Alijebra

Algebra ya pulayimale yikusambira kuti ivyakuyaniska ivi vikuzirwa na vipimo vichi, ivyo vyakupangika mwakugwiliskira nchito za masamu.

Algebra yakudumura yikusambira magulu gha algebra, nga umo mphete ya vipendeko vyambura kugaŵika vikuchemeka

(\mathbb {Z} )

pamoza na nchito za kusazga (

+

) na kwandaniska (

\times

).

Algebra ni chigaŵa cha masamu icho chikukhwaskana na ndondomeko zambura kuwoneka, izo zikuchemeka magulu gha algebra, na kugwiriskira nchito mazgu mukati mwa ndondomeko izo (ivyo vikuchemeka viyelezgero). Ni kusazgirako kwa masamu ghakwamba, icho chikwiza na vyambura-kumanyikwa na nchito zinyake padera pa nchito za masamu ghakwamba, nga kusazga na kwandaniska.

Algebra ya pulayimale ndiyo nthowa yikuru ya algebra iyo yikusambizgika mu masukulu. Iwo ukusanda mazgu gha masamu mwakugwiliskira nchito vyakusintha vya vigaŵa vyambura kumanyikwa, na kukhumba kumanya kuti ni mikhaliro wuli iyo mazgu agha ghali ghaunenesko. Kuti ŵacite nthena, ŵakugwiriskira nchito nthowa zakupambana-pambana zakusinthira vyakuyaniska kuti ŵapature vinthu vyakusintha vya vigaŵa. Algebra ya mzere ni chigaŵa chakukolerana chomene na algebra ya pulayimale, icho chikufufuza vyakuyaniska vya mzere na kunozgeka kwake, ivyo vikuchemeka magulu gha vyakuyaniska vya mzere. Chikupereka nthowa zakusanga vipimo ivyo vikwaniska vyakuyaniska vyose mu gulu pa nyengo yimoza, na kusambira gulu la vigaŵa ivi.

Algebra yakudumura yikusambira magulu gha algebra, ivyo vikupangika na gulu la vinthu vya masamu pamoza na nchito yimoza panji zinandi izo zikulongosoreka pa gulu ilo. Ni kusazgirako kwa algebra ya pulayimale na ya mzere, chifukwa chikuzomerezga vinthu vinyake padera pa vipendeko na nchito zambura kuŵa za masamu ghakwamba. Chikupambaniska pakati pa mitundu yakupambana-pambana ya magulu gha algebra, nga magulu, mphete, na minda, kuyana na unandi wa nchito izo zikugwiriskirika ntchito na madango agho ghakulondezgeka, ghakuchemeka madango ghakuzomerezgeka. Algebra ya wonse na sambiro la mikhaliro vikupereka vipangiro vyakuwaneko kuti vifufuze vipimo vyambura kuwoneka ivyo vikulongosora magulu ghakupambana-pambana gha algebra.

Nthowa za algebra zikasambika dankha mu nyengo yakale kuti zithaske masuzgo ghapadera mu vigaŵa nga geometry. Ichi chikasintha mu vilimika vya m'ma 800 na Muḥammad ibn Mūsā al-Khwārizmī, uyo wakanozga algebra kuŵa chigaŵa chakujipeleka cha masamu, chakupambana na geometry. Ŵasandi ŵakwamba ŵakalongosoranga vyakuyaniska na vyakuvitatuza mwakugwiliskira nchito mazgu na vifupikizgo, m'paka mu vilimika vya m'ma 1500 na 1600, apo nthowa yakukhwima ya viwimbo (symbols) yikapangika. Pakati pa vilimika vya m'ma 1800, uzelu wa algebra ukasazgirapo kuluska sambiro la vyakuyaniska pera na kusazgako mitundu yakupambana-pambana ya nchito na magulu gha algebra. Algebra yikukhwaskana na vigaŵa vinandi vya masamu, nga geometry, topology, sambiro la vipendeko, na calculus, kweniso vigaŵa vinyake, nga logic na kasandiro kakuzenga pa vyakuwonekera.

Ng'anamuro na Muko wa Lizgu

Algebra ni nthambi ya masamu iyo yikusambira magulu gha algebra na nchito izo ghakugwiliskira ntchito.^[1] Gulu la algebra ni gulu la vinthu vya masamu ilo lilije kuŵa lambura kanthu, nga vipendeko vyambura kugaŵika, pamoza na nchito za algebra izo zikulongosoreka pa gulu ilo, nga kusazga na kwandaniska.^[2]^{[lower-alpha 1]} Algebra yikufufuza malango, mikhaliro yose, na mitundu ya magulu gha algebra. Mu magulu ghanyake gha algebra, yikuwoneseska umo vinthu vyakusintha (vyambura-kumanyikwa) vikugwiriskirika ntchito mu vyakuyaniska na umo vingachitikira kuvisintha.^[4]^{[lower-alpha 2]}

Kanandi algebra yikupulikiskika kuŵa kusazgirako kwa masamu ghakwamba.^[8] Masamu ghakwamba ghakusambira nchito nga kusazga, kuwuskapo, kwandaniska, na kugaŵa, mu chigaŵa chinyake cha vipendeko, nga vipendeko vyaunenesko.^[9] Algebra ya pulayimale njiyo mlingo wakwamba wa kudumuka. Nga umo masamu ghakwamba ghakucitira, yikujikanizga ku mitundu yinyake ya vipendeko na nchito. Yikusazgirapo nchito izi mwakuzomerezga unandi wambura kumanyikwa mu mtundu wa vyambura-kumanyikwa padera pa vipendeko.^[10] Mlingo wapachanya wa kudumuka ukusangika mu algebra yakudumura, iyo yikumara yayi pa chigaŵa chinyake ndipo yikuwoneseska magulu gha algebra nga magulu na mphete. Yikuluta kuluska nchito zamasamu ghakwamba mwakusazgaposo mitundu yinyake ya nchito.^[11] Algebra ya wonse njakudumura comenecomene chifukwa yikupulikira yayi magulu gha algebra ghapadera, kweni yikufufuza mikhaliro ya magulu gha algebra mwaunenesko.^[12]

Title page of The Compendious Book on Calculation by Completion and Balancing — Lizgu lakuti *algebra* likufuma ku mutu wa buku la al-Khwarizmi lakuti *Al-Jabr*.^[13]

Lizgu lakuti algebra nyengo zinyake likugwiriskirika ntchito mwakucepa kuti likung'anamurenge waka algebra ya pulayimale panji algebra yakudumura yekha.^[14] Para likugwiriskirika ntchito nga zina lakuŵerengeka, algebra ni mtundu unyake wa gulu la algebra uwo ukukhwaskana na malo gha vector agho ghali na mtundu unyake wa nchito ya viŵiri viŵiri, iyo yikuchemeka nchito yakuŵiriska.^[15] Kulingana na umo vikuyowoyekera, algebra yingayowoyaso za vigaŵa vinyake vya algebra, nga Lie algebra panji algebra yakukolerana.^[16]

Lizgu lakuti algebra likufuma ku lizgu la Chiarabu lakuti الجبر (al-jabr), ilo pakwamba likang'anamuranga opareshoni yakuchizgira viwangwa. Mu vilimika vya m'ma 800 C.E., lizgu ili likaŵa na ng'anamuro la masamu apo Muhammad ibn Musa al-Khwarizmi, uyo wakasambiranga masamu ku Persia, wakagwiriskira ntchito lizgu ili kuti wazunure nthowa yakusinthira vyakuyaniska. Wakagwiriskira ntchito lizgu ili pa mutu wa buku lake, ilo mutu wake ukung'anamulika kuti al-Kitāb al-Mukhtaṣar fī Ḥisāb al-Jabr wal-Muqābalah [Buku Lakulongosora Kupenda mwa Kumalizga na Kulinganiza]. Mutu uwu ukang'anamulika mu Chilatini kuti Liber Algebrae et Almucabola.^{[lower-alpha 3]} Lizgu ili likanjira mu Chingelezi mu vilimika vya m'ma 1500 kufuma ku Chiitaliya, Chisipanishi, na Chilatini cha mu nyengo ya Ŵamitundu.^[18] Pakwamba, ng'anamuro lake likaŵa waka pa fundo yakuti kufundizga vyakuyaniska, ndikuti, uzeru wa kusintha vyakuyaniska vyakuvinandi mwakukhumba kuvitatuza. Ichi chikasintha mu vilimika vya m'ma 1800^{[lower-alpha 4]} apo uzelu wa algebra ukasazgirapo kuti ukwatire kusambira mitundu yakupambana-pambana ya nchito na magulu gha algebra pamoza na madango ghakuzomerezgeka ghawo, malango agho ghakulondezgeka.^[21]

Vigaŵa Vikuru

Algebra ya Pulayimale

Nkhani ikulu: Elementary algebra

Algebra ya pulayimale, iyo yikuchemekaso algebra ya sukulu, ndiyo ni algebra yakale chomene ndiposo yakuzirwa chomene.^[22] Ni generalization ya masamu ghakwamba iyo yikuthemba pa vyakusintha (vyambura-kumanyikwa) na kusanda umo mazgu gha masamu ghangasinthira.^[23]

Masamu ghakwamba ni kasandiro ka nchito za manambara. Ghakufufuza umo manambara ghakusazgikira na kusinthika mwakugwiliskira ntchito kusazga, kuwuskapo, kwandaniska, kugaŵa, kuwoneska (exponentiation), kufumiska misisi (roots), na logarithimu. Mwachiyelezgero, nchito ya kusazga yikusazga manambara ghaŵiri, ghakuchemeka vyakusazgika, kuŵa nambara yachitatu, iyo yikuchemeka muzgoro, nga umo $2+5=7$ .^[9]

Algebra ya pulayimale yikuthemba pa nchito zenezo waka kweni yikuzomerezga vyambura-kumanyikwa padera pa manambara ghaŵaka. Vyambura-kumanyikwa ni viwimbo vya vinthu ivyo vindamanyikwe panji ivyo vyambura kumanyikwa. Vikupangiska kuti wanthu ŵayowoye ukoli uwo munthu wandaumanye mwakukhwima, na kuyowoya malango ghakumanyikwa agho ghali ghaunenesko, kwambura kupwelelera manambara agho ghakugwiriskirika ntchito. Mwachiyelezgero, cakuyaniska $2\times 3=3\times 2$ chikuŵa cha masamu ghakwamba ndipo chikulongora kuyana kwa manambara agha ghekha. Para tikapoka manambara na vyambura-kumanyikwa, tingayowoya dango likuru ilo likulongosoreka ku kunozgeka kose kwa manambara, nga dango la kusintha malo (commutative property) la kwandaniska, ilo likulongosoreka mu cakuyaniska $a\times b=b\times a$ .^[23]

Mazgu gha algebra ghakupangika mwakugwiliskira nchito za masamu ghakwamba kuti akole pamoza vyambura-kumanyikwa na manambara. Mwakuyana na umo vikuŵira, vilembo vichoko nga ni $x$ , $y$ , na $z$ vikwimira vyambura-kumanyikwa. Mu vinthu vinyake, viwimbo vyapasi (subscripts) vikusazgikako kupambaniska vyambura-kumanyikwa, nga $x_{1}$ , $x_{2}$ , na $x_{3}$ . Vilembo vichoko-choko nga ni $a$ , $b$ , na $c$ kanandi ghakugwiriskirika ntchito pa vyambura-kusintha (constants) na vyakusazgirako pa kwandaniska (coefficients).^{[lower-alpha 5]} Mazgu ghakuti $5x+3$ ni mazgu gha algebra ghakupangika mwa kuchulukizga nambara 5 na cambura-kusintha $x$ na kusazgako nambara 3 ku ivyo vikusangika. Vyakuwoneska vinyake vya mazgu gha algebra ni $32xyz$ na $64x_{1}{}^{2}+7x_{2}-c$ .^[25]

Mazgu ghanyake gha algebra ghakuŵa nga mazgu ghakukhwaskana na mazgu ghaŵiri pamoza. Cakuyaniska ni mazgu ghakupangika mwa kuyerezgera mazgu ghaŵiri, kuyowoya kuti ghakuyana waka. Ichi chingalongoreka mwakugwiliskira nchito chiwimbo cha kuyana ( $=$ ), nga $5x^{2}+6x=3y+4$ . Vyambura-kuyana vikukhwaskana na mtundu wakupambana wa kuyerezgera, kuyowoya kuti vigaŵa viŵiri ivi ni vyakupambana. Ichi chingalongoreka mwakugwiliskira ntchito viwimbo nga chiwimbo cha kuchepa ( $<$ ), chiwimbo cha kuluska ( $>$ ), na chiwimbo cha kupambana ( $\neq$ ). Kupambana na mazgu ghanyake, mazgu agha ghangaŵa ghaunenesko panji ghautesi, ndipo unenesko wawo ukuthemba pa mtengo wa vinthu ivyo vikusintha. Mwachiyelezgero, mazgu ghakuti $x^{2}=4$ ni ghaunenesko usange $x$ ni 2 panji −2, ndipo ghautesi mwakupambanako. Ma equation ghakukhala na vyambura-kumanyikwa vingagaŵika mu ma equation ghakumanyikwa (identity equations) na ma equation ghakukhazikika (conditional equations). Ma equation ghakumanyikwa ghali ghaunenesko pa vyose ivyo vingapelekeka ku vyambura-kumanyikwa, nga cakuyaniska $2x+5x=7x$ . Ma equation ghakukhazikika ghakuŵa ghaunenesko waka pa vinyake vyakusintha. Mwachiyelezgero, cakuyaniska $x+4=9$ ni chaunenesko waka usange $x$ ni 5.^[26]

Chakulinga chikuru cha algebra ya pulayimale nkhumanya mtengo uwo mazgu ghakuyowoyeka ghaunenesko. Ici cingacitika mwa kusintha na kugwiliskira nchito mazgu mwakuyana na malango ghanyake. Fundo yikuru njakuti nchito yiliyose iyo yikuchitika ku chigaŵa chimoza cha cakuyaniska yikweneraso kuchitika ku chigaŵa chinyake. Mwachiyelezgero, usange munthu wawuskako 5 ku chigaŵa cha kumaryero, wakwenerato kuwuskako 5 ku chigaŵa cha kumalyero kuti ŵalinganiske vigaŵa vyose viŵiri. Chilato cha ndondomeko izi nkhupatura chinthu icho munthu wakukhumba ku chigaŵa chimoza, ndondomeko iyo yikumanyikwa kuti nkhumazga cakuyaniska kwa chinthu icho chikusintha. Mwachiyelezgero, cakuyaniska $x-7=4$ chingamazgika kwa $x$ mwa kusazgako 7 ku vigaŵa vyose viŵiri, ivyo vikupatura $x$ ku chigaŵa cha kumaryero na kusanga cakuyaniska $x=11$ .^[27]

Pali nthowa zinyake zinandi izo zikugwiriskirika ntchito kumazga ma equation. Kupepuska kukugwiriskirika ntchito kuti chiŵike mazgu ghakusuzga na ghanyake ghakuyana kweni ghapusu. Mwachiyelezgero, mazgu ghakuti $7x-3x$ ghangaŵikika na mazgu ghakuti $4x$ chifukwa $7x-3x=(7-3)x=4x$ mwa dango la kugaŵanya.^[28] Pa mazgu agho ghali na vyambura-kumanyikwa vinandi, kuŵikirapo m'malo mwake (substitution) ni nthowa yakumanyikwa yakusinthira chinthu chimoza na mazgu ghakuyana ghambura kusazgako cambura-kusintha ico. Mwachiyelezgero, usange munthu wamanya kuti $y=3x$ , wangapepuska mazgu ghakuti $7xy$ na kusanga $21x^{2}$ . Munthowa yakuyana, usange munthu wamanya mtengo wa cambura-kusintha chimoza, wangagwiriskira ntchito umo kumanya mtengo wa vinyake.^[29]

Ma equation gha algebra ghangamasulika mu geometry kuti ghalongosore viŵelengero vya malo mu mtundu wa giraf. Vinthu vyakupambana-pambana ivyo vili mu equation vikupulikikwa nga ni maryeruzgo, ndipo vigaŵa ivyo vikumazga equation vikung'anamulika nga ni fundo za giraf. Mwachiyelezgero, usange $x$ yikuŵikika pa zero mu equation $y=0.5x-1$ , mbwenu $y$ yikwenera kuŵa negative one kuti equation yiŵe yaunenesko. Ichi chikung'anamura kuti mafupa gha $(x,y)$ ghakuti $(0,-1)$ ni chigaŵa cha giraf ya equation. Mafupa gha $(x,y)$ ghakuti $(0,7)$ , mwakupambanako, ghakumazga yayi equation ndipo ntheura ghali yayi chigaŵa cha giraf. Grafu yikukhwaska chiŵelengero chose cha x,y paŵiri awo ŵakumazga equation.^[30]

Ma Polynomial

Nkhani ikulu: Polynomial

Polynomial ni mazgu ghakuchitika na chigaŵa chimoza panji vinandi ivyo vikusazgika panji kuwuskika pakati pawo, nga $x^{4}+3xy^{2}+5x^{3}-1$ . Chigaŵa chilichose ni cambura-kusintha, cambura-kumanyikwa, panji muzgoro wa cambura-kusintha na vyambura-kumanyikwa. Cambura-kumanyikwa chilichose chingakwezgeka ku chikwezgo cha cipendeko cambura kupambika (positive integer). Polynomial ya chigaŵa chimoza yikuchemeka monomial, apo polynomial za vigaŵa viŵiri na vitatu zikuchemeka binomial na trinomial. Digiri ya polynomial ni mtengo wapachanya wa muzgoro wa vikwezgo vya vyambura-kumanyikwa (pakati pa vigaŵa vyake) (4 mu chiyelezgero ichi pacanya).^[31] Ma polynomial gha digiri yimoza ghakuchemeka ma polynomial gha mzere. Algebra ya mzere yikusambira magulu gha ma polynomial gha mzere.^[32] Polynomial yikuchemeka univariate panji multivariate, kuyana na usange yikugwiriskira ntchito cambura-kumanyikwa chimoza panji vinandi.^[33]

Kugaŵaniska ni nthowa yakugwiriskira ntchito kupepuska ma polynomial, kupangiska kuti chiŵe chapusu kuvisanda na kumanya vigaŵa ivyo vikuŵa zero. Kugaŵaniska nkhulembaso polynomial kuŵa muzgoro wa vigaŵaniso vinandi. Mwachiyelezgero, polynomial $x^{2}-3x-10$ yingagaŵaniskika kuŵa $(x+2)(x-5)$ . Polynomial yose njizero pekha usange chimoza mwa vigaŵaniso vyake ni zero, ndikuti, usange $x$ ni −2 panji 5.^[34] Pambere vilimika vya m'ma 1800 vindafike, vigaŵa vinandi vya algebra vikaŵanga vya equation ya polynomial, equation iyo yikasangika para munthu wakuyaniska polynomial na zero. Kuyezgayezga kwakwamba kwa kumazga ma equation gha polynomial kukaŵa kwakuti ŵalongore vyakumazga mu mazgu gha misisi ya nth. Muzgoro wa equation ya polynomial ya digiri yachiŵiri ukuperekeka na fomula ya quadratic^[35] $x={\frac {-b\pm {\sqrt {b^{2}-4ac\ }}}{2a}}.$

Mayankho gha madigiri ghatatu na ghanayi ghakupelekeka na mafomula gha cubic na quartic. Palije mayankho ghakumanyikwa ghakumazgira madigiri ghapachanya, nga umo vikawonekera mu vilimika vya m'ma 1800 na fundo ya Abel-Ruffini.^[36] Nanga ni para mayankho ghakumanyikwa ghalipo yayi, mayankho ghakuyana waka ghangasangika mwakugwiliskira nchito vipangizo vya manambara nga nthowa ya Newton-Raphson.^[37]

Fundo yikuru ya algebra yikuti equation yiliyose ya univariate polynomial ya digiri yiwemi iyo yili na coefficients yeneco panji yakusuzga yili na yankho limoza lakusuzga. Ntheura, polynomial yiliyose ya digiri yiwemi yingagaŵika mu ma polynomial gha mzere. Fundo yikuzirwa iyi yikakhozgeka kukwambilira kwa vilimika vya m'ma 1800, kweni yikupereka yayi nthowa yiliyose yakusangirapo vyakumazga.^[38]

Algebra ya Mzere

Nkhani ikulu: Linear algebra

Algebra ya mzere yikwamba na kusambira ndondomeko za ma equation gha mzere.^[39] Equation njakumzere usange yingalembeka mu njila ya $a_{1}x_{1}+a_{2}x_{2}+...+a_{n}x_{n}=b$ , apo $a_{1}$ , $a_{2}$ , ..., $a_{n}$ na $b$ ni vyambura-kusintha. Vyakuyelezgera ni $x_{1}-7x_{2}+3x_{3}=0$ na $\textstyle {\frac {1}{4}}x-y=4$ . Ndondomeko ya ma equation gha mzere ni gulu la ma equation gha mzere agho munthu wakukhumba kusanga vyakumazga vyakuyana pakati pawo.^[40]

Ma matrix ni magulu ghakuŵungana gha manambara mwakachilato (rows na columns) agho ghakambiliranga kwiziliska nga chiwimbo chakufinya na chakuyana mwakukwana cha ndondomeko za ma equation gha mzere.^[41] Mwachiyelezgero, ndondomeko ya ma equation ${\begin{aligned}9x_{1}+3x_{2}-13x_{3}&=0\\2.3x_{1}+7x_{3}&=9\\-5x_{1}-17x_{2}&=-3\end{aligned}}$ yingalembeka nga $AX=B,$ apo $A$ , $X$ na $B$ ni ma matrix $A={\begin{bmatrix}9&3&-13\\2.3&0&7\\-5&-17&0\end{bmatrix}},\quad X={\begin{bmatrix}x_{1}\\x_{2}\\x_{3}\end{bmatrix}},\quad B={\begin{bmatrix}0\\9\\-3\end{bmatrix}}.$

Kuyana na vinjeru vya chiŵelengero cha mizere na mizati, ma matrix ghangasazgika, ghangandaniskika, ndipo nyengo zinyake ghangapiluskika (ghangawendeskeka mwakupiluka).^{[lower-alpha 6]} Nthowa zose zakumazgira ndondomeko za mzere zingalongosoreka nga vyakuwendeska matrix mwakugwiliskira nchito izi. Mwachiyelezgero, kumazga ndondomeko yamucanya nkhupenja matrix yakupiluskika $A^{-1}$ iyo $A^{-1}A=I,$ apo $I$ ni matrix ya identity. Penepapo, kwandaniska ku mkono wamazere mbali zose ziŵiri za equation ya matrix yamucanya na $A^{-1},$ munthu wakusanga muzgoro wa ndondomeko ya ma equation gha mzere nga^[42] $X=A^{-1}B.$

Nthowa zakumazgira ndondomeko za ma equation gha mzere zikwambira ku izo zili zapafupi kumanya, nga kwiza-mu-malo-mwa (substitution)^[43] na kuwuskapo (elimination),^[44] kufika ku vipangizo vyapachanya ivyo vikugwiliskira ntchito ma matrix, nga dango la Cramer, kuwuskapo kwa Gauss, na kugaŵaniska kwa LU.^[45] Ndondomeko zinyake za ma equation nizambura-kuyana (inconsistent),^[46] ndikuti, vyakumazga vilipo yayi chifukwa ma equation ghakususkana wene.^{[lower-alpha 7]} Ndondomeko zakuyana zili na yankho limoza lakupambanapambana panji viŵelengero vyambura kumazgika vya vyakumazga.^[47]^{[lower-alpha 8]}

Kusambira malo gha vector na mapu gha mzere nkhukuru mu algebra ya mzere. Malo gha vector ni gulu la algebra ilo likupangika na gulu (set) lino lili na kusazga uko kukupangiska kuŵa gulu la abelian na kwandaniska na cipendeko (scalar multiplication) uko kukuyanapo na kusazga (wonani malo gha vector kuti mumanye vinandi). Mapu gha mzere ni nchito (function) pakati pa malo gha vector iyo yikuyanapo na kusazga na kwandaniska na cipendeko. Pa nkhani ya malo gha vector ghakumalira, ma vector na mapu gha mzere ghangayimilirika na ma matrix. Ichi chikung'anamura kuti masambiro gha ma matrix na malo gha vector ghakumalira ghakuyana waka. Comenecomene, malo gha vector ghakupereka nthowa yachitatu yakulongosolera na kugwiliskira ntchito ndondomeko za ma equation gha mzere.^[48] Kwiza pa mkhalidwe uwu, matrix ni chithunzithunzi cha mapu gha mzere: usange munthu wakusankha msingi wanyake wakulongosolera ma vector agho ghakusinthika, mbwenu ivyo vikulembeka mu matrix vikupereka vyakufumapo vya kugwiliskira ntchito mapu gha mzere ku ma vector gha msingi.^[49]

Ndondomeko za ma equation zingang'anamulika nga vithunzithunzi vya geometry. Ku ndondomeko zakuŵa na vyambura-kumanyikwa viŵiri, equation yiliyose yikwimira mzere mu malo gha mibaru yiŵiri. Apo mizere yiŵiri yikukhwaskana ndipo ni yankho la ndondomeko yose chifukwa apa ndipapera apo pakumazga equation yakwamba na yachiŵiri pamoza. Ku ndondomeko zambura-kuyana, mizere yiŵiri yikwenda mwakuyana kwambura kukhwaskana, ndikuti, palije yankho chifukwa yikukhwaskana yayi nakale. Usange ma equation ghaŵiri ngambura kujipima paŵiri, ghakulongosora mzere umoza wene, ndikuti, yankho la equation yimoza njakumazgaso equation yinyake. Vyakuyanapo ivi vikupangiska kuti munthu wasange vyakumazga mwakuwona pa giraf, kulemba ma equation na kumanya apo ghakukhwaskana.^[50] Fundo zenezi zikugwira ntchito so ku ndondomeko za ma equation zakuŵa na vyambura-kumanyikwa vinandi, kweni kupambana nkhwakuti ma equation agha ghakulongosora yayi mizere kweni vithunzithunzi vyapachanya kuluska. Mwachiyelezgero, ma equation gha vyambura-kumanyikwa vitatu ghakuyana na mabwalo mu malo gha mibaru yitatu, ndipo apo mabwalo ghose ghakukhwaskana ndipo pakumazga ndondomeko ya ma equation.^[51]

Algebra Yakudumura

Nkhani ikulu: Abstract algebra

Algebra yakudumura, iyo yikuchemekaso algebra ya sono,^[52] njakusambira magulu gha algebra. Gulu la algebra ni chiwongo cha kupulikiska nchito pa vinthu vya masamu, nga kusazga kwa vipendeko. Apo algebra ya pulayimale na algebra ya mzere zikugwira ntchito pakati pa vipimo vya magulu gha algebra ghakupambanapambana, algebra yakudumura yikutora nthowa yikuru kuluska, yakuyaniska umo magulu gha algebra ghakupambanirana na mitundu yose ya magulu gha algebra iyo yiliko, nga magulu, mphete, na minda.^[53] Kupambana kukuru pakati pa mitundu iyi ya magulu gha algebra nkhu chiŵelengero cha nchito izo zikugwiliskira ntchito na madango agho ghakuzomerezgeka.^[54] Mu masambizgo gha masamu, algebra yakudumura yikwimira kalasi yapachanya ya ku yunivesite iyo ŵakusambira masamu ŵakusambira pamanyuma pakumazga makalasi gha algebra ya mzere.^[55]

Pa mkhalidwe wakuzomerezgeka, gulu la algebra ni gulu la vinthu^{[lower-alpha 9]} lakuzikapo, pamoza na nchito yimoza panji zinandi.^{[lower-alpha 10]} Algebra yakudumura yikukhumbisiska comenecomene nchito zaŵiri,^{[lower-alpha 11]} izo zikutora vinthu viŵiri vilivyose mu gulu lakuzikapo kuŵa vyakwiza-mu, ndipo zikuvipeleka ku chinthu chinyake mu gulu ili kuŵa chakufumamo.^[59] Mwachiyelezgero, gulu la algebra $\langle \mathbb {N} ,+\rangle$ lili na vipendeko vyakuŵelenjera ( $\mathbb {N}$ ) kuŵa gulu lakuzikapo na kusazga ( $+$ ) kuŵa nchito yake yaŵiri.^[57] Gulu lakuzikapo lingaŵa na vinthu vinyake kupambuka vipendeko, ndipo nchito zake nizakukakikizgika yayi ku nchito zaumaliro za masamu pera.^[60] Mwachiyelezgero, gulu lakuzikapo la gulu la symmetry la chinthu cha geometry likupangika na kusintha kwa geometry, nga kuzingilizga, apo munsi mwake chinthu chikukhalilira chambura-kusintha. Nchito yake yaŵiri ni kukolerana kwa mapu, iyo yikutora vyakusintha viŵiri kuŵa vyakwiza-mu ndipo yili na chakusintha icho chikufuma pa kugwiliskira ntchito chakusintha chakwamba kulondezgeka na chachiŵiri kuŵa chakufumamo.^[61]

Sambiro la Magulu

Nkhani ikulu: Group theory

Algebra yakudumura yikupatuliska magulu gha algebra kuyana na madango panji maxiom agho nchito zake zikuzomerezga na chiŵelengero cha nchito izo zikugwiliskira ntchito. Mtundu wa chiyambipo comenecomene ni gulu (group), ilo lili na nchito yimoza ndipo likukhumba kuti nchito iyi njakukolerana (associative) ndiposo yili na chinthu cha identity na vinthu vyakupiluska. Nchito njakukolerana usange kalongozgero ka kugwiliskira ntchito kanandi kalije mphampha, ndikuti, usange $(a\circ b)\circ c$ ^{[lower-alpha 12]} nchakuyana na $a\circ (b\circ c)$ ku vinthu vyose. Nchito yili na chinthu cha identity panji chinthu chambura-kusintha usange chinthu chimoza e chilipo icho chikusintha yayi mtengo wa chinthu chilichose chinyake, ndikuti, usange $a\circ e=e\circ a=a$ . Nchito yili na vinthu vyakupiluska usange ku chinthu chilichose $a$ pali chinthu chakuwelengana $a^{-1}$ icho chikuwuskapo vya $a$ . Usange chinthu chikugwira ntchito pa icho chakupiluska chake mbwenu muzgoro ni chinthu chambura-kusintha e, chakulongosoreka mwakuzomerezgeka nga $a\circ a^{-1}=a^{-1}\circ a=e$ . Gulu la algebra lililose ilo likukwaniska vikhumbikwa ivi ni gulu.^[63] Mwachiyelezgero, $\langle \mathbb {Z} ,+\rangle$ ni gulu ilo lapangika na gulu la vipendeko vyambura kugaŵika pamoza na nchito ya kusazga. Chinthu chambura-kusintha ni 0 ndipo chinthu chakupiluska cha cipendeko chilichose $a$ ni $-a$ .^[64] Vipendeko vyakuŵelenjera pamoza na kusazga, kususkana na ivi, vikupanga yayi gulu chifukwa vili pera na vipendeko vyakuzomerezgeka ndipo ntheura vikusoŵa vinthu vyakupiluska.^[65]

Sambiro la magulu likusanda umo magulu ghaliri, pamoza na fundo za musingi nga fundo yikuru ya magulu gha abelian ghakumalira na fundo ya Feit-Thompson.^[66] Iyi yachiŵiri yikaŵa njira yikuru yakwambilira mu chimoza mwa vinthu vikuru comenecomene mu masamu vya vilimika vya m'ma 1900: nchito yakugwirira pamoza, iyo yikatora mapeji ghakuluska 10,000 mu majenali ndipo yikafuma comenecomene pakati pa 1960 na 2004, iyo yikafiskira ku kupatuliska magulu ghapusu ghakumalira kwakukwana.^[67]

Sambiro la Mphete na Sambiro la Munda

Nkhani ikulu: Ring theory and Field (mathematics)

Mphete ni gulu la algebra lino lili na nchito ziŵiri izo zikugwira ntchito mwakuyana na kusazga na kwandaniska kwa vipendeko ndipo zikuchemeka na kulembeka mwakuyana. Mphete ni gulu lakusinthana pasi pa kusazga: kusazga kwa mphete nkhwakukolerana, kwakusinthana, ndipo kuli na chinthu cha identity na vinthu vyakupiluska. Kwandaniska nkhwakukolerana ndiposo kwakugaŵira ku kusazga; ndikuti, $a(b+c)=ab+ac$ na $(b+c)a=ba+ca.$ Kweniso, kwandaniska nkhwakukolerana ndipo kuli na chinthu cha identity icho chikuchemeka comenecomene $1$ .^[68]^{[lower-alpha 13]} Kwandaniska nkhukhumbikwa yayi kuŵa kwakusinthana; usange nkhwakusinthana, munthu wali na mphete yakusinthana.^[70] Mphete ya vipendeko vyambura kugaŵika ( $\mathbb {Z}$ ) ni yimoza mwa mphete zapusu comenecomene izo zili zakusinthana.^[71]

Munda ni mphete yakusinthana iyo $1\neq 0$ ^{[lower-alpha 14]} ndipo chinthu chilichose ichi mbura kuŵa zero chili na chakupiluska cha kwandaniska.^[73] Mphete ya vipendeko vyambura kugaŵika njambura kupanga munda chifukwa yikusoŵa vyakupiluska vya kwandaniska. Mwachiyelezgero, chakupiluska cha kwandaniska cha $7$ ni ${\tfrac {1}{7}}$ , icho ndi cipendeko cambura kugaŵika yayi. Vipendeko vyakugaŵikika, vipendeko vyeneco, na vipendeko vyakusuzga vilivyose vikupanga munda na nchito za kusazga na kwandaniska.^[74]

Sambiro la mphete ni kusambira mphete, kupenja vinthu nga mphete zamunsi, mphete za mgaŵano, mphete za polynomial, na ma ideal pamoza na fundo nga fundo ya msingi ya Hilbert.^[75] Sambiro la munda likukhwaskana na minda, kupenja kutandarizga kwa munda, kujara kwa algebra, na minda yakumalira.^[76] Sambiro la Galois likupenja uzeru pakati pa sambiro la munda na sambiro la magulu, likujizika pa fundo yikuru ya sambiro la Galois.^[77]

Sambiro la Kukhwaskana Pakati pa Magulu gha Algebra

Kupambuka magulu, mphete, na minda, kuli magulu ghanandi ghanyake gha algebra agho ghakusambirika na algebra. Ghakusazgamo ma magma, ma semigroup, ma monoid, magulu gha abelian, mphete zakusinthana, ma module, ma lattice, malo gha vector, algebra pa munda, na magulu ghakukolerana na ghambura kukolerana. Ghakupambana pakati pawo pa mtundu wa vinthu ivyo ghakulongosora na vikhumbikwa ivyo nchito zawo zikufiska. Ghanandi ghakukhwaskana wene pakati pawo mu ntchito yakuti gulu la musingi lingasinthika kuŵa gulu lakuzikirapo comenecomene mwakusazgirapo vikakiro.^[54] Mwachiyelezgero, magma yikuŵa semigroup usange nchito yake njakukolerana.^[78]

Ma homomorphism ni vipangizo vyakupenjera vinjeru vya mzimu mwakuyaniska magulu ghaŵiri gha algebra.^[79] Homomorphism ni mapu ghakufuma ku gulu lakuzikapo la gulu limoza la algebra kuya ku gulu lakuzikapo la gulu linyake la algebra, ghakusunga mikhaliro yinyake ya mzimu. Usange magulu ghaŵiri gha algebra ghakugwiliskira ntchito nchito zaŵiri ndipo ghali mu mtundu $\langle A,\circ \rangle$ na $\langle B,\star \rangle$ mbwenu mapu $h:A\to B$ ni homomorphism usange ghakukwaniska ivi: $h(x\circ y)=h(x)\star h(y)$ . Kuŵako kwa homomorphism kukulongora kuti nchito $\star$ mu gulu lachiŵiri la algebra yikugwira ntchito yakuyana na iyo nchito $\circ$ yikugwira mu gulu lakwamba la algebra.^[80] Ma isomorphism ni mtundu wapadera wa homomorphism uwo ukulongora kuyanako kukuru pakati pa magulu ghaŵiri gha algebra. Isomorphism ni homomorphism yakwenerana pa chimoza-na-chimoza, ndikuti, yikupanga ukhwaskano wa chimoza-na-chimoza pakati pa vinthu vya magulu ghaŵiri gha algebra. Ichi chikung'anamura kuti chinthu chilichose cha gulu lakwamba la algebra chikuya ku chinthu chimoza chapadera mu gulu lachiŵiri, kwambura vinthu vyakusoŵa mapu mu gulu lachiŵiri.^[81]

Chipangizo chinyake cha kuyaniska nkhukhwaskana pakati pa gulu la algebra na chigulu chamunsi cha algebra chake.^[82] Gulu la algebra na chigulu chamunsi chake vikugwiliskira ntchito nchito zakuyana,^{[lower-alpha 15]} izo zikulondezga maxiom ghakuyana. Kupambana kwapera nkhwakuti gulu lakuzikapo la chigulu chamunsi ni chigaŵa cha gulu lakuzikapo la gulu la algebra.^{[lower-alpha 16]} Nchito zose mu chigulu chamunsi zikukhumbikwa kuŵa zakujara mu gulu lake lakuzikapo, ndikuti, zikufumiska pera vinthu ivyo vili mu gulu ili.^[82] Mwachiyelezgero, gulu la vipendeko vyakwenerana pamoza na kusazga ni chigulu chamunsi cha gulu lose la vipendeko vyambura kugaŵika pamoza na kusazga. Ichi nchakuti chifukwa muzgoro wa vipendeko viŵiri vyakwenerana ni cipendeko cakwenerana kavili. Kweni gulu la vipendeko vyambura kwenerana pamoza na kusazga nichigulu chamunsi yayi chifukwa nichambura kujara: kusazga vipendeko viŵiri vyambura kwenerana vikufumiska cipendeko chakwenerana, icho ndi chigaŵa yayi cha gulu ilo lasankhika.^[83]

Algebra ya wonse ni sambiro la magulu gha algebra mwaŵukuru. Nga chigaŵa cha vinjeru vyake vyakuwonera vinthu vyakuzomerezgeka, yikupwelelera yayi vinthu vyeneco ivyo vikupanga magulu ghakuzikapo ndipo yikuwona nchito izo zili na vyakwiza-mu vinandi kuluska viŵiri, nga nchito zitatu. Yikupereka chiwongo cha kupenja vinjeru vya mzimu ivyo magulu ghakupambanapambana gha algebra ghali navyo pamoza.^[85]^{[lower-alpha 17]} Chimoza mwa vinjeru vya mzimu ivi vikukhwaskana na vinthu vyakuyana (identities) ivyo nivyanadi mu magulu ghakupambanapambana gha algebra. Mu mkhalidwe uwu, chinthu chakuyana ni equation yawonse panji equation iyo nyanadi ku vinthu vyose vya gulu lakuzikapo. Mwachiyelezgero, kusinthana (commutativity) ni equation yawonse iyo yikuti $a\circ b$ nichakuyana na $b\circ a$ ku vinthu vyose.^[87] Mtundu wakuyana (variety) ni gulu la magulu ghose gha algebra agho ghakukwaniska vinthu vinyake vyakuyana. Mwachiyelezgero, usange magulu ghaŵiri gha algebra ghakukwaniska kusinthana mbwenu ghose ghaŵiri ni chigaŵa cha mtundu wakuyana uwo ukuyana nawo.^[88]^{[lower-alpha 18]}^{[lower-alpha 19]}

Sambiro la mikhaliro likupenja umo vinthu vya masamu vikukhwaskirana na chinyake mwakugwiliskira ntchito mzimu wa mikhaliro (categories). Mkhaliro ni chiŵungano cha vinthu pamoza na chiŵungano cha vilongozgi panji "mivi" pakati pa vinthu ivi. Viŵungano ivi viŵiri vikwenera kukwaniska vikhumbikwa vinyake. Mwachiyelezgero, vilongozgi vingakolerana, panji ku kolerana: usange muvi ukuŵako kufuma ku chinthu $a$ kuya ku chinthu $b$ , ndipo muvi unyake kufuma ku chinthu $b$ kuya ku chinthu $c$ , mbwenu pakwenera kuŵakoso muvi kufuma ku chinthu $a$ kuya ku chinthu $c$ . Kukolerana kwa vilongozgi kukukhumbikwa kuŵa kwakukolerana, ndipo pakwenera kuŵako "muvi wa identity" ku chinthu chilichose.^[92] Mikhaliro yikugwiliskika comenecomene mu masamu gha sono chifukwa yikupereka chiwongo chakukoleranya kuti chilongosore na kupenja vinjeru vinandi vya musingi vya masamu. Mwachiyelezgero, magulu ya vinthu ghangalongosoreka na mkhaliro wa magulu, ndipo gulu lililose lingawoneka kuŵa vilongozgi vya mkhaliro uwo uli na chinthu chimoza pera.^[93]

Mbiri

Nkhani ikulu: History of algebra and Timeline of algebra

Chiyambi cha algebra chili mu kuyezgayezga kwakumazga masuzgo gha masamu agho ghakukhwaskana na kuŵelenga na viŵelengero vyambura kumanyikwa. Vyakukula ivi vikachitika mu nyengo yakale mu Babylonia, Egypt, Greece, China, na India. Chimoza mwa vyolálemba vyakale comenecomene pa masuzgo gha algebra ni Papyrus ya Rhind Mathematical kufuma ku Egypt yakale, iyo yikalembeka pafupifupi mu 1650 BCE.^{[lower-alpha 20]} Yikudumbiskana vyakumazga vya ma equation gha mzere, nga umo vikulongosoreka mu masuzgo nga "Chinthu; chigaŵa chake chachinayi chikusazgikapo pa icho. Chikuŵa 15. Kasi chinthu ni chiwuli?" Mabuwe gha dongo gha ku Babylon kufuma pa nyengo yeneyiyo ghakulongosora nthowa zakumazga ma equation gha mzere na polynomial gha digiri yachiŵiri, nga nthowa ya kukwaniska cikwezgo.^[95]

Vinandi mwa vinjeru ivi vikafika ku Ŵagiriki ŵakale. Kwambira mu vilimika vya 6 BCE, chikhumbo chawo chikuru chikaŵa geometry m'malo mwa algebra, kweni ŵakagwiliskira ntchito nthowa za algebra kumazga masuzgo gha geometry. Mwachiyelezgero, ŵakasandanga vithunzithunzi vya geometry apo utali na malo ghake vikatoleka kuŵa viŵelengero vyambura kumanyikwa ivyo vikwenera kusangika, nga umo vikawonekera mu nthowa ya Pythagoras ya mphambano wa vikwezgo viŵiri na munyuma mu buku la Euclid's Elements.^[96] Mu vilimika vya 3 CE, Diophantus wakapeleka malongosoro ghakuzura gha umo wangamazgira ma equation gha algebra mu magazini ghakuchemeka Arithmetica. Ndiyo wakaŵa wakwamba kuyezgayezga kugwiliskira ntchito viwimbo (symbolic notation) kulongosora ma polynomial.^[97] Ntchito ya Diophantus yikakhwaskira kukula kwa algebra ku Ŵaarabu, apo nthowa zake zinandi zikawonekera mu vinjeru na vipangizo ivyo vikagwiliskika mu algebra ya Chiarabu ya nyengo za pakati.^[98] Ku China wakale, The Nine Chapters on the Mathematical Art, buku ilo likalembeka pa nyengo yakufuma ku vilimika vya 10 BCE kufika ku vilimika vya 2 CE,^[99] likasandaso nthowa zakupambanapambana zakumazgira ma equation gha algebra, kusazgamo kugwiliskira ntchito vinthu vyakuyana na matrix.^[100]

Nkhwakususkana usange vyakukula ivi vyakale ni chigaŵa cha algebra panji ni vyakwambiliranji pera. Vikapeleka vyakumazga vya masuzgo gha algebra kweni vikaghanaghanira yayi mwamzimu wakudumura na wawonse, m'malo mwake vikakhalilira pa mkhalidwe pakuyowoya na kugwiliskika padera pera.^[101] Ichi chikasintha na msandi wa masamu wa ku Persia, Al-Khwarizmi,^{[lower-alpha 21]} uyo wakafumiska buku lake The Compendious Book on Calculation by Completion and Balancing mu 825 CE. Al-Khwarizmi wakapeleka malongosoro ghakwamba ghakuzomerezgeka ghakumazgira ma equation mwakughaŵika mu mitundu 6 yakuzomerezgeka^{[lower-alpha 22]} ndiposo kupereka ndondomeko zakuzomerezgeka, mwakuyila-kuyila, za vyakuvimazga. Mwakuwuskapo nthowa izi ku vithunzithunzi vyapadera vya geometry na kuwona viŵelengero vyambura kumanyikwa kuŵa vinthu vyawonse vya algebra, wakakhozga chiwongo cha kugwira ntchito chakuzomerezgeka. Kusintha uku, kufuma ku kumazga masuzgo ghapadera kuya ku kukhozga nthowa yawonse, nkhwakasintha algebra kuŵa sambiro lakujimira paokha.^[104] Vyakukhwaskira vinyake vyakuzirwa ku algebra vikafuma ku msandi wa masamu wa Chiarabu Thābit ibn Qurra naye so mu vilimika vya 9 CE na msandi wa masamu wa ku Persia Omar Khayyam mu vilimika vya 11 na 12.^[105]

Ku India, Brahmagupta wakasanda umo wangamazgira ma equation gha digiri yachiŵiri na ndondomeko za ma equation gha vyambura-kumanyikwa vinandi mu vilimika vya 7 CE. Pakati pa vyakukula vyake vikaŵako kugwiliskira ntchito zero na vipendeko vyakusuzga (negative numbers) mu ma equation gha algebra.^[106] Ŵasandi ŵa masamu ŵa ku India Mahāvīra mu vilimika vya 9 na Bhāskara II mu vilimika vya 12 ŵakakhozgaso nthowa na vinjeru vya Brahmagupta.^[107] Mu 1247, msandi wa masamu wa ku China Qin Jiushao wakalemba buku la Mathematical Treatise in Nine Sections, ilo likusazgamo nthowa yakuŵelenga ya kuŵelenga mtengo wa ma polynomial, kusazgamo ma polynomial gha madigiri ghapachanya.^[108]

François Viète (kumalyo) na René Descartes ŵakasanga viwimbo vyakwenerana kulongosora ma equation mwamzimu wakudumura na wachidango.

Msandi wa masamu wa ku Italy Fibonacci wakiza na vinjeru na nthowa za Al-Khwarizmi ku Ulaya mu mabuku ghake, kusazgamo Liber Abaci.^[109] Mu 1545, msandi wa masamu wa mitundu yinandi wa ku Italy Gerolamo Cardano wakafumiska buku lake Ars Magna, ilo likakhwaskira vinthu vinandi mu algebra, likadumbiskana vipendeko vya mung'anamuro, ndipo likaŵa lakwamba kupeleka nthowa zawonse zakumazgira ma equation gha digiri yitatu na digiri yinayi.^[110] Mu vilimika vya 16 na 17, ŵasandi ŵa masamu ŵa ku France François Viète na René Descartes ŵakiza na malembelo na viwimbo kwimira vyambura-kumanyikwa na nchito, kupangiska kuti ma equation ghalembeke mwamzimu wachidango na wakudumura. Awo ŵakaŵako panthazi pawo ŵakajighemezga pa malongosoro ghakuyowoya gha masuzgo na vyakumazga.^[111] Ŵalongozgi ŵanyake ŵa mbiri ŵakuwona kukula uku kuŵa mzuŵa wakuzirwa mu mbiri ya algebra ndipo ŵakuwona ivyo vikaŵako pambere kuŵa mbiri-ntchakwamba ya algebra chifukwa vikasoŵanga mzimu wakudumura uwo ukujimira pa kugwiliskira ntchito viwimbo.^[112]

Mu vilimika vya 17 na 18, kuyezgayezga kwakuluska kukachitika kupenja vyakumazga vyawonse ku ma polynomial gha digiri yankonde na yapachanya. Vyose ivi vikafiliskika.^[36] Kumapeleko gha vilimika vya 18, msandi wa masamu wa ku Germany Carl Friedrich Gauss wakalongosora fundo yikuru ya algebra, iyo yikulongosora kuŵako kwa vizero vya ma polynomial gha digiri yiliyose kwambura kupeleka yankho lawonse.^[19] Kwambilira kwa vilimika vya 19, msandi wa masamu wa ku Italy Paolo Ruffini na msandi wa masamu wa ku Norway Niels Henrik Abel ŵakakhozga kuti palije yankho lawonse ilo lilipo ku ma polynomial gha digiri yankonde na yapachanya.^[36] Kwiza pa na pamanyuma pachoko pa vyakusanga vyawo, msandi wa masamu wa ku France Évariste Galois wakakhozga icho pamanyuma chikachemeka sambiro la Galois, ilo likapeleka kupenja kwakukuzura kwa vyakumazga vya ma polynomial apo so likakhozga msingi wa sambiro la magulu.^[20] Ŵasandi ŵa masamu paukhwakuŵa ŵakamanya kuzirwa kwa sambiro la magulu ku magulu ghanyake gha masamu ndipo ŵakaligwiliskira ntchito ku vipangizo nga geometry na sambiro la vipendeko.^[113]

Kwambira pakati pa vilimika vya 19, chikhumbo mu algebra chikasintha kufuma ku kusambira ma polynomial ivyo vikakhwaskana na algebra ya pulayimale kuya ku kupenja kwakukuru kwa magulu gha algebra, ndipo ichi chikawoneka pakufuma kwa algebra yakudumura. Nthowa iyi yikapenja msingi wa maxiom wa nchito za algebra zakupambanapambana.^[114] Kusangika kwa ndondomeko nyifya za algebra izo zikuzirwa pa nchito na vinthu vyakupambanapambana vikalondezga kukula uku, nga Boolean algebra, algebra ya vector, na algebra ya matrix.^[115] Vyakukula vyakuzirwa vyakwamba mu algebra yakudumura vikapangika na ŵasandi ŵa masamu ŵa ku Germany David Hilbert, Ernst Steinitz, na Emmy Noether pamoza na msandi wa masamu wa ku Austria Emil Artin. Ŵakasanda mitundu yakupambanapambana ya magulu gha algebra ndipo ŵakaghapatuliska kuyana na maxiom ghawo ghakuzikapo kuŵa mitundu, nga magulu, mphete, na minda.^[116]

Mzimu wa nthowa yikuru comenecomene iyo yikukhwaskana na algebra ya wonse ukaghanaghanika na msandi wa masamu wa ku England Alfred North Whitehead mu buku lake la 1898 A Treatise on Universal Algebra. Kwambira mu vya 1930, msandi wa masamu wa ku America Garrett Birkhoff wakatandarizga vinjeru ivi ndipo wakakhozga vinjeru vinandi vya msingi vya gulu ili la sambiro.^[117] Kusangika kwa algebra ya wonse kukalongozgera ku kufuma kwa vipimo vinyake vinyifya ivyo vikang'anamuka pa kupanga algebra ya masamu ghanyake—ndikuti, kugwiliskira ntchito nthowa za algebra ku magulu ghanyake gha masamu. Algebra ya topology yikafuma mu vilimika vyakwamba vya 20, kusambira magulu gha algebra nga magulu gha topology na magulu gha Lie.^[118] Mu vya 1940 na 50, algebra ya homology yikafuma, kugwiliskira ntchito nthowa za algebra kusambira homology.^[119] Pafupifupi nyengo yeneyiyo, sambiro la mikhaliro likafuma ndipo lyakhala likugwira ntchito yikuru mu msingi wa masamu.^[120] Vyakukula vinyake vikaŵa kupangika kwa sambiro la vimodel na kusambira algebra zamahara.^[121]

Kugwiliskika Ntchito

Chikoliro cha algebra nkhutandarika comenecomene, mu masamu mwene na mu kugwiliskika kwake ku magulu ghanyake gha masamu.^[122] Kupanga masamu ghanyake kuŵa gha algebra nkhu ndondomeko ya kugwiliskira ntchito nthowa na vinjeru vya algebra ku magulu ghanyake gha masamu, nga geometry, topology, sambiro la vipendeko, na calculus. Ichi chikuchitika mwakugwiliskira ntchito viwimbo mu mtundu wa vyambura-kumanyikwa kulongosora vinjeru vya masamu pa vipimo vikuru, kupangiska kuti ŵasandi ŵa masamu ŵapange vimodel vyakuzomerezgeka ivyo vikulongosora umo vinthu vikukhwaskirana na kukolerana wene.^[123]

Nchito yimoza, iyo yikusangika mu geometry, ni kugwiliskira ntchito mazgu gha algebra kulongosora vithunzithunzi vya geometry. Mwachiyelezgero, equation $y=3x-7$ yikulongosora mzere mu malo gha mibaru yiŵiri apo equation $x^{2}+y^{2}+z^{2}=1$ yikuyana na sphere mu malo gha mibaru yitatu. Vyakuzirwa comenecomene ku geometry ya algebra ni mitundu yakuyana ya algebra,^{[lower-alpha 23]} izo ni vyakumazga vya ndondomeko za ma equation gha polynomial ivyo vingagwiliskika kulongosora vithunzithunzi vyapasuzgika vya geometry.^[125] Kughanaghanira mwa algebra kungamazgaso masuzgo gha geometry. Mwachiyelezgero, munthu wangamanya usange na apo mzere wakulongosoreka na $y=x+1$ ukukhwaskana na dindi (circle) ilo likulongosoreka na $x^{2}+y^{2}=25$ mwakumazga ndondomeko ya ma equation ghakupangika na ma equation ghaŵiri agha.^[126] Topology yikusambira vinjeru vya vithunzithunzi vya geometry panji malo gha topology ivyo vikukhalilira mwambura kusintha para vichitikirwa kusintha kwakulutilira. Topology ya algebra yikuzirwa pa vinjeru vya algebra nga sambiro la magulu kupatuliska malo gha topology. Mwachiyelezgero, magulu gha homotopy ghakupatuliska malo gha topology kuyana na kuŵako kwa majoro panji maenja mwa iwo.^[127]

Sambiro la vipendeko likukhwaskana na vinjeru na ukhwaskano pakati pa vipendeko vyambura kugaŵika. Sambiro la vipendeko la algebra likugwiliskira ntchito nthowa na vinjeru vya algebra ku chipangizo ichi cha kupenja. Vyakuyelezgera ni kugwiliskira ntchito mazgu gha algebra kulongosora madango ghawonse, nga fundo yaumaliro ya Fermat, na kugwiliskira ntchito magulu gha algebra kusanda mkhalidwe wa vipendeko, nga mphete ya vipendeko vyambura kugaŵika.^[128] Chipangizo chakukhwaskana ichi cha combinatorics chikugwiliskira ntchito nthowa za algebra kumazga masuzgo ghakukhwaskana na kuŵelenga, kunozga, na kukoleranya vinthu vyakupambanapambana. Chiyelezgero mu combinatorics ya algebra ni kugwiliskira ntchito sambiro la magulu kusanda ma graph na symmetries.^[129] Vinjeru vya algebra vikuzirwaso ku calculus, iyo yikugwiliskira ntchito mazgu gha masamu kupenja liŵiro la kusintha na kuwungana. Yikuzirwa pa algebra, mwachiyelezgero, kupulikiska umo mazgu agha ghangasinthika na mzimu uwo vyambura-kumanyikwa vikugwira mwa iwo.^[130] Logic ya algebra yikugwiliskira ntchito nthowa za algebra kulongosora na kusanda magulu na vinjeru ivyo vili musingi wa kughanaghanira kwa logic,^[131] kupenja vyose viŵiri, magulu ghenegho gha masamu na kugwiliskika kwawo ku masuzgo ghakuwoneka gha logic.^[132] Yikusazgamo kusambira Boolean algebra kulongosora propositional logic^[133] pamoza na kupanga na kusanda magulu gha algebra ghakuyana na ndondomeko za logic zapasuzgika kuluska.^[134]

Nthowa za algebra zikugwiliskikaso comenecomene mu magulu ghanyake, nga sayansi za chilengechilenge. Mwachiyelezgero, zikugwiliskika kulongosora madango gha kasandiro na kumazga ma equation mu physics, chemistry, na biology.^[136] Kugwiliskika kwakuyana kukusangikaso mu magulu nga economics, geography, engineering (kusazgamo electronics na robotics), na computer science kulongosora ukhwaskano, kumazga masuzgo, na kupanga vimodel vya ndondomeko.^[137] Algebra ya mzere yikugwira ntchito yikuru mu artificial intelligence na machine learning, mwachiyelezgero, mwakukhozgera kuwendeska na kusanda masetu gha data ghakuru mwakuzomerezgeka.^[138] Magulu ghakupambanapambana ghakuzirwa pa magulu gha algebra ghakusambirika na algebra yakudumura. Mwachiyelezgero, sayansi za vinthu nga crystallography na quantum mechanics zikugwiliskira ntchito comenecomene sambiro la magulu,^[139] ilo likugwiliskikaso kusambira ma puzzle nga Sudoku na Rubik's Cube,^[140] na origami.^[141] Vyose viŵiri sambiro la ma code na cryptology zikuzirwa pa algebra yakudumura kumazga masuzgo ghakukhwaskana na kutuma data, nga kujiwezga na vikwaskiro (noise) na kukhozgera chitetezo cha data.^[142]

Masambizgo

Masambizgo gha algebra ghakuzirwa comenecomene pa algebra ya pulayimale, chimoza mwa vifukwa ivyo algebra ya pulayimale yikuchemekaso "algebra ya sukulu". Comenecomene, yikwambiskika yayi kwambira mu masambizgo ghapachanya chifukwa yikukhumba kumanya makora masamu ghakwamba apo yikwizaso na vyakusuzga vinyake vipya ivyo vikukhwaskana na kughanaghanira kwakudumura na kwawonse.^[144] Yikukhumba kuzgoŵeska ŵasambiri na mbali yakuzomerezgeka ya masamu mwakuŵawovwira kupulikiska viwimbo vya masamu, mwachiyelezgero, umo vyambura-kumanyikwa vingagwiliskika kwimira viŵelengero vyambura kumanyikwa. Kususka kwinyake ku ŵasambiri nkhwakuti, kupambana na kuŵelenga kwaumaliro, mazgu gha algebra ghakusuzga comenecomene kumazgika mwakwenderana. M'malo mwake, ŵasambiri ŵakukhumbikwa kusambira umo ŵangaghasinthira kuyana na madango ghanyake, kanandi kuti ŵamanye chiŵelengero chambura kumanyikwa.^[145]

Vipangizo vinyake vyakwiziska ŵasambiri ku mbali yakudumura ya algebra vikuzirwa pa vimodel vyakuwoneka na kulongora ma equation, kusazgamo vyakuyelezgera vya geometry, vipangizo vyakugwira nawo nchito nga vikoli panji makapu, na "makina gha nchito" (function machines) ivyo vikulongora ma equation kuŵa maonekero gha kuluta patsogolo. Nthowa yimoza yikugwiliskira ntchito vipimiro vyakulinganya kuŵa nthowa yakuwoneka yakovwira ŵasambiri kupulikiska masuzgo ghakwamba gha algebra. Uzito wa vinthu vinyake pa chipimiro nchambura kumanyikwa ndipo ukwimira vyambura-kumanyikwa. Kumazga equation kukuyana na kusazga na kuwuskapo vinthu ku mbali zose ziŵiri mwakuti mbali zikhalilire mu ulinganyi mpaka chinthu chimoza pera icho chikhala pa mbali yimoza ni chinthu chambura kumanyikwa uzito wake.^[146] Masuzgo gha mazgu ni chipangizo chinyake chakulongora umo algebra yikugwiliskika ku vinthu vyaunenesko vya moyo. Mwachiyelezgero, ŵasambiri ŵangapika mu mkhalidwe uwo mbale wa Naomi wali na machiyembe ghakuŵirapo kuŵiri kuluska Naomi. Pakuti ŵaŵiri wose pamoza ŵali na machiyembe 12, ŵasambiri ŵakupikika kuti ŵasange equation ya algebra iyo yikulongosora mkhalidwe uwu ( $2x+x=12$ ) na kumanya machiyembe ghaŵi agho Naomi wali nagho ( $x=4$ ).^[147]

Pa vipimo vya yunivesite, ŵasambiri ŵa masamu ŵakukhwaskana na mitu yapachanya ya algebra kufuma ku algebra ya mzere na algebra yakudumura. Makalasi ghakwamba gha digiri yakwamba mu algebra ya mzere ghakuzirwa pa ma matrix, malo gha vector, na mapu gha mzere. Para ŵamalizga, ŵasambiri comenecomene ŵakwiziskika ku algebra yakudumura, uko ŵakusambira magulu gha algebra nga magulu, mphete, na minda, pamoza na ukhwaskano pakati pawo. Masambiro (curriculum) ghakuzirwa comenecomene ghakukhwaskaso vithuzi vyapadera vya magulu gha algebra, nga ndondomeko za vipendeko vyakugaŵikika, vipendeko vyeneco, na ma polynomial.^[148]

Vinyake Vyakukhwaskana

References

Notes

↑ When understood in the widest sense, an algebraic operation is a function from a Cartesian power of a set into that set, expressed formally as $\omega :A^{n}\to A$ . The addition of real numbers is an example of an algebraic operation: it takes two numbers as input and produces one number as output. It has the form $+:\mathbb {R} ^{2}\to \mathbb {R}$ .^[3]
↑ Algebra is covered by division 512 in the Dewey Decimal Classification^[5] and subclass QA 150-272.5 in the Library of Congress Classification.^[6] It encompasses several areas in the Mathematics Subject Classification.^[7]
↑ The exact meaning of the term al-jabr in al-Khwarizmi's work is disputed. In some passages, it expresses that a quantity diminished by subtraction is restored to its original value, similar to how a bonesetter restores broken bones by bringing them into proper alignment.^[17]
↑ These changes were in part triggered by discoveries that solved many of the older problems of algebra. For example, the proof of the fundamental theorem of algebra demonstrated the existence of complex solutions of polynomials^[19] and the introduction of Galois theory characterized the polynomials that have general solutions.^[20]
↑ Constants represent fixed numbers that do not change during the study of a specific problem.^[24]
↑ Kupiluskika (inversion) nkhung'anamura kusanga matrix yinyake iyo, para yandaniskika na matrix yakwamba, yikupeleka matrix ya "identity"—iyo njakuyana na cipendeko 1 pa nkhani ya kwandaniska manambara.
↑ Mwachiyelezgero, ma equation $x_{1}-3x_{2}=0$ na $x_{1}-3x_{2}=7$ ghakususkana wene chifukwa palije mtengo wa $x_{1}$ na $x_{2}$ uwo ungamazga ma equation ghaŵiri ghara pa nyengo yimoza.^[46]
↑ Usange ndondomeko yakuyana ya ma equation yili na yankho limoza pera, ivi vikukhozga pa chiŵelengero cha vyambura-kumanyikwa na ma equation ghakujipima paŵiri. Ma equation ghakuti ghakujipima paŵiri usange ghakupeleka yayi mzimu umoza ndipo ghangafuma yayi kuchimoza na chinyake. Yankho limoza likuŵako usange chiŵelengero cha vyambura-kumanyikwa nchakuyana na chiŵelengero cha ma equation ghakujipima paŵiri. Ndondomeko zambura kumazgikirathu, kusambazgana na izi, zili na vyambura-kumanyikwa vinandi kuluska ma equation ghakujipima paŵiri ndipo zili na viŵelengero vyambura kumazgika vya vyakumazga usange nizakuyana.^[47]
↑ Gulu la vinthu ni chiŵungano cha vinthu vyakupambanapambana ivyo vilije njila yakukhwaskana, nga vipendeko, ma vector, panji magulu ghanyake. Sambiro la magulu likulongosora madango na mikhaliro ya magulu.^[56]
↑ Kuyana na malongosoro ghanyake, magulu gha algebra ghakusazgapo chinthu chakupambana nga chigaŵa chinyake chakusazgikapo, nga chinthu cha identity pa nkhani ya kwandaniska.^[57]
↑ Magulu ghanyake gha algebra agho ghakusambirika na algebra yakudumura ghali na nchito yimoza kusazgikapo pa nchito zaŵiri. Mwachiyelezgero, malo gha vector ghakuŵa na muyeso ghali na muyeso (norm), uwo ni nchito yimoza iyo yikugwiliskikira comenecomene kukhwaskanya vector na utali wake.^[58]
↑ Viwimbo $\circ$ na $\star$ vikugwiliskika mu nkhani iyi kwimira nchito yiliyose iyo yingayana panji yambakuyana na nchito za masamu.^[62]
↑ Ŵalembi ŵanyake ŵakukhumba yayi kuŵako kwa chinthu cha identity mu kwandaniska. Mphete iyo yilije chinthu cha identity mu kwandaniska nyengo zinyake yikuchemeka rng.^[69]
↑ Ichi chikung'anamura kuti kwandaniska na kusazga vikugwiliskira ntchito vinthu vyakupambanapambana vya identity.^[72]
↑ Kuyana na malongosoro ghanyake, chigulu chamunsi chingaŵaso na nchito zichoko.^[83]
↑ Ichi chikung'anamura kuti vinthu vyose vya gulu lakwamba nivyaso vinthu vya gulu lachiŵiri kweni gulu lachiŵiri lingaŵa na vinthu ivyo vilipo yayi mu gulu lakwamba.^[84]
↑ Nthowa yakupambanako pachoko yikupulikiska algebra ya wonse kuŵa sambiro la mtundu umoza wa magulu gha algebra ghakumanyikwa nga ma algebra ghawonse. Ma algebra ghawonse ghakulongosoreka mwakuzirwa kuti ghasazgemo magulu ghanyake ghanandi gha algebra. Mwachiyelezgero, magulu na mphete ni mitundu yapadera ya algebra ya wonse.^[86]
↑ Mtundu uliwose wa gulu la algebra ukupanga yayi mtundu wakuyana. Mwachiyelezgero, magulu na mphete vyose vikupanga mtundu wakuyana kweni minda njambura kupanga.^[89]
↑ Kupambuka vinthu vyakuyana, algebra ya wonse yikukhumbikwaso na vinjeru vya mzimu ivyo vikukhwaskana na vinthu vyakuyana pachoko (quasi-identities). Chinthu chakuyana pachoko ni chinthu chakuyana icho chikukhumbikwa pera pasi pa vikhumbikwa vinyake (ivyo vikuŵa mu mtundu wa chigaŵa cha Horn^[90]). Ni kutandarizga kwa chinthu chakuyana mu mzimu wakuti chinthu chilichose chakuyana ni chinthu chakuyana pachoko kweni chinthu chilichose chakuyana pachoko ni yayi chinthu chakuyana. Mtundu wakuyana pachoko ni gulu la magulu ghose gha algebra agho ghakukwaniska vinthu vinyake vyakuyana pachoko.^[91]
↑ Zuŵa lenelene nkhwakususkana, ndipo ŵalongozgi ŵanyake ŵa mbiri ŵakuti liŵe pafupifupi mu 1550 BCE.^[94]
↑ Ŵalongozgi ŵanyake ŵa mbiri ŵakumuwona kuŵa "wiske wa algebra" apo ŵanyake ŵakupeleka zina ili ku Diophantus.^[102]
↑ Mitundu iyi ni $bx=c$ , $ax^{2}=bx$ , $ax^{2}=c$ , $ax^{2}+bx=c$ , $ax^{2}+c=bx$ , na $ax^{2}=bx+c$ .^[103]
↑ Mitundu yakuyana ya algebra iyo yikusambirika mu geometry njakupambana na mitundu yakuyana yikuru iyo yikusambirika mu algebra ya wonse.^[124]

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↑
Tanton 2005, p. 9
- Kvasz 2006, p. 290
- Corry 2024, § Problem Solving in Egypt and Babylon
↑
Tanton 2005, p. 9
- Kvasz 2006, p. 290
- Corry 2024, § The Pythagoreans and Euclid
↑
Merzlyakov & Shirshov 2020, § Historical Survey
- Sialaros 2018, p. 55
- Corry 2024, § Diophantus
↑
Hettle 2015, pp. 139–141, 160–161
- Christianidis & Megremi 2019, pp. 16–17
↑ Burgin 2022, p. 10
↑ Higgins 2015, p. 89
↑
Kvasz 2006, pp. 290–291
- Sialaros 2018, p. 55
- Boyer & Merzbach 2011, p. 161
- Derbyshire 2006, p. 31
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Boyer & Merzbach 2011, p. 161
- Derbyshire 2006, p. 31
↑ Bradley 2006, p. 95.
↑
Tanton 2005, p. 10
- Kvasz 2006, pp. 291–293
- Merzlyakov & Shirshov 2020, § Historical Survey
↑
Waerden 2013, pp. 3, 15–16, 24–25
- Jenkins 2010, p. 82
- Pickover 2009, p. 90
↑
Tanton 2005, pp. 9–10
- Corry 2024, § The Equation in India and China
↑
Seshadri 2010, p. 156
- Emch, Sridharan & Srinivas 2005, p. 20
↑
Smorynski 2007, p. 137
- Zwillinger 2002, p. 812
↑
Waerden 2013, pp. 32–35
- Tanton 2005, p. 10
- Kvasz 2006, p. 293
↑
Tanton 2005, p. 10
- Kvasz 2006, p. 293
- Corry 2024, § Cardano and the Solving of Cubic and Quartic Equations
- Miyake 2002, p. 268
↑
Tanton 2005, p. 10
- Kvasz 2006, pp. 291–292, 297–298, 302
- Merzlyakov & Shirshov 2020, § Historical Survey
- Corry 2024, § Viète and the Formal Equation, § Analytic Geometry
↑
Hazewinkel 1994, p. 73
- Merzlyakov & Shirshov 2020, § Historical Survey
↑
Corry 2024, § Applications of Group Theory
- Bueno & French 2018, pp. 73–75
↑
Merzlyakov & Shirshov 2020, § Historical Survey
- Tanton 2005, p. 10
- Corry 2024, § Structural Algebra
- Hazewinkel 1994, pp. 73–74
↑
Merzlyakov & Shirshov 2020, § Historical Survey
- Tanton 2005, p. 10
- Corry 2024, § Matrices, § Quaternions and Vectors
↑
Merzlyakov & Shirshov 2020, § Historical Survey
- Corry 2024, § Hilbert and Steinitz, § Noether and Artin
- Hazewinkel 1994, pp. 73–74
↑
Grätzer 2008, p. vii
- Chang & Keisler 1990, p. 603
- Knoebel 2011, p. 5
- Hazewinkel 1994, pp. 74–75
↑
Hazewinkel 1994, pp. 74–75
- Kleiner 2007, p. 100
- Carlson 2024, § History of topology
↑
Hazewinkel 1994, pp. 74–75
- Weibel 1995, p. xi, 4
↑
Krömer 2007, p. 61
- Laos 1998, p. 100
↑
Hazewinkel 1994, pp. 74–75
- Pratt 2022, § 6. Free Algebras
↑
Houston 2004, p. 319
- Neri 2019, p. xii
- Lidl & Pilz 1997, pp. vii–viii
↑
Kleiner 2007, p. 100
- Pratt 2022, § 5. Algebraization of Mathematics
- Maddocks 2008, p. 130
- Pratt 2022, § 5. Algebraization of Mathematics
- Mancosu 1999, pp. 84–85
↑
Pratt 2022, § 1.4 Cartesian geometry, § 3. Universal Algebra
- Danilov 2006, p. 174
↑
Pratt 2022, § 5.1 Algebraic Geometry
- Danilov 2006, pp. 172, 174
↑ Vince 2007, p. 133
↑
Pratt 2022, § 5.3 Algebraic Topology
- Rabadan & Blumberg 2019, pp. 49–50
- Nakahara 2018, p. 121
- Weisstein 2003, pp. 52–53
↑
Pratt 2022, § 5.2 Algebraic Number Theory
- Jarvis 2014, p. 1
- Viterbo & Hong 2011, p. 127
↑
Gowers, Barrow-Green & Leader 2010, pp. 550, 561
- Godsil 2017, p. viii
- Betten et al. 2013, p. ix
↑
Kilty & McAllister 2018, pp. x, 347, 589
- Bressoud 2021, p. 64
↑
Halmos 1956, p. 363
- Burris & Legris 2021, § 1. Introduction
↑ Andréka, Németi & Sain 2001, pp. 133–134
↑
Andréka, Madarász & Németi 2020, § Concrete Algebraic Logic
- Pratt 2022, § 5.4 Algebraic Logic
- Plotkin 2012, pp. 155–156
- Jansana 2022, Lead section
↑
Andréka, Madarász & Németi 2020, § Abstract Algebraic Logic
- Jansana 2022, § 4. Algebras
↑ Joyner 2008, p. 92
↑
Houston 2004, p. 319
- Neri 2019, p. xii
- Anton & Rorres 2010, p. 327
↑
Neri 2019, p. xii
- Aleskerov, Ersel & Piontkovski 2011, pp. 1–9
- Straffin 1980, p. 269
- Menini & Oystaeyen 2017, p. v
- Lovett 2015, p. ix
- Lidl & Pilz 1997, pp. vii–viii
↑
Wheeler 2023, p. 29, 36–37
- Gallier & Quaintance 2020, pp. 1–2
↑
Klimov 2014, p. ix
- Bengtsson & Życzkowski 2017, pp. 313–353
↑ Terras 2019, pp. 63–64, 142
↑ Hull 2021, p. 180
↑
Lidl & Pilz 1997, pp. 183–184, 239–240
- Carstensen, Fine & Rosenberger 2011, pp. 326–327
↑
Kieran 2006, p. 15
- Kaput 2018, p. 186
- Gardella & DeLucia 2020, pp. 19–22
↑
Arcavi, Drijvers & Stacey 2016, p. xiii
- Dekker & Dolk 2011, p. 69
↑
Arcavi, Drijvers & Stacey 2016, pp. 2–5
- Drijvers, Goddijn & Kindt 2011, pp. 8–10, 16–18
↑
Kieran 2006, p. 15
- Kaput 2018, p. 186
- Gardella & DeLucia 2020, pp. 19–22
- Star et al. 2015, pp. 16–17
↑
Arcavi, Drijvers & Stacey 2016, pp. 58–59
- Drijvers, Goddijn & Kindt 2011, p. 13
↑ Hausberger, Zandieh & Fleischmann 2021, pp. 147–148

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External links

Template:Algebra Template:Polynomials Template:Areas of mathematics

[4] When understood in the widest sense, an algebraic operation is a function from a Cartesian power of a set into that set, expressed formally as $\omega :A^{n}\to A$ . The addition of real numbers is an example of an algebraic operation: it takes two numbers as input and produces one number as output. It has the form $+:\mathbb {R} ^{2}\to \mathbb {R}$ .^[3]

[9] Algebra is covered by division 512 in the Dewey Decimal Classification^[5] and subclass QA 150-272.5 in the Library of Congress Classification.^[6] It encompasses several areas in the Mathematics Subject Classification.^[7]

[20] The exact meaning of the term al-jabr in al-Khwarizmi's work is disputed. In some passages, it expresses that a quantity diminished by subtraction is restored to its original value, similar to how a bonesetter restores broken bones by bringing them into proper alignment.^[17]

[24] These changes were in part triggered by discoveries that solved many of the older problems of algebra. For example, the proof of the fundamental theorem of algebra demonstrated the existence of complex solutions of polynomials^[19] and the introduction of Galois theory characterized the polynomials that have general solutions.^[20]

[29] Constants represent fixed numbers that do not change during the study of a specific problem.^[24]

[47] Kupiluskika (inversion) nkhung'anamura kusanga matrix yinyake iyo, para yandaniskika na matrix yakwamba, yikupeleka matrix ya "identity"—iyo njakuyana na cipendeko 1 pa nkhani ya kwandaniska manambara.

[53] Mwachiyelezgero, ma equation $x_{1}-3x_{2}=0$ na $x_{1}-3x_{2}=7$ ghakususkana wene chifukwa palije mtengo wa $x_{1}$ na $x_{2}$ uwo ungamazga ma equation ghaŵiri ghara pa nyengo yimoza.^[46]

[55] Usange ndondomeko yakuyana ya ma equation yili na yankho limoza pera, ivi vikukhozga pa chiŵelengero cha vyambura-kumanyikwa na ma equation ghakujipima paŵiri. Ma equation ghakuti ghakujipima paŵiri usange ghakupeleka yayi mzimu umoza ndipo ghangafuma yayi kuchimoza na chinyake. Yankho limoza likuŵako usange chiŵelengero cha vyambura-kumanyikwa nchakuyana na chiŵelengero cha ma equation ghakujipima paŵiri. Ndondomeko zambura kumazgikirathu, kusambazgana na izi, zili na vyambura-kumanyikwa vinandi kuluska ma equation ghakujipima paŵiri ndipo zili na viŵelengero vyambura kumazgika vya vyakumazga usange nizakuyana.^[47]

[65] Gulu la vinthu ni chiŵungano cha vinthu vyakupambanapambana ivyo vilije njila yakukhwaskana, nga vipendeko, ma vector, panji magulu ghanyake. Sambiro la magulu likulongosora madango na mikhaliro ya magulu.^[56]

[67] Kuyana na malongosoro ghanyake, magulu gha algebra ghakusazgapo chinthu chakupambana nga chigaŵa chinyake chakusazgikapo, nga chinthu cha identity pa nkhani ya kwandaniska.^[57]

[69] Magulu ghanyake gha algebra agho ghakusambirika na algebra yakudumura ghali na nchito yimoza kusazgikapo pa nchito zaŵiri. Mwachiyelezgero, malo gha vector ghakuŵa na muyeso ghali na muyeso (norm), uwo ni nchito yimoza iyo yikugwiliskikira comenecomene kukhwaskanya vector na utali wake.^[58]

[74] Viwimbo $\circ$ na $\star$ vikugwiliskika mu nkhani iyi kwimira nchito yiliyose iyo yingayana panji yambakuyana na nchito za masamu.^[62]

[82] Ŵalembi ŵanyake ŵakukhumba yayi kuŵako kwa chinthu cha identity mu kwandaniska. Mphete iyo yilije chinthu cha identity mu kwandaniska nyengo zinyake yikuchemeka rng.^[69]

[86] Ichi chikung'anamura kuti kwandaniska na kusazga vikugwiliskira ntchito vinthu vyakupambanapambana vya identity.^[72]

[98] Kuyana na malongosoro ghanyake, chigulu chamunsi chingaŵaso na nchito zichoko.^[83]

[100] Ichi chikung'anamura kuti vinthu vyose vya gulu lakwamba nivyaso vinthu vya gulu lachiŵiri kweni gulu lachiŵiri lingaŵa na vinthu ivyo vilipo yayi mu gulu lakwamba.^[84]

[103] Nthowa yakupambanako pachoko yikupulikiska algebra ya wonse kuŵa sambiro la mtundu umoza wa magulu gha algebra ghakumanyikwa nga ma algebra ghawonse. Ma algebra ghawonse ghakulongosoreka mwakuzirwa kuti ghasazgemo magulu ghanyake ghanandi gha algebra. Mwachiyelezgero, magulu na mphete ni mitundu yapadera ya algebra ya wonse.^[86]

[107] Mtundu uliwose wa gulu la algebra ukupanga yayi mtundu wakuyana. Mwachiyelezgero, magulu na mphete vyose vikupanga mtundu wakuyana kweni minda njambura kupanga.^[89]

[110] Kupambuka vinthu vyakuyana, algebra ya wonse yikukhumbikwaso na vinjeru vya mzimu ivyo vikukhwaskana na vinthu vyakuyana pachoko (quasi-identities). Chinthu chakuyana pachoko ni chinthu chakuyana icho chikukhumbikwa pera pasi pa vikhumbikwa vinyake (ivyo vikuŵa mu mtundu wa chigaŵa cha Horn^[90]). Ni kutandarizga kwa chinthu chakuyana mu mzimu wakuti chinthu chilichose chakuyana ni chinthu chakuyana pachoko kweni chinthu chilichose chakuyana pachoko ni yayi chinthu chakuyana. Mtundu wakuyana pachoko ni gulu la magulu ghose gha algebra agho ghakukwaniska vinthu vinyake vyakuyana pachoko.^[91]

[114] Zuŵa lenelene nkhwakususkana, ndipo ŵalongozgi ŵanyake ŵa mbiri ŵakuti liŵe pafupifupi mu 1550 BCE.^[94]

[123] Ŵalongozgi ŵanyake ŵa mbiri ŵakumuwona kuŵa "wiske wa algebra" apo ŵanyake ŵakupeleka zina ili ku Diophantus.^[102]

[125] Mitundu iyi ni $bx=c$ , $ax^{2}=bx$ , $ax^{2}=c$ , $ax^{2}+bx=c$ , $ax^{2}+c=bx$ , na $ax^{2}=bx+c$ .^[103]

[147] Mitundu yakuyana ya algebra iyo yikusambirika mu geometry njakupambana na mitundu yakuyana yikuru iyo yikusambirika mu algebra ya wonse.^[124]

[1] Merzlyakov & Shirshov 2020, Lead section
Gilbert & Nicholson 2004, p. 4

[mwBnM] Gilbert & Nicholson 2004, p. 4

[2] Fiche & Hebuterne 2013, p. 326
Merzlyakov & Shirshov 2020, § The Subject Matter of Algebra, Its Principal Branches and Its Connection with Other Branches of Mathematics.
Gilbert & Nicholson 2004, p. 4

[mwBoQ] Merzlyakov & Shirshov 2020, § The Subject Matter of Algebra, Its Principal Branches and Its Connection with Other Branches of Mathematics.

[mwBoc] Gilbert & Nicholson 2004, p. 4

[3] Baranovich 2023, Lead section

[5] Pratt 2022, Lead section, § 1. Elementary Algebra, § 2. Abstract Algebra, § 3. Universal Algebra
Merzlyakov & Shirshov 2020, § The Subject Matter of Algebra, Its Principal Branches and Its Connection with Other Branches of Mathematics.

[mwBpg] Merzlyakov & Shirshov 2020, § The Subject Matter of Algebra, Its Principal Branches and Its Connection with Other Branches of Mathematics.

[6] Higham 2019, p. 296

[7] Library of Congress, p. 3

[8] zbMATH Open 2024

[10] Maddocks 2008, p. 129
Burgin 2022, p. 45

[mwBrQ] Burgin 2022, p. 45

[auto1-11] 1 2 Romanowski 2008, pp. 302–303
HC Staff 2022
MW Staff 2023
Bukhshtab & Pechaev 2020

[mwBsQ] HC Staff 2022

[mwBsY] MW Staff 2023

[mwBsg] Bukhshtab & Pechaev 2020

[12] Maddocks 2008, pp. 129–130
Pratt 2022, Lead section, § 1. Elementary Algebra
Wagner & Kieran 2018, p. 225

[mwBtU] Pratt 2022, Lead section, § 1. Elementary Algebra

[mwBtc] Wagner & Kieran 2018, p. 225

[13] Maddocks 2008, pp. 131–132
Pratt 2022, § 2. Abstract Algebra
Wagner & Kieran 2018, p. 225

[mwBuY] Pratt 2022, § 2. Abstract Algebra

[mwBug] Wagner & Kieran 2018, p. 225

[14] Pratt 2022, § 3. Universal Algebra
Grillet 2007, p. 559
Denecke & Wismath 2018, p. v
Cohn 2012, p. xiii

[mwBvU] Grillet 2007, p. 559

[mwBvk] Denecke & Wismath 2018, p. v

[mwBv4] Cohn 2012, p. xiii

[15] Cresswell 2010, p. 11
OUP Staff
Menini & Oystaeyen 2017, p. 722

[mwBw0] OUP Staff

[mwBw8] Menini & Oystaeyen 2017, p. 722

[16] Weisstein 2003, p. 46
Walz 2016, Algebra

[mwBx0] Walz 2016, Algebra

[17] Weisstein 2003, p. 46
Brešar 2014, p. xxxiii
Golan 1995, pp. 219–227

[mwByo] Brešar 2014, p. xxxiii

[mwBy4] Golan 1995, pp. 219–227

[18] EoM Staff 2017

[19] Oaks & Alkhateeb 2007, pp. 45–46, 58
Gandz 1926, p. 437

[mwBz4] Gandz 1926, p. 437

[21] Cresswell 2010, p. 11
OUP Staff
Menini & Oystaeyen 2017, p. 722
Hoad 1993, p. 10

[mwB0w] OUP Staff

[mwB04] Menini & Oystaeyen 2017, p. 722

[mwB1I] Hoad 1993, p. 10

[auto7-22] 1 2 Tanton 2005, p. 10
Kvasz 2006, p. 308
Corry 2024, § The Fundamental Theorem of Algebra

[mwB14] Kvasz 2006, p. 308

[mwB2E] Corry 2024, § The Fundamental Theorem of Algebra

[auto2-23] 1 2 Kvasz 2006, pp. 314–345
Merzlyakov & Shirshov 2020, § Historical Survey
Corry 2024, § Galois Theory, § Applications of Group Theory

[mwB2w] Merzlyakov & Shirshov 2020, § Historical Survey

[mwB28] Corry 2024, § Galois Theory, § Applications of Group Theory

[25] Tanton 2005, p. 10
Corry 2024, § Structural Algebra
Hazewinkel 1994, pp. 73–74

[mwB3s] Corry 2024, § Structural Algebra

[mwB30] Hazewinkel 1994, pp. 73–74

[26] Arcavi, Drijvers & Stacey 2016, p. 2
Benson 2003, pp. 111–112

[mwB40] Benson 2003, pp. 111–112

[auto4-27] 1 2 Maddocks 2008, p. 129
Berggren 2015, Lead section
Pratt 2022, § 1. Elementary Algebra
Merzlyakov & Shirshov 2020, § 1. Historical Survey

[mwB50] Berggren 2015, Lead section

[mwB58] Pratt 2022, § 1. Elementary Algebra

[mwB6E] Merzlyakov & Shirshov 2020, § 1. Historical Survey

[28] Sobolev 2015

[30] Maddocks 2008, pp. 129–130
Young 2010, p. 999
Majewski 2004, p. 347
Pratt 2022, § 1. Elementary Algebra
Sorell 2000, p. 19

[mwB68] Young 2010, p. 999

[mwB7M] Majewski 2004, p. 347

[mwB7Y] Pratt 2022, § 1. Elementary Algebra

[mwB7g] Sorell 2000, p. 19

[31] Musser, Peterson & Burger 2013, p. 16
Goodman 2001, p. 5
Williams 2022

[mwB8c] Goodman 2001, p. 5

[mwB8s] Williams 2022

[32] Maddocks 2008, p. 130
McKeague 1986, pp. 51–54
Pratt 2022, § 1. Elementary Algebra
Merzlyakov & Shirshov 2020, § 1. Historical Survey

[mwB9c] McKeague 1986, pp. 51–54

[mwB9s] Pratt 2022, § 1. Elementary Algebra

[mwB90] Merzlyakov & Shirshov 2020, § 1. Historical Survey

[33] Tan, Steeb & Hardy 2012, p. 306
Lamagna 2019, p. 150

[mwB-w] Lamagna 2019, p. 150

[34] Berggren 2015, § Solving Systems of Algebraic Equations
McKeague 2014, p. 386
McKeague 1986, p. 148

[mwB_g] McKeague 2014, p. 386

[mwB_w] McKeague 1986, p. 148

[35] Maddocks 2008, pp. 130–131
Rohde et al. 2012, p. 89
Walz 2016, Algebra

[mwCAo] Rohde et al. 2012, p. 89

[mwCA4] Walz 2016, Algebra

[36] Bracken & Miller 2014, pp. 386–387
Kaufmann & Schwitters 2011, p. 220
Markushevich 2015

[mwCBw] Kaufmann & Schwitters 2011, p. 220

[mwCCA] Markushevich 2015

[37] Sahai & Bist 2002, p. 21
Maddocks 2008, p. 131
Barrera-Mora 2023, pp. ix, 1–2

[mwCC4] Maddocks 2008, p. 131

[mwCDE] Barrera-Mora 2023, pp. ix, 1–2

[38] Geddes, Czapor & Labahn 2007, p. 46

[39] Lukas 2022, pp. 47–49
Berggren 2015, § Algebraic Expressions, § Solving Algebraic Equations

[mwCEg] Berggren 2015, § Algebraic Expressions, § Solving Algebraic Equations

[40] Berggren 2015, § Solving algebraic equations
Corry 2024, § Classical algebra

[mwCFI] Corry 2024, § Classical algebra

[auto3-41] 1 2 3 Tanton 2005, p. 10
Merzlyakov & Shirshov 2020, § Historical Survey
Corry 2024, § Impasse with Radical Methods

[mwCGI] Merzlyakov & Shirshov 2020, § Historical Survey

[mwCGU] Corry 2024, § Impasse with Radical Methods

[42] Igarashi et al. 2014, p. 103

[43] Berggren 2015, § Solving algebraic equations
Tanton 2005, p. 10
Kvasz 2006, p. 308
Corry 2024, § The Fundamental Theorem of Algebra

[mwCHY] Tanton 2005, p. 10

[mwCHk] Kvasz 2006, p. 308

[mwCHw] Corry 2024, § The Fundamental Theorem of Algebra

[44] Maddocks 2008, p. 131
Barrera-Mora 2023, pp. ix, 1–2,

[mwCIg] Barrera-Mora 2023, pp. ix, 1–2,

[45] Anton & Rorres 2013, pp. 2–3
Maddocks 2008, p. 131
Voitsekhovskii 2011

[mwCJc] Maddocks 2008, p. 131

[mwCJo] Voitsekhovskii 2011

[46] Saikia 2008, p. 1
Lal 2017, p. 31
Mirakhor & Krichene 2014, p. 107

[mwCKc] Lal 2017, p. 31

[mwCKs] Mirakhor & Krichene 2014, p. 107

[48] Brown 2015, pp. 30–31
Waerden 2003, pp. 70–72

[mwCLs] Waerden 2003, pp. 70–72

[49] Young 2010, pp. 697–698
Maddocks 2008, p. 131
Sullivan 2010, pp. 53–54

[mwCMk] Maddocks 2008, p. 131

[mwCMw] Sullivan 2010, pp. 53–54

[50] Anton & Rorres 2013, pp. 7–8
Sullivan 2010, pp. 55–56
Atanasiu & Mikusinski 2019, p. 75

[mwCNs] Sullivan 2010, pp. 55–56

[mwCN8] Atanasiu & Mikusinski 2019, p. 75

[51] Maddocks 2008, p. 131
Anton & Rorres 2013, pp. 7–8, 11, 491

[mwCO4] Anton & Rorres 2013, pp. 7–8, 11, 491

[auto10-52] 1 2 Anton & Rorres 2013, pp. 3–7
Mortensen 2013, pp. 73–74
Young 2023, pp. 714–715

[mwCP8] Mortensen 2013, pp. 73–74

[mwCQM] Young 2023, pp. 714–715

[auto11-54] 1 2 Maddocks 2008, p. 131
Harrison & Waldron 2011, p. 464
Anton 2013, p. 255

[mwCRM] Harrison & Waldron 2011, p. 464

[mwCRg] Anton 2013, p. 255

[56] Valenza 2012, p. vii
Chahal 2018, § 1.1 What is Linear Algebra?
Solomon 2014, pp. 57–58, 61–62
Ricardo 2009, p. 389

[mwCSc] Chahal 2018, § 1.1 What is Linear Algebra?

[mwCSo] Solomon 2014, pp. 57–58, 61–62

[mwCS4] Ricardo 2009, p. 389

[57] Solomon 2014, p. 57
Ricardo 2009, pp. 395–396

[mwCT0] Ricardo 2009, pp. 395–396

[58] Anton & Rorres 2013, pp. 3–5
Young 2010, pp. 696–697
Sneyd, Fewster & McGillivray 2022, p. 211

[mwCUw] Young 2010, pp. 696–697

[mwCVA] Sneyd, Fewster & McGillivray 2022, p. 211

[59] Anton & Rorres 2013, pp. 3–5
Young 2010, p. 713
Sneyd, Fewster & McGillivray 2022, p. 211

[mwCWA] Young 2010, p. 713

[mwCWQ] Sneyd, Fewster & McGillivray 2022, p. 211

[60] Gilbert & Nicholson 2004, p. 1
Dominich 2008, p. 19

[mwCXU] Dominich 2008, p. 19

[61] Maddocks 2008, pp. 131–132
Pratt 2022, Lead section, § 2. Abstract Algebra
Gilbert & Nicholson 2004, pp. 1–3
Dominich 2008, p. 19

[mwCYM] Pratt 2022, Lead section, § 2. Abstract Algebra

[mwCYU] Gilbert & Nicholson 2004, pp. 1–3

[mwCYo] Dominich 2008, p. 19

[auto9-62] 1 2 Pratt 2022, Lead section, § 2. Abstract Algebra
Merzlyakov & Shirshov 2020, The Subject Matter of Algebra, Its Principal Branches and Its Connection with Other Branches of Mathematics.
Bourbaki 1998, pp. 428–430, 446

[mwCZg] Merzlyakov & Shirshov 2020, The Subject Matter of Algebra, Its Principal Branches and Its Connection with Other Branches of Mathematics.

[mwCZs] Bourbaki 1998, pp. 428–430, 446

[63] Hausberger 2020, Abstract Algebra Teaching and Learning

[64] Tanton 2005, p. 460
Murthy 2012, p. 1.3

[mwCas] Murthy 2012, p. 1.3

[auto5-66] 1 2 Ovchinnikov 2015, p. 27

[68] Grillet 2007, p. 247

[70] Whitelaw 1995, p. 61
Nicholson 2012, p. 70
Fiche & Hebuterne 2013, p. 326
Pratt 2022, Lead section, § 2. Abstract Algebra

[mwCcg] Nicholson 2012, p. 70

[mwCcw] Fiche & Hebuterne 2013, p. 326

[mwCdE] Pratt 2022, Lead section, § 2. Abstract Algebra

[71] Maddocks 2008, pp. 131–132
Pratt 2022, Lead section, § 2. Abstract Algebra

[mwCd0] Pratt 2022, Lead section, § 2. Abstract Algebra

[72] Olver 1999, pp. 55–56
Abas & Salman 1994, pp. 58–59
Häberle 2009, p. 640

[mwCeo] Abas & Salman 1994, pp. 58–59

[mwCe8] Häberle 2009, p. 640

[73] Gilbert & Nicholson 2004, p. 4

[75] Kargapolov & Merzlyakov 2016, § Definition
Khattar & Agrawal 2023, pp. 4–6
Maddocks 2008, pp. 131–132
Pratt 2022, Lead section, § 2. Abstract Algebra
Neri 2019, p. 258

[mwCgM] Khattar & Agrawal 2023, pp. 4–6

[mwCgg] Maddocks 2008, pp. 131–132

[mwCgs] Pratt 2022, Lead section, § 2. Abstract Algebra

[mwCg0] Neri 2019, p. 258

[76] Khattar & Agrawal 2023, pp. 6–7
Maddocks 2008, pp. 131–132
Adhikari & Adhikari 2013, p. 72

[mwCho] Maddocks 2008, pp. 131–132

[mwCh0] Adhikari & Adhikari 2013, p. 72

[77] McWeeny 2002, p. 6
Kramer & Pippich 2017, p. 49

[mwCi0] Kramer & Pippich 2017, p. 49

[78] Tanton 2005, p. 242
Bhattacharya, Jain & Nagpaul 1994, p. 141
Weisstein 2003, p. 1020

[mwCjw] Bhattacharya, Jain & Nagpaul 1994, p. 141

[mwCkE] Weisstein 2003, p. 1020

[79] Elwes 2006
Wilson 2009, p. 2

[mwCk0] Wilson 2009, p. 2

[80] Weisstein 2003, p. 2579
Maxwell 2009, pp. 73–74
Pratt 2022, § 2.3 Rings

[mwClo] Maxwell 2009, pp. 73–74

[mwCl4] Pratt 2022, § 2.3 Rings

[81] Silverman 2022, p. 64

[83] Geddes, Czapor & Labahn 2007, p. 24

[84] Smith 2015, p. 161

[85] Moh 1992, pp. 102–103

[87] Geddes, Czapor & Labahn 2007, p. 24
Weisstein 2003, pp. 1047, 2579
Pratt 2022, § 2.4 Fields

[mwCoc] Weisstein 2003, pp. 1047, 2579

[mwCoo] Pratt 2022, § 2.4 Fields

[88] Irving 2004, pp. 77, 236
Weisstein 2003, pp. 1047, 2579
Hohn 2013, pp. 83–84

[mwCpY] Weisstein 2003, pp. 1047, 2579

[mwCpk] Hohn 2013, pp. 83–84

[89] Serovajsky 2020, § Room 4B.5 Rings
Kleiner 2007, p. 63
Kline 1990, p. 1153

[mwCqU] Kleiner 2007, p. 63

[mwCqk] Kline 1990, p. 1153

[90] Waerden 2003, pp. 110–114, 231, 246
Karpilovsky 1989, p. 45
Kleiner 2007, p. 63

[mwCrc] Karpilovsky 1989, p. 45

[mwCrs] Kleiner 2007, p. 63

[91] Lang 2005, pp. 261–262
Cox 2012, pp. 161–162

[mwCso] Cox 2012, pp. 161–162

[92] Cooper 2011, p. 60

[93] Rowen 2006, p. 12
Pratt 2022, § 3.3 Birkhoff's Theorem
Grätzer 2008, p. 34

[mwCuA] Pratt 2022, § 3.3 Birkhoff's Theorem

[mwCuI] Grätzer 2008, p. 34

[94] Pratt 2022, § 3.3 Birkhoff's Theorem
Rowen 2006, p. 12
Gowers, Barrow-Green & Leader 2010, pp. 27–28
Adhikari 2016, pp. 5–6

[mwCu0] Rowen 2006, p. 12

[mwCvE] Gowers, Barrow-Green & Leader 2010, pp. 27–28

[mwCvY] Adhikari 2016, pp. 5–6

[95] Neri 2019, pp. 278–279
Ivanova & Smirnov 2012
Deo 2018, p. 295
Ono 2019, p. 84

[mwCwU] Ivanova & Smirnov 2012

[mwCwg] Deo 2018, p. 295

[mwCww] Ono 2019, p. 84

[auto8-96] 1 2 Indurkhya 2013, pp. 217–218
Pratt 2022, § 3.3 Birkhoff's Theorem
Grätzer 2008, p. 34

[mwCx0] Pratt 2022, § 3.3 Birkhoff's Theorem

[mwCx8] Grätzer 2008, p. 34

[auto6-97] 1 2 Indurkhya 2013, pp. 217–218

[99] Efimov 2014

[101] Pratt 2022, § 3. Universal Algebra
Cohn 2012, p. xiii

[mwCzY] Cohn 2012, p. xiii

[102] Smirnov 2020
Grätzer 2008, pp. 7–8
Bahturin 2013, p. 346

[mwC0A] Grätzer 2008, pp. 7–8

[mwC0M] Bahturin 2013, p. 346

[104] Pratt 2022, § 3.2 Equational Logic
Mal’cev 1973, pp. 210–211

[mwC08] Mal’cev 1973, pp. 210–211

[105] Mal’cev 1973, pp. 210–211
Cohn 2012, p. 162
Rosen 2012, p. 779
Hazewinkel 1994, p. 406

[mwC1w] Cohn 2012, p. 162

[mwC2A] Rosen 2012, p. 779

[mwC2Q] Hazewinkel 1994, p. 406

[106] Cohn 1995, p. 8

[108] Mal’cev 1973, p. 211

[109] Mal’cev 1973, pp. 210–211
Pratt 2022, § 3. Universal Algebra
Artamonov 2003, p. 873

[mwC3o] Pratt 2022, § 3. Universal Algebra

[mwC3w] Artamonov 2003, p. 873

[111] Weisstein 2003, pp. 347–348
Gowers, Barrow-Green & Leader 2010, pp. 6, 165
Cheng 2023, p. 102

[mwC4o] Gowers, Barrow-Green & Leader 2010, pp. 6, 165

[mwC48] Cheng 2023, p. 102

[112] Gowers, Barrow-Green & Leader 2010, pp. 6, 165
Borceux 1994, p. 20
Laos 1998, p. 100
Cheng 2023, pp. 128–131

[mwC54] Borceux 1994, p. 20

[mwC6I] Laos 1998, p. 100

[mwC6Y] Cheng 2023, pp. 128–131

[113] Corry 2024, § Problem-Solving in Egypt and Babylon
Brezinski, Meurant & Redivo-Zaglia 2022, p. 34

[mwC68] Brezinski, Meurant & Redivo-Zaglia 2022, p. 34

[115] Tanton 2005, p. 9
Kvasz 2006, p. 290
Corry 2024, § Problem Solving in Egypt and Babylon

[mwC74] Kvasz 2006, p. 290

[mwC8E] Corry 2024, § Problem Solving in Egypt and Babylon

[116] Tanton 2005, p. 9
Kvasz 2006, p. 290
Corry 2024, § The Pythagoreans and Euclid

[mwC80] Kvasz 2006, p. 290

[mwC9A] Corry 2024, § The Pythagoreans and Euclid

[117] Merzlyakov & Shirshov 2020, § Historical Survey
Sialaros 2018, p. 55
Corry 2024, § Diophantus

[mwC9s] Sialaros 2018, p. 55

[mwC98] Corry 2024, § Diophantus

[118] Hettle 2015, pp. 139–141, 160–161
Christianidis & Megremi 2019, pp. 16–17

[mwC-s] Christianidis & Megremi 2019, pp. 16–17

[119] Burgin 2022, p. 10

[120] Higgins 2015, p. 89

[121] Kvasz 2006, pp. 290–291
Sialaros 2018, p. 55
Boyer & Merzbach 2011, p. 161
Derbyshire 2006, p. 31

[mwDAc] Sialaros 2018, p. 55

[mwDAs] Boyer & Merzbach 2011, p. 161

[mwDBA] Derbyshire 2006, p. 31

[122] Boyer & Merzbach 2011, p. 161
Derbyshire 2006, p. 31

[mwDB0] Derbyshire 2006, p. 31

[FOOTNOTEBradley2006[httpbooksgooglecombooksidEIdtVPeD7GcCpgPA95_95]-124] Bradley 2006, p. 95.

[126] Tanton 2005, p. 10
Kvasz 2006, pp. 291–293
Merzlyakov & Shirshov 2020, § Historical Survey

[mwDDE] Kvasz 2006, pp. 291–293

[mwDDQ] Merzlyakov & Shirshov 2020, § Historical Survey

[127] Waerden 2013, pp. 3, 15–16, 24–25
Jenkins 2010, p. 82
Pickover 2009, p. 90

[mwDEE] Jenkins 2010, p. 82

[mwDEU] Pickover 2009, p. 90

[128] Tanton 2005, pp. 9–10
Corry 2024, § The Equation in India and China

[mwDFM] Corry 2024, § The Equation in India and China

[129] Seshadri 2010, p. 156
Emch, Sridharan & Srinivas 2005, p. 20

[mwDGA] Emch, Sridharan & Srinivas 2005, p. 20

[130] Smorynski 2007, p. 137
Zwillinger 2002, p. 812

[mwDHA] Zwillinger 2002, p. 812

[131] Waerden 2013, pp. 32–35
Tanton 2005, p. 10
Kvasz 2006, p. 293

[mwDH4] Tanton 2005, p. 10

[mwDIE] Kvasz 2006, p. 293

[132] Tanton 2005, p. 10
Kvasz 2006, p. 293
Corry 2024, § Cardano and the Solving of Cubic and Quartic Equations
Miyake 2002, p. 268

[mwDI4] Kvasz 2006, p. 293

[mwDJE] Corry 2024, § Cardano and the Solving of Cubic and Quartic Equations

[mwDJM] Miyake 2002, p. 268

[133] Tanton 2005, p. 10
Kvasz 2006, pp. 291–292, 297–298, 302
Merzlyakov & Shirshov 2020, § Historical Survey
Corry 2024, § Viète and the Formal Equation, § Analytic Geometry

[mwDKE] Kvasz 2006, pp. 291–292, 297–298, 302

[mwDKQ] Merzlyakov & Shirshov 2020, § Historical Survey

[mwDKc] Corry 2024, § Viète and the Formal Equation, § Analytic Geometry

[134] Hazewinkel 1994, p. 73
Merzlyakov & Shirshov 2020, § Historical Survey

[mwDLQ] Merzlyakov & Shirshov 2020, § Historical Survey

[135] Corry 2024, § Applications of Group Theory
Bueno & French 2018, pp. 73–75

[mwDL8] Bueno & French 2018, pp. 73–75

[136] Merzlyakov & Shirshov 2020, § Historical Survey
Tanton 2005, p. 10
Corry 2024, § Structural Algebra
Hazewinkel 1994, pp. 73–74

[mwDM0] Tanton 2005, p. 10

[mwDNA] Corry 2024, § Structural Algebra

[mwDNI] Hazewinkel 1994, pp. 73–74

[137] Merzlyakov & Shirshov 2020, § Historical Survey
Tanton 2005, p. 10
Corry 2024, § Matrices, § Quaternions and Vectors

[mwDN8] Tanton 2005, p. 10

[mwDOI] Corry 2024, § Matrices, § Quaternions and Vectors

[138] Merzlyakov & Shirshov 2020, § Historical Survey
Corry 2024, § Hilbert and Steinitz, § Noether and Artin
Hazewinkel 1994, pp. 73–74

[mwDO0] Corry 2024, § Hilbert and Steinitz, § Noether and Artin

[mwDO8] Hazewinkel 1994, pp. 73–74

[139] Grätzer 2008, p. vii
Chang & Keisler 1990, p. 603
Knoebel 2011, p. 5
Hazewinkel 1994, pp. 74–75

[mwDP4] Chang & Keisler 1990, p. 603

[mwDQM] Knoebel 2011, p. 5

[mwDQc] Hazewinkel 1994, pp. 74–75

[140] Hazewinkel 1994, pp. 74–75
Kleiner 2007, p. 100
Carlson 2024, § History of topology

[mwDRY] Kleiner 2007, p. 100

[mwDRo] Carlson 2024, § History of topology

[141] Hazewinkel 1994, pp. 74–75
Weibel 1995, p. xi, 4

[mwDSc] Weibel 1995, p. xi, 4

[142] Krömer 2007, p. 61
Laos 1998, p. 100

[mwDTY] Laos 1998, p. 100

[143] Hazewinkel 1994, pp. 74–75
Pratt 2022, § 6. Free Algebras

[mwDUU] Pratt 2022, § 6. Free Algebras

[144] Houston 2004, p. 319
Neri 2019, p. xii
Lidl & Pilz 1997, pp. vii–viii

[mwDVI] Neri 2019, p. xii

[mwDVY] Lidl & Pilz 1997, pp. vii–viii

[145] Kleiner 2007, p. 100
Pratt 2022, § 5. Algebraization of Mathematics
Maddocks 2008, p. 130
Pratt 2022, § 5. Algebraization of Mathematics
Mancosu 1999, pp. 84–85

[mwDWY] Pratt 2022, § 5. Algebraization of Mathematics

[mwDWg] Maddocks 2008, p. 130

[mwDWs] Pratt 2022, § 5. Algebraization of Mathematics

[mwDW0] Mancosu 1999, pp. 84–85

[146] Pratt 2022, § 1.4 Cartesian geometry, § 3. Universal Algebra
Danilov 2006, p. 174

[mwDXc] Danilov 2006, p. 174

[148] Pratt 2022, § 5.1 Algebraic Geometry
Danilov 2006, pp. 172, 174

[mwDYM] Danilov 2006, pp. 172, 174

[149] Vince 2007, p. 133

[150] Pratt 2022, § 5.3 Algebraic Topology
Rabadan & Blumberg 2019, pp. 49–50
Nakahara 2018, p. 121
Weisstein 2003, pp. 52–53

[mwDZY] Rabadan & Blumberg 2019, pp. 49–50

[mwDZs] Nakahara 2018, p. 121

[mwDZ8] Weisstein 2003, pp. 52–53

[151] Pratt 2022, § 5.2 Algebraic Number Theory
Jarvis 2014, p. 1
Viterbo & Hong 2011, p. 127

[mwDao] Jarvis 2014, p. 1

[mwDa4] Viterbo & Hong 2011, p. 127

[152] Gowers, Barrow-Green & Leader 2010, pp. 550, 561
Godsil 2017, p. viii
Betten et al. 2013, p. ix

[mwDb8] Godsil 2017, p. viii

[mwDcM] Betten et al. 2013, p. ix

[153] Kilty & McAllister 2018, pp. x, 347, 589
Bressoud 2021, p. 64

[mwDdI] Bressoud 2021, p. 64

[154] Halmos 1956, p. 363
Burris & Legris 2021, § 1. Introduction

[mwDeA] Burris & Legris 2021, § 1. Introduction

[155] Andréka, Németi & Sain 2001, pp. 133–134

[156] Andréka, Madarász & Németi 2020, § Concrete Algebraic Logic
Pratt 2022, § 5.4 Algebraic Logic
Plotkin 2012, pp. 155–156
Jansana 2022, Lead section

[mwDfQ] Pratt 2022, § 5.4 Algebraic Logic

[mwDfY] Plotkin 2012, pp. 155–156

[mwDfo] Jansana 2022, Lead section

[157] Andréka, Madarász & Németi 2020, § Abstract Algebraic Logic
Jansana 2022, § 4. Algebras

[mwDgU] Jansana 2022, § 4. Algebras

[158] Joyner 2008, p. 92

[159] Houston 2004, p. 319
Neri 2019, p. xii
Anton & Rorres 2010, p. 327

[mwDhk] Neri 2019, p. xii

[mwDh0] Anton & Rorres 2010, p. 327

[160] Neri 2019, p. xii
Aleskerov, Ersel & Piontkovski 2011, pp. 1–9
Straffin 1980, p. 269
Menini & Oystaeyen 2017, p. v
Lovett 2015, p. ix
Lidl & Pilz 1997, pp. vii–viii

[mwDi0] Aleskerov, Ersel & Piontkovski 2011, pp. 1–9

[mwDjI] Straffin 1980, p. 269

[mwDjY] Menini & Oystaeyen 2017, p. v

[mwDjs] Lovett 2015, p. ix

[mwDj8] Lidl & Pilz 1997, pp. vii–viii

[161] Wheeler 2023, p. 29, 36–37
Gallier & Quaintance 2020, pp. 1–2

[mwDk8] Gallier & Quaintance 2020, pp. 1–2

[162] Klimov 2014, p. ix
Bengtsson & Życzkowski 2017, pp. 313–353

[mwDl8] Bengtsson & Życzkowski 2017, pp. 313–353

[163] Terras 2019, pp. 63–64, 142

[164] Hull 2021, p. 180

[165] Lidl & Pilz 1997, pp. 183–184, 239–240
Carstensen, Fine & Rosenberger 2011, pp. 326–327

[mwDnw] Carstensen, Fine & Rosenberger 2011, pp. 326–327

[166] Kieran 2006, p. 15
Kaput 2018, p. 186
Gardella & DeLucia 2020, pp. 19–22

[mwDow] Kaput 2018, p. 186

[mwDpA] Gardella & DeLucia 2020, pp. 19–22

[167] Arcavi, Drijvers & Stacey 2016, p. xiii
Dekker & Dolk 2011, p. 69

[mwDqA] Dekker & Dolk 2011, p. 69

[168] Arcavi, Drijvers & Stacey 2016, pp. 2–5
Drijvers, Goddijn & Kindt 2011, pp. 8–10, 16–18

[mwDrA] Drijvers, Goddijn & Kindt 2011, pp. 8–10, 16–18

[169] Kieran 2006, p. 15
Kaput 2018, p. 186
Gardella & DeLucia 2020, pp. 19–22
Star et al. 2015, pp. 16–17

[mwDsA] Kaput 2018, p. 186

[mwDsQ] Gardella & DeLucia 2020, pp. 19–22

[mwDsk] Star et al. 2015, pp. 16–17

[170] Arcavi, Drijvers & Stacey 2016, pp. 58–59
Drijvers, Goddijn & Kindt 2011, p. 13

[mwDtQ] Drijvers, Goddijn & Kindt 2011, p. 13

[171] Hausberger, Zandieh & Fleischmann 2021, pp. 147–148

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