Tafiya zuwa duniyar lissafi mara gaskiya

Na rubuta wannan labarin a daya daga cikin mahallin, bayan lacca da aiki a kwalejin kimiyyar kwamfuta. Ina kare kaina daga sukar da ake yi wa daliban wannan makaranta, iliminsu, halayensu ga kimiyya da kuma mafi mahimmanci, kwarewar koyarwa. Wannan...ba wanda yake koya musu.

Me ya sa nake karewa haka? Don dalili mai sauƙi - Ina cikin shekarun da, mai yiwuwa, duniyar da ke kewaye da mu ba a fahimta ba tukuna. Watakila ina koya musu yadda za su yi amfani da dawakai su kwance damara, ba tuƙi mota ba? Wataƙila in koya musu rubutu da alƙalami? Ko da yake ina da ra'ayi mafi kyau game da mutum, na ɗauki kaina "bin", amma ...

Har kwanan nan, a makarantar sakandare, sun yi magana game da lambobi masu rikitarwa. Kuma a wannan Laraba ne na dawo gida, na daina - kusan babu ɗaya daga cikin ɗaliban da ya koyi menene da kuma yadda ake amfani da waɗannan lambobin. Wasu suna kallon duk ilimin lissafi kamar Goose a ƙofar fenti. Amma na yi mamakin gaske sa’ad da suka gaya mini yadda zan koya. A taƙaice, kowace sa’a ta lacca aikin gida ne na sa’o’i biyu: karanta littafi, koyan yadda ake warware matsaloli a kan wani maudu’i, da dai sauransu. Bayan da muka shirya ta wannan hanya, mun zo da darussan, inda muka inganta duk abin da ... Abin farin ciki, dalibai, a fili, tunanin cewa zaune a lacca - mafi sau da yawa neman fitar da taga - riga ya tabbatar da shigar da ilimi a cikin kai.

Tsaya! Ya isa haka. Zan bayyana amsar dana bayar ga wata tambaya da na samu a lokacin darasi tare da masu rike da asusun ajiyar yara na kasa, cibiyar da ke tallafawa yara masu hazaka daga ko'ina cikin kasar. Tambayar (ko kuma wajen shawarar) ita ce:

- Za ku iya gaya mana wani abu game da lambobi marasa gaske?

"Tabbas," na amsa. 

Gaskiyar lambobi

"Aboki wani ni ne, abokantaka shine rabon lambobi 220 da 284," in ji Pythagoras. Abin lura a nan shi ne, jimillar masu rabon lamba 220 284 ne, jimillar masu rabon lamba 284 kuwa 220 ne;

1 + 2 + 4 + 71 + 142 = 220 ma'ana

1 + 2 + 4 + 5 + 10 = 11 + 20 + 22 + 44 + 55 + 110 = 284

Wani daidaituwa mai ban sha'awa tsakanin lambobi 220 da 284 shine wannan: manyan lambobi goma sha bakwai mafi girma sune 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53 , kuma 59.

Adadin su shine 2x220, jimlar murabba'in shine 59x284.

Na farko. Babu ra'ayi na "lamba na gaske". Kamar bayan karanta wani labarin game da giwaye, sai ka tambayi, "Yanzu za mu nemi wadanda ba giwaye ba." Akwai gabaɗaya kuma ba cikakke ba, masu hankali da rashin hankali, amma babu gaskiya. Musamman: Lambobin da ba na gaske ba ba a kiran su marasa inganci. Akwai nau'ikan "lambobi" da yawa a cikin ilimin lissafi, kuma sun bambanta da juna, kamar - don ɗaukar kwatancen dabbobi - giwa da tsutsotsi.

Na biyu, za mu yi ayyuka waɗanda ƙila kun riga kun san an hana su: cire tushen murabba'in lambobi mara kyau. To, ilimin lissafi zai shawo kan irin wannan shinge. Shin yana da ma'ana ko? A cikin ilimin lissafi, kamar yadda yake a kowane kimiyya, ko ka'idar ta shiga har abada a cikin ma'ajin ilimi ya dogara ... akan aikace-aikacensa. Idan kuma ba shi da amfani, to sai ya koma cikin shara, sannan a cikin wasu sharar tarihin ilmi. Ba tare da lambobin da na yi magana game da su a ƙarshen wannan labarin ba, ba shi yiwuwa a bunkasa ilimin lissafi. Amma bari mu fara da wasu ƙananan abubuwa. Menene ainihin lambobi, kun sani. Suna cika layin lamba da yawa kuma ba tare da gibi ba. Hakanan kun san menene lambobi na dabi'a: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, ...... - dukkansu ba za su dace ba. memory ko da mafi girma. Suna kuma da kyakkyawan suna: na halitta. Suna da kaddarorin ban sha'awa da yawa. Yaya kuke son wannan:

1 + 15 + 42 + 98 + 123 + 179 + 206 + 220 = 3 + 11 + 46 + 92 + 129 + 175 + 210 + 218

1² + 15² + 42² + 98² + 123² + 179² + 206² + 220² = 3² + 11² + 46² + 92² + 129² + 175² + 210² + 218²

1³ + 15³ + 42³ + 98³ + 123³ + 179³ + 206³ + 220³ = 3³ + 11³ + 46³ + 92³ + 129³ + 175³ + 210³ + 218³

1⁴ + 15⁴ + 42⁴ + 98⁴ + 123⁴ + 179⁴ + 206⁴ + 220⁴ = 3⁴ + 11⁴ + 46⁴ + 92⁴ + 129⁴ + 175⁴ + 210⁴ + 218⁴

1⁵ + 15⁵ + 42⁵ + 98⁵ + 123⁵ + 179⁵ + 206⁵ + 220⁵ = 3⁵ + 11⁵ + 46⁵ + 92⁵ + 129⁵ + 175⁵ + 210⁵ + 218⁵

1⁶ + 15⁶ + 42⁶ + 98³ + 123⁶ + 179⁶ + 206⁶ + 220⁶ = 3⁶ + 11⁶ + 46⁶ + 92⁶ + 129⁶ + 175⁶ + 210⁶ + 218⁶

1⁷ + 15⁷ + 42⁷ + 98³ + 123⁷ + 179⁷ + 206⁷ + 220⁷ = 3⁷ + 11⁷ + 46⁷ + 92⁷ + 129⁷ + 175⁷ + 210⁷ + 218⁷

Karl Lindenholm, da Leopold Kronecker (1823-1891) suka ce: “Yana da kyau a yi sha’awar adadin lambobi.” Rukunin juzu'i (wanda ake kira lambobi masu hankali ta masu ilimin lissafi) suma suna da kaddarorin ban mamaki:

kuma cikin daidaito:

Kuna iya, farawa daga gefen hagu, shafa ƙari kuma maye gurbin su tare da alamun haɓaka - kuma daidaito zai kasance gaskiya:

Da sauransu.

Kamar yadda ka sani, ga juzu'in a/b, inda a da b suke lamba, da b ≠ 0, sun ce. lambar hankali. Amma a cikin Yaren mutanen Poland kawai suna kiran kansu da hakan. Suna jin Turanci, Faransanci, Jamusanci da Rashanci. lambar hankali. A cikin Ingilishi: lambobi masu ma'ana. Lambobin rashin hankali rashin hankali ne, rashin hankali. Har ila yau, muna magana da Yaren mutanen Poland game da ra'ayoyin da ba su dace ba, ra'ayoyi da ayyuka - wannan hauka ne, tunaninsa, wanda ba zai iya bayyanawa ba. Suna cewa mata suna tsoron beraye - shin wannan ba rashin hankali bane?

A zamanin da, lambobi suna da rai. Kowannensu yana nufin wani abu, kowanne yana nuna alamar wani abu, kowanne yana nuna ɓangarorin wannan jituwa ta sararin samaniya, wato, a cikin Hellenanci, Cosmos. Kalmar "cosmos" tana nufin daidai "tsari, tsari". Mafi mahimmanci su ne shida (cikakkiyar lamba) da goma, jimlar lambobi a jere 1+2+3+4, sun ƙunshi wasu lambobi waɗanda alamarsu ta wanzu har yau. Don haka Pythagoras ya koyar da cewa lambobi sune farkon da tushen komai, kuma kawai ganowa lambobi marasa hankali ya juya motsin Pythagorean zuwa ga lissafi. Mun san dalili daga makaranta cewa

√2 lamba ce mara hankali

Don ɗauka cewa akwai: kuma wannan juzu'in ba za a iya rage shi ba. Musamman, duka p da q ba su da kyau. Mu yi murabba'i: 2q²=p². Lambar p ba za ta iya zama m ba, tun daga lokacin p² Hakanan zai kasance, kuma gefen hagu na daidaito shine ma'auni na 2. Saboda haka, p yana da ma, watau p = 2r, saboda haka p.²= shekaru 4². Mun rage ma'auni 2q²= shekaru 4² ta 2. Muna samun q²= shekaru 2² kuma mun ga cewa q dole ne ya kasance ma, wanda muka ɗauka ba haka ba ne. Sakamakon sabani ya kammala hujja - Ana iya samun wannan dabara sau da yawa a cikin kowane littafin lissafi. Wannan tabbataccen hujja shine abin da masana sophists suka fi so.

Pythagoreans ba za su iya fahimtar wannan girman ba. Duk abin dole ne a iya kwatanta shi ta lambobi, kuma diagonal na murabba'i, wanda kowa zai iya zana tare da sanda a fadin yashi, ba shi da, wato, ma'auni, tsayi. "Imaninmu banza ne," kamar yadda Pythagoreans suka ce. Ta yaya haka? Yana da irin... rashin hankali. Kungiyar ta yi kokarin ceton kanta ta hanyoyin bangaranci. Duk wanda ya kuskura ya bayyana samuwarsa lambobi marasa hankali, za a yanke masa hukuncin kisa, kuma, a fili, maigidan da kansa ya zartar da hukuncin farko.

Amma "tunanin ya wuce babu matsala." Zaman zinare ya iso. Girkawa sun ci Farisawa (Marathon 490, Block 479). An ƙarfafa dimokuradiyya, sababbin cibiyoyin tunanin falsafa da sababbin makarantu sun tashi. Pythagoreans har yanzu suna fama da lambobi marasa ma'ana. Wasu sun yi wa'azi: ba za mu fahimci wannan asiri ba; za mu iya yin tunani kawai kuma mu yi mamakin Uncharted. Na ƙarshe sun fi yin aiki tuƙuru kuma ba sa mutunta Asirin. A lokacin, gine-ginen tunani guda biyu sun bayyana waɗanda suka ba da damar fahimtar lambobi marasa ma'ana. Gaskiyar cewa mun fahimce su da kyau a yau na Eudoxus ne (ƙarni na XNUMX BC), kuma a ƙarshen karni na XNUMX ne masanin lissafin Jamus Richard Dedekind ya ba ka'idar Eudoxus ingantaccen ci gaba daidai da buƙatun tsauri. ilimin lissafi dabaru.

Yawan adadi ko azabtarwa

Za ku iya rayuwa ba tare da lambobi ba? Ko da yaya rayuwa za ta kasance... Dole ne mu je kantin sayar da takalma don siyan takalma da sanda, wanda a baya mun auna tsawon ƙafa. "Ina son apples, ah, ga shi!" - za mu nuna masu sayarwa a kasuwa. "Yaya nisa daga Modlin zuwa Nowy Dwur Mazowiecki"? "Kyawawan kusa!"

Ana amfani da lambobi don aunawa. Tare da taimakonsu, muna kuma bayyana wasu ra'ayoyi da yawa. Misali, ma'aunin taswirar yana nuna yadda yankin ƙasar ya ragu. Ma'auni biyu zuwa ɗaya, ko 2 kawai, yana bayyana gaskiyar cewa an ninka wani abu a girman. Bari mu ce da lissafi: kowane kamanni yayi daidai da lamba - sikelin sa.

Aikin. Mun yi kwafin xerographic, muna ɗaukaka hoton sau da yawa. Sa'an nan kuma ƙara girman guntu ya sake kara girman b sau. Menene ma'aunin haɓakawa gabaɗaya? Amsa: a × b ana ninka su da b. Wadannan ma'auni suna buƙatar ninkawa. Lambar "rasa ɗaya", -1, yayi daidai da daidaici ɗaya wanda ke tsakiya, watau juyawa digiri 180. Wane lamba yayi daidai da juyi digiri 90? Babu irin wannan lamba. Shi ne, shi ne… ko maimakon haka, zai kasance nan ba da jimawa ba. Shin kuna shirye don azabtar da ɗabi'a? Yi ƙarfin hali kuma ɗauki tushen murabba'in ban da ɗaya. Ina sauraro? Me ba za ku iya ba? Bayan haka, na ce ku yi jaruntaka. Cire shi! Hey, da kyau, ja, ja ... Zan taimaka ... Anan: -1 Yanzu da muke da shi, bari mu yi ƙoƙari mu yi amfani da shi ... Tabbas, yanzu za mu iya cire tushen duk lambobi marasa kyau, don misali.:

√-4 = 2-1, √-16 = 4-1

"Ko da kuwa bacin ran da yake ciki." Wannan shi ne abin da Girolamo Cardano ya rubuta a cikin 1539, yana ƙoƙari ya shawo kan matsalolin tunani da ke tattare da su - kamar yadda ba da daɗewa ba aka kira shi - ƙididdiga masu yawa. Ya yi la'akari da wadannan ...

...Aikin. Raba 10 zuwa kashi biyu, samfurin wanda shine 40. Na tuna cewa daga labarin da ya gabata ya rubuta wani abu kamar haka: Tabbas ba zai yiwu ba. Duk da haka, bari mu yi haka: raba 10 zuwa kashi biyu daidai, kowanne daidai yake da 5. Riba su - ya zama 25. Daga sakamakon 25, yanzu cire 40, idan kuna so, kuma kuna samun -15. Yanzu duba: √-15 da aka ƙara kuma aka cire daga 5 yana ba ku samfurin 40. Waɗannan su ne lambobi 5-√-15 da 5 + √-15. Cardano ne ya tabbatar da sakamakon kamar haka:

“Komai ciwon zuciya da yake ciki, ninka 5 + √-15 ta 5-√-15. Muna samun 25 - (-15), wanda yake daidai da 25 + 15. Don haka, samfurin shine 40 .... Gaskiya yana da wahala."

To, nawa ne: (1 + √-1) (1-√-1)? Mu yawaita. Ka tuna cewa √-1 × √-1 = -1. Mai girma. Yanzu aiki mai wahala: daga a + b√-1 zuwa ab√-1. Me ya faru? Tabbas, kamar haka: (a + b√-1) (ab√-1) = a²+b²

Menene ban sha'awa game da wannan? Alal misali, gaskiyar cewa za mu iya sarrafa maganganun da ba mu "ba su sani ba a da." Ƙididdigar yawan adadin da aka taƙaita don²-b² Kuna tuna da dabara don²+b² ba, domin ba zai iya zama ba. A cikin yankin na ainihin lambobi, da yawa²+b² ba shi yiwuwa. Bari mu nuna “mu” tushen murabba’in “rasa ɗaya” tare da harafin i.²= -1. Babban lamba ce "mara gaske". Kuma wannan shine abin da ke kwatanta jujjuyawar jirgin sama na digiri 90. Me yasa? Bayan haka,²= -1, da kuma hada daya 90-digiri juyi da wani 180-digiri juyawa yana ba da 45-digiri juyawa. Wane nau'i na juyawa ake bayyana? Babu shakka juya digiri XNUMX. Menene ma'anar -i? Ya ɗan fi rikitarwa:

(-I)² = -i × (-i) = +i² = -1

Don haka -i kuma yana bayyana jujjuyawar digiri 90, kawai a cikin kishiyar jujjuyawa na. Wanne ya rage kuma wanne ne daidai? Dole ne ku yi alƙawari. Muna ɗauka cewa lambar da na ƙididdige jujjuyawa a cikin alkiblar da masana ilimin lissafi suka ɗauka mai kyau: gaba da agogo. Lambar -i tana kwatanta juyawa a cikin hanyar da masu nuni ke motsawa.

Amma akwai lambobi kamar ni da -i? Ba! Mun dai kawo su a rai. Ina sauraro? Cewa sun wanzu ne kawai a cikin kawunanmu? To me ake jira? Duk sauran lambobi kuma suna wanzu ne kawai a cikin tunaninmu. Muna bukatar mu ga ko adadin jariran mu sun tsira. Ƙari daidai, ko zane yana da ma'ana kuma ko za su kasance da amfani ga wani abu. Da fatan za a ɗauki maganata cewa komai yana cikin tsari kuma waɗannan sabbin lambobin suna da taimako sosai. Lambobi kamar 3+i, 5-7i, gabaɗaya: a+bi ana kiransu hadaddun lambobi. Na nuna muku yadda za ku iya samun su ta hanyar karkatar da jirgin. Ana iya shigar da su ta hanyoyi daban-daban: a matsayin maki a cikin jirgin sama, kamar yadda wasu nau'i-nau'i iri-iri, a matsayin wasu nau'i na lambobi ... kuma kowane lokaci suna daya: equation x² +1=0 babu wani abu... hocus pocus ya riga ya kasance!!!! Muyi murna da murna!!!

Ƙarshen yawon shakatawa

Wannan ya kawo karshen rangadinmu na farko na kasar lambobi na jabu. Daga cikin sauran lambobin da ba a san su ba, zan kuma ambaci waɗanda ke da adadi marasa iyaka a gaba, ba a baya ba (ana kiran su 10-adic, a gare mu p-adic sun fi mahimmanci, inda p shine lambar farko), don misali X = … … … 96109004106619977392256259918212890625

Mu kirga X don Allah². Domin? Idan muka lissafta murabba'in lamba fa da adadin lambobi marasa iyaka? To, mu ma mu yi haka. Mun san cewa x² = X.

Bari mu sami wani irin wannan lamba tare da adadi mara iyaka a gaba wanda ya gamsar da ma'auni. Alamomi: murabba'in lamba da ta ƙare a shida ita ma tana ƙarewa cikin shida. Madaidaicin lamba wanda ya ƙare a 76 shima yana ƙarewa da 76. Ƙaƙƙarfan lambar da ta ƙare a cikin 376 kuma ta ƙare a 376. Ƙarshen lambar da ta ƙare a 9376 kuma ta ƙare a 9376. Ƙarshen lambar da ta ƙare a cikin XNUMX. XNUMX na… Hakanan akwai lambobi waɗanda suke ƙanana waɗanda, kasancewa tabbatacce, sun kasance ƙasa da kowane adadi mai kyau. Suna da ƙanƙanta ta yadda wani lokaci ya isa a yi murabba'in su don samun sifili. Akwai lambobi waɗanda basu gamsar da yanayin a × b = b × a. Hakanan akwai lambobi marasa iyaka. Lambobin halitta nawa ne akwai? Da yawa marasa iyaka? E, amma nawa? Ta yaya za a iya bayyana wannan a matsayin lamba? Amsa: mafi ƙarancin lambobi marasa iyaka; an yi masa alama da kyakkyawan harafi: A kuma an ƙara shi da sifili A₀ , aleph-zero.

Akwai kuma lambobin da ba mu san akwai su ba... ko kuma waɗanda za ku iya gaskatawa ko kafirta yadda kuke so. Kuma magana game da irin wannan: Ina fatan har yanzu kuna son Lambobin da ba na gaske ba, Lambobin Fantasy Species.