Number

Lamba

An yi amfani da shi don ƙidaya, aunawa, da lakabi

kidayayyun lambobi

Lamba abu ne na lissafi da ake amfani da shi don ƙidaya, aunawa, da lakabi. Mafi mahimmancin misalan su ne lambobi na halitta 1, 2, 3, 4, da sauransu. Ana iya wakilta lambobi cikin harshe tare da kalmomin lamba. Fiye da duk duniya, ana iya wakilta lambobi ɗaya da alamomi, da ake kira lambobi; misali “5” lamba ce da ke wakiltar lamba biyar. Kamar yadda ƙananan alamomi kawai za a iya haddace su, ainihin lambobi yawanci ana tsara su a cikin tsarin ƙididdigewa, wanda hanya ce mai tsari don wakiltar kowace lamba. Mafi yawan tsarin lambobi shine tsarin lambobi na Hindu-Larabci, wanda ke ba da damar wakilcin kowace lamba mara kyau ta amfani da haɗe-haɗe na manyan alamomin lamba goma, da ake kira lambobi. Baya ga amfani da su wajen kirgawa da aunawa, ana yawan amfani da lambobi don tambari (kamar yadda ake da lambobin waya), don yin oda (kamar yadda ake da lambobin serial), da lambobin (kamar yadda suke da ISBNs). A cikin amfani na gama gari, lamba ba ta bambanta a fili da lambar da take wakilta ba. A cikin ilimin lissafi, an tsawaita ra'ayin lamba a cikin ƙarni don haɗa da sifili (0), lambobi mara kyau, lambobi masu ma'ana kamar rabi ɗaya. ( 1 2 ) {\displaystyle \hagu({\tfrac {1}{2}}\dama)}, lambobi na gaske kamar tushen murabba'i na 2 ( 2 ) {\displaystyle \ hagu({\sqrt {2}}\ dama)} da π, da kuma hadaddun lambobi waɗanda ke tsawaita ainihin lambobi tare da tushen murabba'i na -1 (da haɗe-haɗe tare da lambobi na gaske ta hanyar ƙara ko rage yawan adadinsa). Ana yin ƙididdiga tare da lambobi tare da ayyukan lissafi, wanda aka fi sani da ƙari shine ƙari, ragi, ninkawa, rarrabuwa, da ƙari. Nazarin su ko amfani da su ana kiran su arithmetic, kalma wanda kuma yana iya nufin ka'idar lamba, nazarin kaddarorin lambobi.

Bayan amfaninsu na amfani, lambobi suna da mahimmancin al'adu a duk faɗin duniya. Misali, a cikin al'ummar Yamma, ana ɗaukar lamba 13 a matsayin rashin sa'a, kuma "miliyan" na iya nufin "mai yawa" maimakon ainihin adadin. Ko da yake a yanzu ana ɗaukarsa azaman pseudoscience, imani da mahimmancin sufanci na lambobi, wanda aka sani da numerology, ya mamaye tunani na da da na da. Ƙididdigar ƙididdiga ta yi tasiri sosai wajen haɓakar ilimin lissafi na Girkanci, yana ƙarfafa binciken matsalolin da yawa a ka'idar lamba waɗanda har yanzu suna da sha'awa a yau.

A cikin karni na 19, masu ilmin lissafi sun fara haɓaka ƙididdiga daban-daban waɗanda ke raba wasu kaddarorin lambobi, kuma ana iya ganin su suna faɗaɗa ra'ayi. Daga cikin na farko akwai lambobi masu rikitarwa, waɗanda suka ƙunshi haɓaka daban-daban ko gyare-gyare na tsarin lambobi masu rikitarwa. A cikin ilimin lissafi na zamani, tsarin lamba ana ɗaukar mahimman misalai na musamman na ƙarin tsarin algebra gaba ɗaya kamar zobe da filaye, kuma aikace-aikacen kalmar "lamba" wani al'amari ne na al'ada, ba tare da mahimmancin mahimmanci ba.

Tarihi

Amfani da lambobi na farko

Kidayyan Romawa.

Babban labarin: Tarihin tsoffin tsarin lambobi An gano kasusuwa da sauran kayan tarihi tare da yanke alamomin da mutane da yawa suka yi imani da cewa alamomi ne. Wataƙila an yi amfani da waɗannan alamomin don kirga lokacin da suka wuce, kamar adadin kwanaki, zagayowar wata ko adana bayanai na adadi, kamar na dabbobi.

Tsarin ƙididdigewa ba shi da ra'ayi na ƙimar wuri (kamar yadda yake a cikin ƙayyadaddun ƙayyadaddun ƙayyadaddun ƙayyadaddun ƙayyadaddun ƙididdiga na zamani), wanda ke iyakance wakilcin adadi masu yawa. Duk da haka, ana ɗaukar tsarin ƙididdigewa a matsayin nau'in tsarin ƙididdiga na farko.

Tsarin da aka sani na farko tare da ƙimar wuri shine tsarin Mesopotamiya tushe na 60 (c. 3400 BC) kuma sanannen tushe 10 tsarin kwanan wata zuwa 3100 BC a Masar.

Lambobi Ya kamata a bambanta lambobi daga lambobi, alamomin da ake amfani da su don wakiltar lambobi. Masarawa sun ƙirƙiro tsarin ƙididdiga na farko, kuma Girkawa sun bi taswirar ƙidayar su akan haruffan Ionian da Doric. Lambobin Roman, tsarin da ya yi amfani da haɗin haruffa daga haruffan Roman, ya kasance mai rinjaye a Turai har zuwa yaduwar tsarin lambobi mafi girma na Hindu-Larabci a kusa da ƙarshen karni na 14, kuma tsarin lambobi na Hindu-Larabci ya kasance mafi yawan tsarin wakilci don wakiltar. lambobi a duniya a yau.[mafi kyawun tushe da ake buƙata] Makullin tasirin tsarin shine alamar sifili, wanda tsoffin masana lissafin Indiya suka haɓaka a cikin 500 AD.

Sifili

Sifili

Sanannen farkon da aka rubuta amfani da kwanakin sifili zuwa AD 628, kuma ya bayyana a cikin Brāhmasphuṭasiddhānta, babban aikin masanin lissafi na Indiya Brahmagupta. Ya ɗauki 0 azaman lamba kuma ya tattauna ayyukan da suka haɗa da shi, gami da rarraba. A wannan lokaci (karni na 7) ra'ayin ya kai ga Cambodia a matsayin lambobin Khmer, kuma rubuce-rubuce sun nuna ra'ayin daga baya ya yadu zuwa kasar Sin da duniyar Musulunci.

Lambar 605 a cikin lambobin Khmer, daga rubutu daga 683 AD. Fara amfani da sifili azaman adadi na goma. Brāhmasphuṭasiddhānta na Brahmagupta shine littafi na farko da ya ambaci sifili a matsayin lamba, don haka Brahmagupta yawanci ana ɗaukarsa a matsayin farkon wanda ya tsara manufar sifili. Ya ba da ka'idoji na amfani da sifili tare da lambobi marasa kyau da masu kyau, kamar "sifili da lamba mai kyau ita ce lamba mai kyau, kuma lamba mara kyau da sifili ita ce mummunan lamba". Brāhmasphuṭasiddhānta shine rubutun farko da aka sani don ɗaukar sifili azaman lamba a kansa, maimakon a matsayin kawai lamba mai riƙewa a wakiltar wata lamba kamar yadda Babila suka yi ko kuma a matsayin alama don ƙarancin yawa kamar yadda Ptolemy ya yi kuma Romawa.

Amfani da 0 a matsayin lamba ya kamata a bambanta daga amfani da shi azaman adadin ma'auni a tsarin ƙimar wuri. 0. Nassosin Babila da na Masar sun yi amfani da shi. Masarawa sun yi amfani da kalmar nfr don nuna ma'aunin sifili a cikin lissafin shigarwa sau biyu. Rubutun Indiya sun yi amfani da kalmar Sanskrit Shunye ko shunya don komawa ga manufar banza. A cikin rubutun lissafi wannan kalma sau da yawa tana nufin lamba sifili. A cikin irin wannan jijiya, Pāṇini (ƙarni na 5 BC) ya yi amfani da ma'aikacin null (sifili) a cikin Ashtadhyayi, misali na farko na nahawun algebra don harshen Sanskrit (kuma duba Pingala).

Akwai sauran amfani da sifili kafin Brahmagupta, kodayake takaddun ba su cika ba kamar yadda yake a cikin Brāhmasphuṭasiddhānta.

Bayanai sun nuna cewa Girkawan tsohuwar sun kasance kamar ba su da tabbas game da matsayin 0 a matsayin lamba: sun tambayi kansu "Ta yaya 'babu' zai zama wani abu?" yana haifar da ilimin falsafa mai ban sha'awa kuma, ta lokacin Medieval, muhawarar addini game da yanayi da wanzuwar 0 da vacuum. Matsalolin Zeno na Elea sun dogara da wani ɓangare na fassarar rashin tabbas na 0. (Tsarin Helenawa ma sun yi tambaya ko 1 lamba ce.)

Marigayi Olmec na kudancin tsakiyar Mexico sun fara amfani da alamar sifili, glyph, a cikin Sabuwar Duniya, mai yiwuwa a karni na 4 BC amma tabbas ta 40 BC, wanda ya zama wani ɓangare na lambobi na Maya da kalandar Maya. . Mayan ilmin lissafi sunyi amfani da tushe 4 da tushe na 5 da aka rubuta azaman tushe na 20. George I. Sánchez a cikin 1961 ya ba da rahoton tushe 4, tushe 5 "yatsa" abacus.[mafi kyawun tushe da ake bukata].

A shekara ta 130 AD, Ptolemy, wanda Hipparchus da Babila suka rinjayi, yana amfani da alamar 0 (ƙaramin da'ira mai tsayi mai tsayi) a cikin tsarin adadi na jima'i in ba haka ba yana amfani da lambobin haruffa na Helenanci. Domin an yi amfani da shi shi kaɗai, ba a matsayin mai riƙe da wuri kawai ba, wannan sifili na Hellenistic shine farkon rubuce-rubucen amfani da sifilin gaskiya a cikin Tsohuwar Duniya. A cikin rubuce-rubucen Byzantine daga baya na Syntaxis Mathematica (Almagest), sifilin Hellenistic ya rikiɗe zuwa harafin Helenanci Omicron (in ba haka ba ma'ana 70).

An yi amfani da wani sifili na gaskiya a cikin tebur tare da lambobin Roman ta 525 (wanda Dionysius Exigus yayi amfani da shi na farko), amma a matsayin kalma, nulla yana nufin kome ba, ba a matsayin alama ba. Lokacin da rabo ya samar da 0 a matsayin saura, nihil, kuma ma'anar kome ba, an yi amfani da shi. Waɗannan sifilai na tsakiya duk masu ƙididdigewa na tsakiya na gaba (masu ƙididdiga na Easter) ne suka yi amfani da su. An yi amfani da keɓantaccen amfani da farkon su, N, a cikin tebur na lambobi na Roman ta Bede ko abokin aiki kusan 725, alamar sifili na gaskiya. Marigayi Olmec na kudancin tsakiyar Mexico sun fara amfani da alamar sifili, glyph, a cikin Sabuwar Duniya, mai yiwuwa a karni na 4 BC amma tabbas ta 40 BC, wanda ya zama wani ɓangare na lambobi na Maya da kalandar Maya. . Mayan ilmin lissafi sunyi amfani da tushe 4 da tushe na 5 da aka rubuta azaman tushe na 20. George I. Sánchez a cikin 1961 ya ba da rahoton tushe 4, tushe 5 "yatsa" abacus.[mafi kyawun tushe da ake bukata].

A shekara ta 130 AD, Ptolemy, wanda Hipparchus da Babila suka rinjayi, yana amfani da alamar 0 (ƙaramin da'ira mai tsayi mai tsayi) a cikin tsarin adadi na jima'i in ba haka ba yana amfani da lambobin haruffa na Helenanci. Domin an yi amfani da shi shi kaɗai, ba a matsayin mai riƙe da wuri kawai ba, wannan sifili na Hellenistic shine farkon rubuce-rubucen amfani da sifilin gaskiya a cikin Tsohuwar Duniya. A cikin rubuce-rubucen Byzantine daga baya na Syntaxis Mathematica (Almagest), sifilin Hellenistic ya rikiɗe zuwa harafin Helenanci Omicron (in ba haka ba ma'ana 70).

An yi amfani da wani sifili na gaskiya a cikin tebur tare da lambobin Roman ta 525 (wanda Dionysius Exigus yayi amfani da shi na farko), amma a matsayin kalma, nulla yana nufin kome ba, ba a matsayin alama ba. Lokacin da rabo ya samar da 0 a matsayin saura, nihil, kuma ma'anar kome ba, an yi amfani da shi. Waɗannan sifilai na tsakiya duk masu ƙididdigewa na tsakiya na gaba (masu ƙididdiga na Easter) ne suka yi amfani da su. An yi amfani da keɓantaccen amfani da farkon su, N, a cikin tebur na lambobi na Roman ta Bede ko abokin aiki kusan 725, alamar sifili na gaskiya.

Lambobi masu bashi

Alaman bashi a kidaya

Ƙarin bayani: Tarihin lambobi masu bashi An gane maƙasudin ra'ayi masu bashi na lambobi a farkon 100-50 BC a China. Babi tara akan fasahar lissafi sun ƙunshi hanyoyin gano wuraren ƙididdiga; An yi amfani da jajayen sanduna don nuna madaidaitan ƙididdiga, baƙar fata ga korau. Magana ta farko a cikin aikin yammacin duniya shine a karni na 3 AD a Girka. Diophantus yayi magana akan lissafin daidai da 4x + 20 = 0 (maganin ba shi da kyau) a cikin Arithmetica, yana mai cewa lissafin ya ba da sakamako mara kyau.

A cikin 600s, an yi amfani da lambobi masu bashi a Indiya don wakiltar bashi. Masanin ilimin lissafi na Indiya Brahmagupta ya tattauna batun da Diophantus ya yi a baya, a cikin Brāhmasphuṭasiddhānta a cikin 628, wanda ya yi amfani da lambobi mara kyau don samar da tsarin ƙididdiga na gaba ɗaya wanda ya rage amfani da shi a yau. Duk da haka, a cikin karni na 12 a Indiya, Bhaskara ya ba da tushe mara kyau ga ma'auni guda hudu amma ya ce mummunan darajar "a cikin wannan yanayin ba za a dauka ba, domin bai isa ba; mutane ba su yarda da tushen mara kyau ba".

Masana ilimin lissafi na Turai, a mafi yawan lokuta, sun yi tsayayya da ra'ayi na lambobi mara kyau har zuwa karni na 17, kodayake Fibonacci ya ba da izinin magance matsalolin kudi a cikin matsalolin kudi inda za a iya fassara su a matsayin bashi (babi na 13 na Liber Abaci, 1202) kuma daga baya a matsayin hasara (a cikin Flos). ). René Descartes ya kira su tushen ƙarya yayin da suke girma cikin algebra polynomials duk da haka ya sami hanyar musanya tushen gaskiya da tushen ƙarya kuma. A lokaci guda, Sinawa suna nuna munanan lambobi ta hanyar zana bugun jini ta hanyar dama-mafi yawan lambobi marasa sifili na madaidaicin lambar tabbataccen lamba. Farkon amfani da lambobi mara kyau a cikin aikin Turai shine Nicolas Chuquet a cikin ƙarni na 15. Ya yi amfani da su a matsayin masu magana, amma ya kira su da "lambobi marasa hankali".

Kwanan nan kamar karni na 18, al'ada ce ta yau da kullun don yin watsi da duk wani sakamako mara kyau da aka dawo ta hanyar daidaitawa akan zaton ba su da ma'ana.

Lambobi masu ma'ana

Lambobi masu ma'ana

Wataƙila manufar lambobi masu ɓarna sun kasance tun zamanin da. Masarawa na d ¯ a sun yi amfani da juzu'in su na Masar don lambobi masu ma'ana a cikin rubutun lissafi kamar Rhind Mathematical Papyrus da Kahun Papyrus. Masana lissafin Girka na gargajiya da na Indiya sun yi nazarin ka'idar lambobi masu hankali, a matsayin wani ɓangare na nazarin ka'idar lamba. Mafi sanannun waɗannan su ne abubuwan Euclid, waɗanda suka yi kusan 300 BC. Daga cikin rubutun Indiya, wanda ya fi dacewa shine Sthananga Sutra, wanda kuma ya shafi ka'idar lamba a matsayin wani ɓangare na nazarin lissafi na gaba ɗaya.

Tunanin juzu'i na ƙima yana da alaƙa da alaƙa da ƙima-darajar wuri-disimal; su biyun kamar sun ci gaba a dunkule. Misali, ya zama ruwan dare ga Jain math sutra ya haɗa da ƙididdige ƙididdiga na ƙayyadaddun juzu'i zuwa pi ko tushen murabba'in 2. Hakazalika, rubutun lissafin Babila sun yi amfani da ɓangarorin sexagesimal (tushe 60) tare da mitoci mai yawa.

Lambobin marasa ma'ana Ƙarin bayani: Tarihin lambobi marasa ma'ana Sanin farkon amfani da lambobi marasa ma'ana shine a cikin Sulba Sutras na Indiya wanda aka haɗa tsakanin 800 da 500 BC. (mafi yiwuwa na geometrical) hujja na rashin hankali na tushen murabba'in 2. Labarin ya nuna cewa Hippasus ya gano lambobi marasa ma'ana yayin ƙoƙarin wakiltar tushen murabba'in 2 a matsayin juzu'i. Duk da haka, Pythagoras ya gaskanta da cikar lambobi, kuma ya kasa yarda da wanzuwar lambobi marasa ma'ana. Ba zai iya karyata wanzuwarsu ta hanyar tunani ba, amma ya kasa yarda da lambobi marasa ma'ana, don haka, ana zarginsa da rahotanni akai-akai, ya yanke wa Hippasus hukuncin kisa ta hanyar nutsewa, don hana yada wannan labari mai ban tsoro.

Ƙarni na 16 ya kawo karɓuwar Turai ta ƙarshe na lambobi mara kyau da na ɓarna. Ya zuwa karni na 17, masu ilmin lissafi gabaɗaya sun yi amfani da juzu'i na ƙima tare da bayanin zamani. Sai dai, sai a karni na 19, masanan lissafi suka raba marasa hankali zuwa algebra da sassa masu wuce gona da iri, kuma sun sake yin nazarin kimiyya na rashin hankali. Ya kasance kusan kwance tun Euclid. A cikin 1872, an gabatar da buga ka'idodin Karl Weierstrass (da almajirinsa E. Kossak), Eduard Heine, Georg Cantor, da Richard Dedekind. A cikin 1869, Charles Méray ya ɗauki matsayi ɗaya na tashi da Heine, amma ka'idar gabaɗaya ana magana da ita zuwa shekara ta 1872. Salvatore Pincherle (1880) ya tsara hanyar Weierstrass gaba ɗaya, kuma Dedekind's ya sami ƙarin shahara ta hanyar aikin marubucin daga baya. (1888) da kuma amincewa da Paul Tannery (1894). Weierstrass, Cantor, da Heine sun kafa ra'ayoyinsu akan jerin marasa iyaka, yayin da Dedekind ya samo nasa akan ra'ayin yanke (Schnitt) a cikin tsarin lambobi na ainihi, yana raba duk lambobi masu ma'ana zuwa ƙungiyoyi biyu suna da takamaiman halaye. Batun ya sami gudummawa daga baya a hannun Weierstrass, Kronecker, da Méray.

Binciken tushen ma'auni na quintic da mafi girma digiri shine muhimmin ci gaba, ka'idar Abel-Ruffini (Ruffini 1799, Abel 1824) ya nuna cewa ba za a iya magance su ta hanyar masu tsattsauran ra'ayi ba (ka'idodin da suka shafi ayyukan lissafi da tushen kawai). Don haka ya zama dole a yi la'akari da faɗin saitin lambobi na algebra (duk hanyoyin magance ma'auni masu yawa). Galois (1832) ya danganta ma'auni masu yawa zuwa ka'idar rukuni wanda ke haifar da fagen ka'idar Galois.

Ci gaba da juzu'i, masu alaƙa da lambobi marasa ma'ana (kuma saboda Cataldi, 1613), sun sami kulawa a hannun Euler, kuma a farkon karni na 19 an kawo su cikin shahara ta hanyar rubuce-rubucen Joseph Louis Lagrange. Druckenmüller (1837), Kunze (1857), Lemke (1870), da Günther (1872) sun ba da wasu gudummawar da suka dace. Ramus ya fara haɗa batun tare da masu tantancewa, sakamakon, tare da gudunmawar Heine, Möbius, da Günther na gaba, a cikin ka'idar Kettenbruchdeterminanten.

Lambobin wucewa da haqiqanin gaske Ƙarin bayani: Tarihin π (pi) Liouville ne ya fara kafa kasancewar lambobi masu wucewa (1844, 1851). Hermite ya tabbatar a cikin 1873 cewa e transcendental ne kuma Lindemann ya tabbatar a cikin 1882 cewa π ya wuce gona da iri. A ƙarshe, Cantor ya nuna cewa saitin duk ainihin lambobi ba su da iyaka amma saitin duk lambobin algebra ba su da iyaka, don haka akwai adadi mara iyaka na lambobi masu wucewa.

Rashin iyakan Lambobi

Alaman rashin iyakan Lambobi

Ƙarin bayani: Tarihin rashin iyaka Tunanin farko da aka sani game da rashin iyaka ya bayyana a cikin Yajur Veda, wani tsohon rubutun Indiya, wanda a wani lokaci ya ce, "Idan ka cire wani bangare daga rashin iyaka ko ƙara wani sashi zuwa rashin iyaka, har yanzu abin da ya rage shi ne rashin iyaka." Infinity sanannen jigo ne na nazarin falsafa tsakanin masanan Jain c. 400 BC. Sun bambanta tsakanin nau'ikan marasa iyaka guda biyar: mara iyaka a cikin kwatance ɗaya da biyu, mara iyaka a cikin yanki, mara iyaka a ko'ina, kuma mara iyaka na har abada. Alamar ∞ {\displaystyle {\text{∞}}} ana yawan amfani dashi don wakiltar adadi mara iyaka.

Aristotle ya bayyana ra'ayin gargajiya na yammacin duniya na rashin iyaka. Ya bambanta tsakanin ainihin rashin iyaka da rashin iyaka mai yuwuwa - gabaɗayan ijma'i shine cewa kawai na ƙarshe yana da ƙimar gaske. Sabbin Kimiyyar Kimiyya Biyu na Galileo Galilei sun tattauna ra'ayin wasiƙa ɗaya zuwa ɗaya tsakanin saiti marasa iyaka. Amma babban ci gaba na gaba a cikin ka'idar Georg Cantor ya yi; a cikin 1895 ya buga littafi game da sabuwar ka'idar sa, gabatar da, a tsakanin sauran abubuwa, lambobi masu canzawa da tsara hasashen ci gaba.

A cikin 1960s, Abraham Robinson ya nuna yadda ƙila za a iya fayyace ƙaƙƙarfan lambobi masu girma da ƙima da amfani da su don haɓaka fagen bincike mara inganci. Tsarin lambobi na zahiri yana wakiltar tsayayyen hanyar magance ra'ayoyin game da lambobi marasa iyaka da marasa iyaka waɗanda masana lissafi, masana kimiyya, da injiniyoyi suka yi amfani da su ba tare da izini ba tun lokacin da Newton da Leibniz suka ƙirƙira ƙididdiga marasa iyaka.

An ba da sigar ma'auni na zamani na infinity ta hanyar lissafi mai ƙima, wanda ke gabatar da "mahimman maki a rashin iyaka", ɗaya don kowane shugabanci na sarari. Kowane iyali na layi ɗaya a cikin wani jagorar da aka ba da shi an sanya shi don haɗuwa zuwa madaidaicin manufa. Wannan yana da alaƙa kusa da ra'ayin ɓarna a cikin zanen hangen nesa.

Lambobi masu rikitarwa

Lambobi masu rikitarwa

Ƙarin bayani: Tarihin hadaddun lambobi Maganar farko mai wucewa ga tushen murabba'ai na lambobi mara kyau sun faru a cikin aikin masanin lissafi kuma mai ƙirƙira Heron na Alexandria a cikin ƙarni na 1 AD, lokacin da ya yi la'akari da ƙarar abin takaici na dala. Sun yi fice sosai yayin da a cikin karni na 16 rufaffiyar tsarin tushen tushen tsarin digiri na uku da na hudu aka gano ta hanyar masana lissafin Italiya kamar Niccolò Fontana Tartaglia da Gerolamo Cardano. Nan da nan aka gane cewa waɗannan dabarun, koda kuwa mutum yana da sha'awar ainihin mafita, wani lokacin yana buƙatar yin amfani da tushen murabba'in lambobi mara kyau.

Wannan ya kasance mai tayar da hankali sau biyu tun da ba su ma la'akari da lambobi marasa kyau a matsayin tushe a lokacin ba. Lokacin da René Descartes ya ƙirƙira kalmar "haske" ga waɗannan adadi a cikin 1637, ya yi nufin hakan a matsayin wulakanci. (Dubi lambar hasashe don tattaunawa akan "gaskiya" na hadaddun lambobi.) Wani ƙarin abin da ya haifar da ruɗani shine cewa lissafin.

( - 1 ) 2 = - 1 - 1 = - 1 {\displaystyle \hagu({\sqrt {-1}}\ dama)^{2}={\sqrt {-1}}{\sqrt {-1}}=-1} ya zama kamar bai dace da ainihin algebraic ba

a b = a b , {\displaystyle {\sqrt {a}}{\sqrt {b}}={\sqrt {ab}},} wanda ke aiki don tabbataccen lambobi na ainihi a da b, kuma an yi amfani da su a cikin hadadden lissafin lamba tare da ɗayan a, b tabbatacce da sauran mara kyau. Amfani da wannan shaidar ba daidai ba, da kuma alaƙar da ke da alaƙa

1 a = 1 a {\displaystyle {\frac {1}{\sqrt {a}}}={\sqrt {\frac {1}{a}}} a cikin yanayin lokacin da duka a da b ba su da kyau har ma da Euler. Wannan wahala daga ƙarshe ta