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Istoria Matematicii IstoriaMatematicii file:///C:/Programele%20Mele/IstoriaMatematicii/IstoriaMatematicii.html Istoria Matematicii Cuprins: Introducere Numere și reprezentarea lor Aritmetică Algebră Geometrie Analiză matematică Logică Matematică aplicată Matematică computațională Programarea calculatoarelor Repere istorice Introducere Din totdeauna, matematica a făcut apanajul potentaților vremii, a fost un instrument cu ajutorul căruia oamenii și-au măsurat bogăția, strălucirea sau puterea. Acest material nu își propune o înșiruire de date istorice sau nume ale unor matematicieni aranjate cronologic, ci mai degrabă o călătorie în timp prin universul matematicii, pentru a redescoperi ordinea în care s-au formulat principalele probleme și soluțiile lor. Dintre personalitățile matematicii, atenția se va abate mai mult asupra celor care au avut ceva de socotit, decât asupra acelora care au făcut socotelile. Oricare dintre realizările lor poate fi astăzi un bun exercițiu pentru scrie proiectul cu mijloace moderne, sau pentru calculul necesarului. Cel care poate alcătui proiectul și caietul de sarcini pentru Piramida lui Keops, sau pentru Zidul Chinezesc, va putea proiecta mai ușor o locuință modernă sau o anexă. Poate fi un exercițiu bun și calculația pentru un proiect ce utilizează doar tehnologia și resursele existente la vremea respectivă. Fiecare dintre subiectele propuse poate face subiectul unei disertații, al unui eseu sau al unei ore de curs facultativ. Nedorind a amalgama prea mult noțiunile și așa destul de complexe, șirul povestirii este deșirat în mai multe capitole, structurate pentru diferitele ramuri ale matematicii. Textul nu are caracter didactic dar poate inspira viitorii profesori de matematică. Este bun un pretext pentru o călătorie în timp, un fel de temă de casă a unui elev mai întârziat ca vârstă. Acest text nu are pretenția de a epuiza subiectul, ci doar de a deschide apetitul altora. Întreaga piramidă a cunoasterii se ridică asemănător piramidelor mayașe, din bolovani aglomerați unul peste altul. Fiecare nou savant, se cațără pe munca precedenților, rostogolind la vale pietrele mai vechi, pentru a duce în vârf piatra lui filozofală. Și procesul continuă apoi, secole după secole, milenii după milenii. Numere și reprezentarea lor 1 of 143 6/3/2020 6:24 PM IstoriaMatematicii file:///C:/Programele%20Mele/IstoriaMatematicii/IstoriaMatematicii.html Denumirea de matematică provine din limba greacă de la cuvântul mathema (matnoua) cu semnificația de "ceea ce trebuie studiat". Matematica s-a născut din necesitatea de a estima dimensiunile unor entități fizice sau relațiile de ordin calitatativ dintre fenomenele naturale. Primii gânditori ai Antichității au utilizat astfel de estimări pentru a reconstitui evenimente trecute, sau pentru a planifica evenimente prezente și viitoare. În primul rând a fost vorba despre măsurători și relația dintre obiectele măsurate. Pentru a simplifica operațiile cu obiecte a căror natură este diferită, cum sunt de exemplu arborii dintr-o pădure în raport cu animalele ce viețuiesc în ea, matematicienii au inventat niște obiecte abstracte denumite numere. Indiferent de natura unui anumit element, o unitate din acel element poate fi înlocuită printr-un număr. Astfel putem aduna merele cu perele sau arborii cu păsările cântătoare. Cu alte cuvinte, numerele sunt niște convenții, sau niște coduri, ce țin locul unui obiect real sau imaginar. Pentru a fi ușor de recunoscut, numerele sunt reprezentate prin simboluri grafice simple și prin cuvinte intuitive, denumite numerale. În ansamblul lor, toate numeralele utilizate pentru a reprezenta numere sunt denumite sistem de numerație. Termenul de notație matematică include și simbolurile utilizate pentru operații cu numere sau pentru anumite atribute ale numerelor. În cursul istoriei, diferitele popoare au inventat și apoi au abandonat numeroase sisteme numerale. Dintre acestea merită amintite cele inventate de: babilonieni (cuneiforme), egipteni (hieroglife, hieratice), chinezi, japonezi și coreeni (ideograme), fenicieni (ugaritice), romani (latine), evrei (aramaice), indieni (devanagari, bengali, tamil), greci (ionice, milesiene, alexandriene), armeni (erkatagir), khmeri (thai, lao), arabi (abjad, aryabahta, bakhshali), slavi (chirilice), mayași (maiașe), sau azteci (kipu). Sisteme noi de numerație se dezvoltă și în prezent, cel mai bun exemplu fiind cele utilizate pentru tehnica modernă de calcul (Hex, ASCII). Încă din cele mai vechi timpuri, pentru a exprima grupuri și mulțimi de elemente, numerele au fost grupate în clustere (familii de numere). Toate numerele aparținând unui astfel de grup aveau un atribut comun, sau respectau o anumită regulă. În antichitate, calculele matematice depășeau foarte rar numărul miilor. Ca urmare, numeralele aflate în uz erau și ele destul de sumare. Primele numere cu adevărat mari au devenit banale doar începând cu secolul al XIII-lea, o dată cu apariția primelor bănci și a primului sistem rudimetar de contabilitate. Europenii au menținut sistemul decimal roman, dar au adoptat numeralele arabe. Principalul avantaj a constat din ușurimea cu care se puteau reprezenta numerele mari. Adăugând cifra zero la sfârșitul unui număr, valoarea acestuia creștea de zece ori. Acest sistem a permis reprezentarea numerelor mari cu ajutorul unor relații matematice, de exemplu ca puteri ale lui 10. Primele numerale au fost simple semne scrijelite pe ceva (lemn, os, piatră, lut) pentru a ține o evidență contabilă simplă. Nu întotdeauna contabilitatea s-a ținut prin semne grafice. De exemplu în Babilonia, femeile obișnuiau să planteze lângă casă câte un palmier datier pentru fiecare copil nou născut. Acești palmieri nu numai că produceau mai multe fructe, dar ofereau și o reprezentare naturală a familiei: câți membri, cât de vârstnici și cât de apropiați între ei. La moartea unui membru al familiei, palmierul lui era sacrificat. Deși primitiv, acest sistem natural de codificare s-a păstrat până în zilele noastre. Pentru operații matematice, babilonienii utilizau un sistem numeral denumit sexagesimal, cu numerale cuprinse între 1 și 60. 2 of 143 6/3/2020 6:24 PM IstoriaMatematicii file:///C:/Programele%20Mele/IstoriaMatematicii/IstoriaMatematicii.html Orice mulțime mai mare de 60 era divizată în grupuri de până la 60. Necesitățile lor de calcul depășeau doar foarte rar 60 x 60 = 3600 de elemente. Pentru astfel de situații, pur și simplu mai adăugau o coloană formată din alte 60 de elemente. Nu întotdeauna calculele se făceau de la mai mic spre mai mare. S-a născut astfel necesitatea de a diviza valorile mari în valori mai mici. A apărut însă o nouă problemă. Nu toate numerele întregi puteau fi divizate în grupuri mai mici, egale între ele. De exemplu, o turmă de oi nu putea fi împărțită în mod egal între mai mulți frați. După numeroase calcule și frustrări, au fost identificate și izolate acele numere care nu se divid exact decât prin sine. Astfel de numere se numesc numere prime și formează o serie care merge spre infinit. Primele din serie sunt: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97 (cam un sfert din primele o sută). Cea mai veche reprezentare a unor numere prime apare pe un papirus din Egipt, datat aproximativ în jurul anului 1500 îen, unde se utilizau fracții diferite pentru numerele prime față de numerele compuse. Cea mai veche abordare sistematică a problemei a fost consemnată în lucrarea denumită Elemente, aparținând lui Euclid din Alexandria (Eucleidou), un arhitect din timpul faraonului Ptolomeu Soter (323-283 îen), cunoscut ca "părinte al Geometriei " și expert în măsurarea parcelelor de teren. Teoria sa despre numerele întregi include o demonstrație simplă potrivit căreia trebuie să existe un număr infinit de numere prime. În rezumat, demonstrația lui este următoarea: "Considerând o listă oarecare de numere prime p1, p2 ...pn, indiferent cât de lungă este lista trebuie să existe un alt număr prim mai mare decât ultimul din listă. Acest număr este produsul tuturor numerelor prime din listă plus 1. Numărul rezultat nu poate fi decât un număr prim deoarece restul de 1 nu poate fi divizat prin nici un alt număr întreg" . Numerele prime sunt într-adevăr infinite, însă cu cât devin mai mari, cu atât sunt din ce în ce mai rare, prin creșterea numărului de divizori posibili. Plecând de la problema divizării unor suprafețe de teren, Euclid a izolat și o altă serie de numere, aflată la polul opus față de numerele prime. Aceste numere sunt formate expres din suma divizorilor lor, adică pot fi împărțite în mod egal în mai multe feluri. Din acest considerent au fost denumite numere perfecte, pentru a sublinia cât este de ușor de lucrat cu ele. Primele patru numere perfecte din serie sunt: 6 = 1 + 2 + 3 28 = 1 + 2+ 3 + 4 + 5 + 6 + 7 496 = 1 + 2 + 3 + ... + 29 + 30 + 31 8128 = 1 + 2 + 3 + ... + 125 + 126 + 127 În Antichitate, matematicienii nu au avut mijloace pentru a calcula următorul număr perfect (33550336). Au concluzionat că doar aceste patru numere au puteri magice și fac parte din Casa Domnului (Lumea a fost creată în 6 zile, Luna are o orbită de 28 de zile ...). Euclid însă a identificat o relație matematică prin care a legat cele patru numere între ele. Potrivit acestei relații q (q+1)/2 este un număr perfect par de forma 2 la puterea p minus 1, unde p trebuie să fie un număr prim (2, 3, 5 , 7 ...). Două milenii mai târziu, matematicianul elvețian Leonhard Euler a formulat o nouă teoremă (Euclid-Euler) potrivit căreia un număr este perfect doar atunci când suma respectă ecuația: Nr
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