si-17, no igo'ö ia sotöi [[Gottfried Leibniz]] ba wa'omasinia samösa, eluahania tenga ba döi gangowuloa ilau da'ö ba zi 10 fakhe fa'ara aefa da'ö, sitobali ohitö dödö ba wangalui hewisa zinöndra ba ''sistem persamaan linear simultan'' sangoguna'ö ''Matriks (matematika)''. Hiza, Gabriel Cramer ilau mohalöŵö sifakhai ba matriks faoma determinan ba abad si-18. Permutasi i'alui lala hewisa wangasiwaisi ngawalö wanofu sotöi [[Joseph-Louis de Lagrange]] ba mbuku nisurania me döfi si 1770 nibe'enia högönia ''Réflexions sur la résolution algébrique des équations'' (Refleksi ba resolusi aljabar ba zisambua wamagölösi), si'oföna sibai ba wotatugöi hewisa lala wangasiwaisi fanofu zisambua ''persamaan aljabar'', hiza no ifaehagö [[resolven (teori Galois)|resolven Lagrange]]. Paolo Ruffini ya'ia da'ö niha si'oföna sibai sofanöi teori moroi ba ''grup permutasi'', hulö zimane nifalui ndra matematikawan ba mböröta, tola göi ba wangalui sinöndra ''persamaan aljabar''. == Angolifa Matematika Sangoguna'ö Ngawua Wehede Aljabar ba Döira ''(Bidang matematika dengan kata aljabar pada nama mereka)'' == Ba zi ha'uga gangolifa matematika si no ahono ba zifakhai ba Aljabar ''(klasifikasi aljabar abstrak)'' mamaogö fehede aljabar ba döira; simane ''aljabar linear'' ha sambua duma-duma da'a. Ba hiza so göi zi lö'ö fao döira, simane: ''teori grup'', ''teori gelanggang'', faoma ''Medan (matematika)'' ma ''teori bidang''. Ba da'a tou so gangolifa matematika sangoguna'ö fehede "aljabar" ba döira. * '''Aljabar elementer''', angolifa moroi ba aljabar sito'ölö nifahaö ba ''perkuliahan dasar Matematika''. * '''Aljabar abstrak''', hiza ba ''struktur aljabar'' simane ''Grup (matematika)'', ''Gelanggang (matematika)'' faoma ''Medan (matematika)'' nifo'eluaha faoma ''sistem aksioma'' ma asese lamane ''aksiomatis''. * '''Aljabar linear''', hiza no oya dandra-tandra moroi ba ''persamaan linear'', ''Ruang vektor'' faoma ''Matriks (matematika)'' nitema niha ba wamahaö. * '''Aljabar Boolean''', angolifa aljabar samahatö ''komputasi'' faoma sahatö ba ''nilai kebenaran'' nifaehagö faoma fehede; ''atulö/benar'' faoma ''falasa/salah''. * '''Aljabar komutatif''', famahaö sifakhai ba ''gelanggang komutatif''. * '''Komputasi simbolik''' mazui '''Aljabar komputer''', famatöfaigö metode-metode aljabar simane; ''algoritme'' faoma ''program komputer''. * '''Aljabar homologis''', famahaö fenaeta ''aljabar'' sitobali dane-dane ba wangi'ila ''ruang topologi''. * '''Aljabar semesta''', tandra-tandra sito'ölö ba zi'ogötö'ö fenaeta ''aljabar'' nitema ba wamahaö. * '''Teori bilangan aljabar''', tandra-tandra ''bilangan'' nitema wamahaö molo'ö wamaigi moroi ba aljabar. * '''Geometri aljabar''', sambua gangolifa moroi ba geometri, ba wamaigi moroi meföna ba wotatugöi ''kurva'' faoma tete sitobali lala ba wangalui sinöndra wamagölösi ''persamaan polinomial''. * '''Kombinatorika aljabar''', da'a sambua ''metode-metode'' aljabar ni'oguna'ö ba wamaigi fanofu ''kombinatorika''. * '''Aljabar relasional''': sambua gangowuloa moroi ba ''relasi berhingga'' sitolumö barö zisambua ''operator''. Oya wenaeta matematika nifotöi '''aljabar''': * '''Aljabar atas medan''' mazui töra simane ''aljabar di atas gelanggang''.<br>Oya nahia ba aljabar ba dete danölafa mazui yaŵa ba nahia sangokhögö töi nitatugoi ''(spesifik)'': ** '''Aljabar asosiatif''' ** '''Aljabar takasosiatif''' ** '''Aljabar Lie''' ** '''Aljabar Hopf''' ** '''Aljabar-C*''' ** '''Aljabar simetri''' ** '''Aljabar eksterior''' ** '''Aljabar tensor''' * Ba ''teori ukuran'', ** '''Aljabar sigma''' ** Algebra ba dete ''Medan himpunan'' * Bakha ba ''teori kategori'' ** '''''Aljabar-F''''' faoma '''''Koaljabar-F''''' ** '''''Aljabar-T''''' * Bakha ba ''logika'', ** '''Aljabar relasi''', aljabar Boolean beresidu, ni'ebolo'ö faoma fa'abua nifotöi faoma ''konvers''. ** '''Aljabar Boolean (struktur)''', '''kisi distributif''' faoma '''kisi komplemen''' na la'adogo'ö ia lamane '''komplemen'''. ** '''Aljabar Heyting''' == Aljabar elementer == [[Berkas:algebraic equation notation.svg|jmpl|ka|Notasi ekspresi aljabar:<br/>&nbsp; 1 – pangkat (''power'')<br/>&nbsp; 2 – koefisien<br/>&nbsp; 3 – suku (''term'')<br/>&nbsp; 4 – operator<br/>&nbsp; 5 – suku konstanta<br/>&nbsp; ''x'' ''y'' ''c'' – variabel/konstanta]] '''Aljabar elementer''' ya'ia da'ö angolifa aljabar nibali'ö dane-dane. Aljabar elementer nifahaö ba ndraono sekola ma ''siswa/mahasiswa'' nifaigi walö hadöi fangi'ila ba zifakhai ba matematika, ba hiza töra ma'ifu moroi ba nifaehagö si sabata, hiza tobali ia dane-dane ba wangi'ila ngawalö zedöna latema wamahaö matematika ya'ia da'ö dane-dane moroi ba aritmetika. Bakha ba aritmetika, ha ''bilangan'' faoma ''operasi aritmetika'' (simane; +, −, ×, ÷) ni'ila niha. Bakha ba aljabar, numero asese lasura ia faoma famaedo, nitötöi faoma fanguma'ö ''variabel (matematika)'' (simane; ''a'', ''n'', ''x'', ''y'', mazui ''z''). Da'a moguna, börö me: * Tetehegö ba wame'e lala nitörö moroi ba zi no latatugöi ba ''hukum-hukum aritmetika'' (simane; ''a'' + ''b'' = ''b'' + ''a'' ero-ero; ''a'' faoma ''b''), ba hiza da'a no sambua bosi si'oföna numalö ba wangi'ila ma ''eksplorasi sistematis'' ba dandra-tandra numero ''sistem bilangan real''. * Tetehegö hezo lahalö ia gumbu zifakhai ba numero "anu", fame'e lala ma ''perumusan'' ba wamagölösi ma ''persamaan'' faoma ba wangalui sinöndra. (Simane duma-dumania, "Carilah bilangan ''x'' irege 3''x'' + 1 = 10" ma na tatohugö ia; "Carilah bilangan ''x'' irege tobalia ia; ''ax'' + ''b'' = ''c''". Bosi da'a numalö ba wurugö wa tenga dandra-tandra moroi ba zisambua numero zi tola mu'alui zinöndrania, hiza moroi ba ''operasi'' nilataeni khönia. * Tetehegö ba wangalui wamosinöndra zifakhai ba ''fungsi (matematika)''. (Simane duma-dumania, "Na ö amawa ''x'' karcis, wa hare nisöndrau ya'ia da'ö fa'ebuania; 3''x'' − 10 rupiah, mazui ''f''(''x'') = 3''x'' − 10, na tafaigi wa; ''f'' sambua ''fungsi'', faoma ''x'' ya'ia da'ö ni'erai ba ''fungsi'' nilaeteni khönia. === Polinomial === [[Berkas:Polynomialdeg3.svg|[[Grafik fungsi|Grafik]] fungsi polinomial berderajat 3.|jmpl|ka]] '''Polinomial''' ma asese latötöi faoma fehede; '''suku banyak''' ya'ia da'ö sambua ''ekspresi (matematika)'' fa'oya ni'erai sitola ni'erai moroi ba ''suku (logika)'' ma ''suku-suku'' sitenga ''nol'' (tak-nol), ero-ero ''suku'' so khönia wokali nilai konstanta faoma wa'oya zitola ni'erai ''variabel (matematika)'' nifa'aso faoma fefu ''pangkat bilangan''. Duma-dumania; ''x''² + 2''x'' − 3 ya'ia da'ö; polinomial bakha ba variabel tunggal ''x''. Sambua '''ekspresi polinomial''' ya'ia da'ö; ekspresi sitola lafomendrua wanura ya'ia sitobali polinomial, sangoguna'ö ''sifat-sifat komutativitas'', ''asosiativitas'', faoma ''distributivitas'' famonönö faoma fokalikö. Simane duma-dumania;, (''x'' − 1)(''x'' + 3) da'a no sambua ''ekspresi polinomial''. Sambua '''fungsi polinomial''' tola göi lamane ''fungsi'' nifo'eluaha polinomial, mazui, molo'ö ''ekivalen'', moroi ba zi sambua ''ekspresi polinomial''. Dombua duma-duma da'ö yaŵa mamaehagö ''fungsi polinomial'' sifagölö. Dombua wanofu soguna sibai ba faoma fakhai bakha ba aljabar ya'ia da'ö; ''faktorisasi polinomial'', famotokhi ya'ia simane ya'ia da'ö; mamaehagö ''polinomial'' sitobali fokali moroi ba ''polinomial-polinomial'' tanöbö'ö si tebai lagoligö sa'ae ma tebai nibagi, ba ''komputasi'' ma ''(faktor persekutuan terbesar polinomial)''. Duma-duma polinomial da'ö yaŵa sitola ni goligö sitobali; (''x'' − 1)(''x'' + 3). Sambua wenaeta wanofu sifakhai ya'ia da'ö ba wangalui ''ekspresi aljabar'' ba wamosinöndra ''akar fungsi'' zi sambua polinomial bakha ba variabel si mosindro samösa. === Famahaö ''(Pendidikan)'' === No lafaehagö wa ''aljabar elementer'' andre ena'ö sambua lala wamahaö ba ndraono sekola si no irugi 11 fakhe,<ref>{{Cite web |title=Hull's Algebra |work=New York Times |date=1904-07-16 |url=https://query.nytimes.com/mem/archive-free/pdf?res=F10714FB395E12738DDDAF0994DF405B848CF1D3 |format=[[pdf]] |accessdate=2012-09-21}}</ref> heŵa'ae ba gafuriata andre so ma'ifu wa'atedou ba wamahaö ''aljabar elementer'' ba ndraono [[SMP]] ma sodadao ba kalasi 1 [[SMP]] (≈&nbsp;13&nbsp;tahun&nbsp;±) di Amerika Serikat.<ref>{{Cite web |last=Quaid |first=Libby |title=Kids misplaced in algebra |work=[[Associated Press]] |date=2008-09-22 |url=https://www.usatoday.com/news/nation/2008-09-22-357650952_x.htm |format=Report |accessdate=2012-09-23}}</ref> Heŵa'ae simanö, ba zekola si so ba [[Amerika Serikat]], aljabar no lafahaö ba kalasi 3 SMP. I'otarai me döfi si 1997, ''Institut Politeknik'' faoma ''Universitas Negeri Virginia'' ba [[universitas]] tanöbö'ö, no labörötaigö mangoguna'ö sambua lala wamahaö aljabar si no laföfögö ba wame'e fa'aboto ba dödö ba nifahaö moroi ba no lahaogö ''peranti lunak komputer'' ba wangahatö lala wangi'ila ba aljabar, ena'ö lö oya tetibo ginötö faoma soguna tanöbö'ö sangoguna'ö kefe.<ref>{{cite news|url=https://www.nytimes.com/2012/09/07/us/ut-arlington-adopts-new-way-to-tackle-algebra.html|title=THE TEXAS TRIBUNE; U.T.-Arlington Adopts New Way to Tackle Algebra|last=Hamilton|first=Reeve|date=2012-09-07|work=The New York Times|accessdate=2012-09-10}}</ref> == Aljabar abstrak == '''Aljabar abstrak''' mangebolo'ö ngawalö zifakhai ba ''aljabar elementer'' faoma [[Aritmetika]] ''bilangan'' numalö ba mbosi-bosi sito'ölö la'oguna'ö. Ba da'a tou so mbosi-bosi ma lala nitörö bakha ba ''aljabar abstrak''. '''Himpunan Matematika''': tenga ha sito'ölö töra ma'ifu wangasiwaisi ngawalö wanofu sifakhai ba wangeraisi sifabö'ö-bö'ö, aljabar abstrak fakhai ba gangorudua ma nibe'e fanötöi ; ''himpunan'' na lafo'eluaha ia ya'ia da'ö; angorudua si so amakhaita ba wangerai si tola ni'orudugö. Fefu gangorudua da'a latötöi faoma ''himpunan''. Duma-duma simane; ''Matriks (matematika)'' dua-kali-dua(2x2), angorudua ''polinomial'' nibe'e faka 2 ma ''(berderajat 2 (''ax''² + ''bx'' + ''c'')'', angorudua ''Vektor (spasial)'' dua dimensi ba zisambua nahia, ba tanöbö'ö soya sibai simane ''grup siklis'', sifao bakha ba ''aritmetika modular'' ma ''modulo'' bilangan bulat ''n''. ''Teori himpunan'' andre ya'ia da'ö sambua gangolifa moroi ba ''logika'' ba molo'ö lala wangalui sinöndrania ya'ia da'ö tenga fenaeta moroi ba aljabar. '''Operasi biner''': amozua wamonönö ma ''perjumlahan'' (+) la ''abstraksi'' ia ba wame'e sambua khönia ''operasi biner'', Tola lamane ∗amozua moroi ba ''operasi biner'' lö gunania na lö hadöi nahia gangorudua lala wangerai sedöna lafalua. Ba zi dombua ''elemen'' moroi ba ''a'' faoma ''b'' bakha ba gangorudua ''S'', ''a'' ∗ ''b'' ya'ia da'ö ''elemen'' tanöbö'ö bakha ba ni'orudugö; fangerai da'a latötöi ia fa'ateduhö matematika ma ba [[li Indonesia]] sasese latötöi ba matematika lamane; ''ketertutupan (matematika)''. Famonönö (+), Fangeheta (−), Fokali (×), faoma Fombagi (÷) tola tobali ''operasi biner'' na tefaehagö ia ba gangorudua zifabö'ö, duma-dumania famonönö faoma fokali ''matriks, vektor'', faoma ''polinomial''. '''Elemen identitas''', ya'ia da'ö: Numero ''nol'' faoma sambua ni ''abstraksi'' ba wamo'eluaha zisambua ''elemen identitas'' ba zisambua fangerai. Nol no sambua ''elemen identitas'' ba wamonönö ba sambua ia göi ba wokali. Ba ''operator biner'' si no oi la'ila niha sato misa ∗ elemen identitas ''e'' lötolalö'ö ena'ö so khönia; ''a'' ∗ ''e'' = ''a'' faoma ''e'' ∗ ''a'' = ''a'', ba mangerai samösa ia, na so ni'erai khönia. Da'a te'oguna'ö ia ba wamonönö sitobali; ''a'' + 0 = ''a'' faoma 0 + ''a'' = ''a'' ba simanö göi ba wokali; ''a'' × 1 = ''a'' faoma 1 × ''a'' = ''a''. Lö fefu gangorudua faoma fangerai sifabö'ö-bö'ö oi so ''elemen identitas''; simane, angorudua ''bilangan asli positif'' (1, 2, 3,&nbsp;...) lö hadöi khönia ''elemen identitas'' ba wamonönö. '''Elemen invers''' mazui '''unsur balikan''', ya'ia da'ö; ''Bilangan negatif'' samaehagö lala wangerai; ''elemen invers''. Ba wamonönö, invers ''a'' lasura ia faoma −''a''; na ba wokali lasura ia faoma ''a''⁻¹. Elemen invers te'oguna'ö ba zidombua lala wangerai ''a''⁻¹ asala so khönia zikhala wa ''a'' ∗ ''a''⁻¹ = ''e'' faoma ''a''⁻¹ ∗ ''a'' = ''e'', hiza i ''e'' no sambua ''elemen identitas''. '''Sifat asosiatif'' ma''Asosiativitas'''la fo'eluaha ia: famonönö ''bilangan bulat'' sangokhögö sikhala nifotöi ''asosiativitas''. Ya'ia da'ö, famöfögö ''bilangan'' na la fonönö ia lö tebulö zinöndrania. Simane duma-dumania: {{nowrap|1=(2 + 3) + 4 = 2 + (3 + 4)}}. Na sito'ölönia, da'a tobali; (''a'' ∗ ''b'') ∗ ''c'' = ''a'' ∗ (''b'' ∗ ''c''). Sikhala wangerai da'a tefalua göi ia ba lala wangerai ''operasi biner'', hiza i tenga ba wangerai fangeheta, fombagi, mazuo fokali ''oktonion''. '''Sifat komutatif''', ya'ia da'ö; Famonönö faoma fokali ''bilangan real'' sifagölö sangokhögö sikhala ''komutatif''. Ya'ia da'ö, fenaera numero ni'erai si lö mamulö'ö zinöndra moroi ba wangerai ya'ia. Duma-dumania; 2 + 3 = 3 + 2. Molo'ö fangerai sito'ölö, tola la'erai ia; ''a'' ∗ ''b'' = ''b'' ∗ ''a''. Sikhala da'a lö la'oguna'ö ia ba wangerai ''operasi biner''. Simane, fokali ''matriks'' faoma fokali ''kuaternion'', börö me dombua wangerai da'a tenga angolifa zikhala wangerai ''komutatif''. === Angorudua ''(Grup)'' === Fangorudugö lala wangerai da'ö yaŵa mame'e fangi'ila hewisa ia wenaeta soguna sibai bakha ba matematika ma abölö enahöi lamane ba; '''Grup (matematika)'''. Nifotöi Grup ba da'a ya'ia da'ö; famaruka sambua angorudua ''S'' ba sambua lala wangerai ''operasi biner'' ∗, lafo'eluaha ia bakha ba lala wangerai nifili, hiza böi fasala molo'ö lala simane ba da'a tou; * So sambua khönia ''elemen identitas'' ''e'', irege ero sambua ni'erai khönia simane; ''a'' moroi ba ''S'', ''e'' ∗ ''a'' faoma ''a'' ∗ ''e'' dombua lala da'a fagölö ira; ''a''. * Ero-ero ''elemen'' sokhönia ''invers'': ero ngawua; ''a'' moroi ba ''S'', sokhönia ngawua ''a''⁻¹ irege ''a'' ∗ ''a''⁻¹ faoma ''a''⁻¹ ∗ ''a'' da'a fagölö ira ''elemen identitas''. * Lala wangerai sikhala ''asosiatif'': Na ''a'', ''b'', faoma ''c'' angolifa moroi ba ''S'', hiza (''a'' ∗ ''b'') ∗ ''c'' fagölö ira ''a'' ∗ (''b'' ∗ ''c''). Na'atö angorudua da'a ''komutatif'', ya'ia da'ö ero dombua gangolifania ''a'' faoma ''b'' moroi ba ''S'', ''a'' ∗ ''b'' fagölö ira ''b'' ∗ ''a''—ba hiza angorudua da'a tola latötöi ia faoma fanötöi ''grup abelian''. Sitobali duma-duma, angorudua moroi ba ''bilangan bulat'' barö wangerai famonönö no sambua angorudua ma ''grup''. Ba gangorudua da'a, ''elemen identitas'' ya'ia da'ö; 0 ba hiza ''invers'' moroi ba ''elemen'', ''a'' ya'ia da'ö ''negasi'', −''a''. Si lötolalö'ö te'a'asogö ba wangerai ''asosiativitas'', me ero ''bilangan bulat'', ''a'', ''b'', faoma ''c'', (''a'' + ''b'') + ''c'' = ''a'' + (''b'' + ''c'') '''Bilangan rasional'' sitenga Nol tehaogö khönia gangorudua ma ''grup'' barö wangerai fokali. Ba da'a, ''elemen identitas'' ya'ia da'ö 1, me 1 × ''a'' = ''a'' × 1 = ''a'' ero ''bilangan rasional'' ''a''. Invers moroi ba ''a'' ya'ia da'ö 1/''a'', börö me ''a'' × 1/''a'' = 1. Heŵa'ae simanö, ''bilangan bulat'' barö wangerai fokali lö tehaogö khönia gangorudua ma grup. Da'a börö me ''invers'' fokali ''bilangan bulat'' lö mamosinöndra tobali ''bilangan bulat''. Simane duma-duma, 4 no ta'ila wa sambua ''bilangan bulat'', hiza i ''invers'' fokali ya'ia, ya'ia da'ö; ¼, ba no tefa'ele'ö wa tenga ''bilangan bulat''. Lala wangerai sifakhai ba gangorudua nitötöi faoma fehede ''grup'' ba wamahaö andre, no mozawili misa wamahaö ya'ia si no oi la'ila niha ya'ia da'ö; ''teori grup''. Nisöndra moroi ba wamahö andre ya'ia da'ö hewisa ba wama'oli gangorudua-gangorudua wangerai ma fanötöi ba li matematika ''klasifikasi grup-grup sederhana berhingga'', hiza abölö ebua nifatumbu'ö ba gotalua ndröfi 1955 faoma döfi si 1983, samöfögö ''grup sederhana'', ''himpunan berhingga'' tobali ba zi 30 ngawalö dane-dane wamahaö. ''Semigrup'', ''kuasigrup'', faoma ''monoid'' ya'ia da'ö fenaeta-fenaeta sifagölö ira ''grup'', hiza i sikhalania sifagölötö ba danö bö'ö. No lahaogö sambua gangorudua faoma sambua wangerai ''operasi biner'' si tekusi, ha ya'i lö moguna sibai so khönia goi-goi nihonogöi. ''Semigrup'' so khönia ''operasi biner'' simane; ''asosiatif'', ba lö hadöi ''elemen identitas''. Na ''Monoid'' ya'ia da'ö ''semigrup'' sangokhögö ''elemen identitas'', ba lö hadöi khönia ''invers'' ero-ero ''elemen''. Fabö'ö ira na ''Kuasigrup'', so zi lötolalö'ö tefa'aso khönia me gofu hadia manö ''elemen'' tola tefangamöi tobali ''elemen'' tanöbö'ö moroi ba wokali kabera ma zui kambölö; ha ya'i ''operasi biner''nia tola manö lö khönia sikhala ''asosiatif''. Fefu gangorudua ma ''grup'' tola lamane fao ba ''monoid'', ba fefu ''monoid'' ya'ia da'ö; ''semigrup''. {| class="wikitable" |+'''D u m a - D u m a''' |- !Himpunan | colspan=2|[[Bilangan asli]] '''N''' | colspan=2|[[Bilangan bulat]] '''Z''' | colspan=4|[[Bilangan rasional]] '''Q''' (juga [[Bilangan real]] '''R''' faoma [[Bilangan kompleks|kompleks]] '''C''') | colspan=2|[[Aritmetika modular|Modulo]] bilangan bulat 3: '''Z'''₃ = {0, 1, 2} |- !Operasi | + | × (tak-nol) | + | × (tak-nol) | + | − | × (tak-nol) | ÷ (tak-nol) | + | × (tak-nol) |- !Tertutup | Ya | Ya | Ya | Ya | Ya | Ya | Ya | Ya | Ya | Ya |- | Identitas | 0 | 1 | 0 | 1 | 0 | Lö hadöi | 1 | Lö hadöi | 0 | 1 |- | Invers | Lö hadöi | Lö hadöi | −''a'' | Lö hadöi | −''a'' | Lö hadöi | 1/''a'' | Lö hadöi | ero-ero: 0, 2, 1 | ero-ero: Lö hadöi, 1, 2 |- | Asosiatif | Ya | Ya | Ya | Ya | Ya | Lö'ö | Ya | Lö'ö | Ya | Ya |- | Komutatif | Ya | Ya | Ya | Ya | Ya | Lö'ö | Ya | Lö'ö | Ya | Ya |- | Struktur | [[monoid]] | [[monoid]] | [[grup abelian]] | [[monoid]] | [[grup abelian]] | [[kuasigrup]] | [[grup abelian]] | [[kuasigrup]] | [[grup abelian]] | [[grup abelian]] ('''Z'''₂) |} === ''Gelanggang'' faoma ''Lapangan'' === Grup ha sambua khönia ''operasi biner''. Famaehagö ngawalö lala wangerai sifabö'ö-bö'ö, fangerai si so fenaeta dombua lala wangerai lötolalö'ö lafanöi lala wangi'ila ba wamahaö aljabar andre. Sabölö moguna ya'ia da'ö ba wangi'ila hadia ia nifotöi ''Gelanggang'' (matematika), faoma ''Medan''(matematika). Sambua '''Gelanggang'''(matematika) so dombua khönia ''Operasi biner'' ya'ia da'ö; (+) faoma (×), simanö göi × distributif ba dete wamonönö (+). Fangerai si'oföna ya'ia da'ö (+), te faehagö khönia ''grup abelian''. Fangerai aefa da'ö, fokali (×), sikhalania ya'ia sa'ae nifotöi ''asosiatif'', ha ya'i lö khönia ''identitas'' mazui ''invers'', irege lö moguna la bologö ero fenaeta. ''Elemen identitas'' wamonönö (+) lasura ia faoma numero; 0 ba ''invers'' famonönö moroi ba ''a'' lasura ia faoma; −a. Böi olifu wa lala wangerai da'a tola lamane fangerai si ''abstrak'', fefu zedöna la faehagö. '''Distributif'' ma sikhala moroi ba ''distributif'' la'osara'ö ba zito'ölö la'erai niha ya'ia da'ö; ''hukum distributif'' fangerai numero. Na ba ''bilangan bulat'' {{nowrap|1=(''a'' + ''b'') × ''c'' = ''a'' × ''c'' + ''b'' × ''c''}} faoma {{nowrap|1=''c'' × (''a'' + ''b'') = ''c'' × ''a'' + ''c'' × ''b'',}} faoma × latötöi ia; ''distributif'' ba dete +. ''Bilangan bulat'' sambua duma-duma moroi ba nifotöi ''gelanggang''. ''Bilangan bulat'' sikhalania so khönia wamonönö solohe ya'ia tobali '''domain integral''', ma lamane; '''daerah bilangan bulat''', mazui '''ranah bilangan bulat,'''. Sambua ''medan (matematika)'' ma ''lapangan]'' ya'ia da'ö; ''gelanggang'' sikhala wamonönö wa fefu ''elemen'' tak-nol mamaehagö ''grup abelian'' barö ×. Sikhala wokali (×) lasura faoma numero 1 ba ''invers'' wokali moroi ba ''a'' lasura faoma ''a''⁻¹. ''Bilangan rasional, bilangan real'', faoma ''bilangan kompleks'' sambua duma-duma moroi ba ''lapangan''. == Umbu == [[Kategori:Nias]] 3gbb2m38gth826nsu9qwhtldxazaga5