ࡱ> 7 9bjbjUU 7|7|:l  ****,V$ vjdSSSsnunununununun$rw yRn) S@SSSn )uS snSsn\Wz@ gLz rP d*i|DjgL "u0vDryygL Sumerian Civilizacia Sumerian a nflorit cu 4.000 ani .C. n fertila cmpie dintre Tigru _i Eufrat. Era o civilizacie avansat care construia ora_e _i sisteme de irigacie, care a realizat un sistem legislativ, care avea un sistem administrativ performant _i chiar un serviciu po_tal. Cu peste 3.500 ani .C., sumerienii scriau pe tblice de lut. Obiecte diferite erau reprezentate prin simboluri diferite, iar numrul acestora era era prezentat prin repeticie. Prin anul 3.200 .C. Sumerul este cucerit de akkadieni. Cele dou civilizacii _i unesc cuno_tincele n toate domeniile. Acest sistem avea dou mari inconveniente. n primul rnd, pentru fiecare obiect trebuia s existe un simbol caracteristic, simboluri care - evident - trebuia memorate. Al doilea inconvenient era legat de reprezentarea cantitcii. Pentru a reprezenta trei butoaie de ulei se repeta de trei ori simbolul acestuia. Dar dac numrul acestora este mai mare, scrierea nu mai este a_a de simpl _i poate conduce la erori. Dezvoltarea economic a impus crearea unui alt sistem de reprezentare. Prima mare inovacie dup inventarea scrisului a fost abstractizarea numrului de obiecte de acela_i fel. Astfel, trei butoaie cu ulei erau reprezentate prin simbolul pentru trei urmat de simbolul pentru butoi cu ulei. La fel se puteau reprezenta 3 oi, 3 vaci, n general 3 obiecte de acela_i fel. Un astfel de sistem este metrologic (a_a cum scriem 3kg, 3h, 3m). Astfel, simbolul pentru "trei" nu este n totalitate abstract, dar a reprezentat un salt uria_ n dezvoltarea reprezentrii numerelor _i a calculului. Sumerienii foloseau 60 de simboluri numerice, dar nu pentru orice fel de numere. Astfel, aveau un set de simboluri (o tabl) pentru numrarea obiectelor discrete (cum ar fi oi, butoaie etc.) _i o alt tabl pentru calcularea ariilor sau volumelor. Pentru a numra obiectele discrete (de ex. oi, capre, pe_ti) simbolul pentru un singur obiect era un mic con. Zece conuri erau nlocuite printr-un cerc mic. ^ase cercuri mici se nlocuiau printr-un con mare. Zece conuri mari erau reprezentate printr-un con mare cu un cerc mic n interior. ^ase conuri mari se nlocuiau printr-un cerc mare. n fine, 10 cercuri mari erau reprezentate printr-un cerc mare n interiorul cruia era plasat un cerc mic. Astfel, ultima unitate numra 10 6 10 6 10 = 36.000 obiecte. 10610610 INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/sum-36000.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/sum-3600.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/sum-600.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/sum-60.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/sum-10.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/sum-1.gif" \* MERGEFORMATINET Scrierea pe tblicele de lut se putea face foarte u_or: cercul era creat prin apsarea vertical a unui cui, conul prin aplicarea oblic a cuiului pe tblic. Sumerienii foloseau _i un sistem bisexagesimal n care factorii de multiplicare erau 10, 6, 2, 10 _i 6, astfel c simbol pentru cantitatea cea mai mare (un cerc mare cu dou cercuri mici interioare) avea valoarea de 6102610 = 7.200 unitci de baz. Un alt sistem era folosit pentru msurarea grnelor. n acest sistem, factorii de multiplicare erau 5, 10, 3, _i 10, astfel nct unitatea cea mai mare (un con mare cu un cerc mic n interior) avea valoarea de 103105=1.500 unitci de baz. Astfel, acela_i semn putea fi folosit n diferite sisteme, valoarea sa depinznd de sistemul respectiv. De exemplu, cercul mic putea nsemna 6, 10 or 18 conuri mici. Treptat, n decursul mileniului 3 .C., aceste semne au fost nlocuite de echivalentul lor cuneiform. Pe la sfr_itul mileniului 3 .C. a fost introdus sistemul de numeracie sexagesimal pozicional. Numrul de simboluri a fost redus la dou:  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/cun-1.gif" \* MERGEFORMATINET , derivat din conul mic, _i  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/cun-10.gif" \* MERGEFORMATINET derivat din cercul mic, care avea valoarea de 10 unitci de baz. Sistemul arta acum astfel: 10610610... INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/cun-10.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/cun-1.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/cun-10.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/cun-1.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/cun-10.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/cun-1.gif" \* MERGEFORMATINET _i putea continua indefinit. Apare totu_i un inconvenient: simbolul  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/cun-1.gif" \* MERGEFORMATINET poate reprezenta 1, 60 (610), 3.600 (6060) etc. unitci de baz, valoarea sa depinznd de pozicia pe care o ocup. Sistemul sexagesimal pozicional a u_urat foarte mult efectuarea calculelor, numai c el era folosit exclusiv la efectuarea calculelor. Rezultatele obcinute erau apoi transformate n vechile sisteme metrologice. Babilonian Civilizacia babilonian a nlocuit-o pe cea sumerian ncepnd cu 2.000 .C.. Babilonienii au mo_tenit cuno_tincele pe care le aveau sumerienii _i akadienii. De_i au mprumutat scrierea numerelor _i baza de numeracie de la ace_tia, sistemul de numeracie a evoluat devenind pozicional. Babilonienii stabiliser unitci de msur pentru lungime, mas _i volum, timp (mprciser ziua n 24 de ore, ora n 60 de minute _i minutul n 60 de secunde), creaser un calendar foloseau mprcirea cercului n 360 de grade. Babilonienii aveau cuno_tince astronomice avansate, putnd s prevad eclipsele de soare _i de lun. Foloseau fracciile, ptratul unui numr, rdcina ptrat. Au inventat un sistem de scriere pozicional cu baza 60. Aveau un semn pentru unu INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/cun-1.gif" \* MERGEFORMATINET , care repetat ddea doi INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/cun-2.gif" \* MERGEFORMATINET , trei INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/cun-3.gif" \* MERGEFORMATINET  _i a_a mai departe, pn la zece, pentru care exista un alt semn INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/cun-10.gif" \* MERGEFORMATINET . Combinnd semnele reprezentnd pe unu _i pe zece se obcin 11, 12, ..., 59. Pentru _aizeci se folosea acela_i semn ca pentru unu, dar valoarea sa era dat de coloana n care se gsea. Se putea continua avnd posibilitatea reprezentrii oricrui numr. 12345678910 INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/cun-1.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/cun-2.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/cun-3.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/cun-4.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/cun-5.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/cun-6.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/cun-7.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/cun-8.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/cun-9.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/cun-10.gif" \* MERGEFORMATINET 11121314151617181920 INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/cun-11.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/cun-12.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/cun-13.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/cun-14.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/cun-15.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/cun-16.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/cun-17.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/cun-18.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/cun-19.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/cun-20.gif" \* MERGEFORMATINET 30405060 INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/cun-30.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/cun-40.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/cun-50.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/cun-1.gif" \* MERGEFORMATINET Pentru a scrie numere mai mari dect 60, mesopotamienii foloseau aceste reprezentri n sensul actual de cifr. NumrulTrecerea n baza 10 1 15 INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/cun-1.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/cun-15.gif" \* MERGEFORMATINET 1601+15600=60+15=751 40 INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/cun-1.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/cun-40.gif" \* MERGEFORMATINET 1601+40600=60+4=10016 43 INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/cun-16.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/cun-40.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/cun-3.gif" \* MERGEFORMATINET 16601+43600=960+43=1.00344 26 40 INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/cun-40.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/cun-4.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/cun-20.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/cun-6.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/cun-40.gif" \* MERGEFORMATINET 44602+26601+40600=158.400+1.560+40=160.0001 24 51 10 INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/cun-1.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/cun-20.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/cun-4.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/cun-50.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/cun-1.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/cun-10.gif" \* MERGEFORMATINET 1603+24602+51601+10600=216.000+86.400+3.060=305.470Sistemul avea un inconvenient: deoarece nu exista reprezentare pentru cifra 0, mesopotamienii n locul acesteia lsau un loc liber. Dar nu totdeauna !. Astfel, nu este clar dac  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/cun-1.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/cun-1.gif" \* MERGEFORMATINET nseamn 2, 61, 3601 sau 3660. Totu_i, n practic cifra 0 n sexagesimal apare destul de rar. Mai trziu, cnd astronomii au avut nevoie de foarte multe calcule, au introdus un semn special pentru a nlocui spaciul (cifra 0). Scrierea pozicional permite reprezentarea u_oar a fracciilor. Pentru separarea prcii ntregi de cea zecimal noi folosim virgula zecimal, anglo-saxonii punctul zecimal. Mesopotamienii nu foloseau nimic. Stabilirea faptului c un numr este ntreg sau zecimal se fcea "prin inspeccie". De exemplu,  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/cun-40.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/cun-4.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/cun-20.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/cun-6.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/cun-40.gif" \* MERGEFORMATINET poate nsemna 16.000 sau 1/81. Pentru unele fraccii uzuale, mesopotamienii foloseau notacii speciale:  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/cun-1pe2.gif" \* MERGEFORMATINET 1/2 INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/cun-1pe3.gif" \* MERGEFORMATINET 1/3 INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/cun-2pe3.gif" \* MERGEFORMATINET 2/3 INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/cun-5pe3.gif" \* MERGEFORMATINET 5/6Fiind pozicional, sistemul este u_or de folosit deoarece utilizeaz acela_i semn pe diferite locuri, valoarea sa intrinsec rmnnd aceea_i, dar valoarea efectiv depinznd de pozicia pe care o ocup. Nu au fost descoperite table pentru adunare sau scdere. Se presupune c scribii nvcau s adune _i s scad odat cu nvcarea cititului _i scrisului, a_a c tablele pentru adunare _i scdere nu-_i aveau rostul. n schimb, exist o mulcime de table de multiplicare. Pe la 2.300 .C. au inventat abacul _i au creat metode pentru adunarea, scdere, nmulcire _i mprcire. Babilonienii au creat table pentru nmulcire sub dou forme: table simple _i table combinate. Tablele simple concin produsele unui singur numr, numit numr principal (de ex.  HYPERLINK "http://math4all.home.ro/aritmetica/bab_x05.htm" 5,  HYPERLINK "http://math4all.home.ro/aritmetica/bab_x10.htm" 10). Deoarece baza de numeracie este 60, s-ar prea c tabla trebuia s concin 58 de linii (de la 2 la 59). n realitate, tablele concineau liniile cu produsele de la 2 la 20, apoi cu 30, 40 _i 50. Dac se dorea produsul cu 39 (de ex.) se adunau multiplul lui 30 cu multiplul lui 9. Uneori tablele se ncheiau cu ptratul numrului principal. Tablele combinate concin mai multe numere principale, fiind de fapt, formate din mai multe table simple (de ex. cu  HYPERLINK "http://math4all.home.ro/aritmetica/bab_12-30.htm" 12-30, cu  HYPERLINK "http://math4all.home.ro/aritmetica/bab_44-26-40.htm" 44-26-40). Aproape toate tablele care apar n table combinate se gsesc _i separat, ca table simple. Pentru a putea calcula mai u_or un produs foloseau formula: ab = [(a + b)2  a2  b2]/2 sau o formul chiar mai eficient: ab = [(a + b)2  (a  b)2]/4 A fost descoperit o tabl a ptratelor numerelor pn la 59 _i una a cuburilor pn la 32. Nu exist table pentru mprcire, n schimb a fost creat o  HYPERLINK "http://math4all.home.ro/aritmetica/bab_inv.htm" tabl de inverse. Inversul numrului n este fraccia 1/n. n loc s mpart un numr la n, babilonienii l nmulceau cu cu inversul lui n. Ca _i n sistemul nostru de numeracie, _i n sistemul babilonian existau fraccii sexagesimale infinite. Evident, singurele inverse care erau fraccii sexagesimale finite erau cele care nu concineau alci factori afar de puteri ale lui 2, 3 _i 5. Mai exist _i cteva table pentru rdcina ptrat _i cubic. Exist _i table pentru rezolvarea unor probleme financiare. n fine, au fost gsite _i cteva table de conversie pentru unitci de msur. Exist o tabl de corespondenc ntre lungimea  HYPERLINK "http://math4all.home.ro/aritmetica/bab_pat.htm" diagonalei _i latura ptratului. Matematica babilonienilor se ocupa de lucruri practice, n special de calcule. Nu se punea problema unei demonstracii. Interesul pentru studiul geometriei era, de asemenea, minor. De_i foloseau construccii geometrice, problemele conduceau la calcule aritmetice. Problemele erau formulate cu date concrete, din viaca de zi cu zi. Elevilor li se cerea s afle lungimi de canale, masa unor stnci, aria unor terenuri, numrul de crmizi folosite ntr-o construccie etc. De obicei se cerea aflarea lungimii laturii sau diagonalei unui ptrat, determinarea ariei sau a volumului. Pe unele tblice erau desenate figuri geometrice standard cum ar fi ptrat, dreptunghi, triunghi, trapez, cerc etc. Studiul corpurilor geometrice era dominat de calcul de crmizi _i planuri nclinate, dar apar _i cilindri, trunchiuri de con _i piramide. Egiptean Egiptul a fost probabil prima civilizacie n care interesul pentru _tiince a fost major. Au excelat n medicin _i matematici aplicate, dar _i n astronomie, mecanic, chimie, fizic, administracie. Chiar numele de chimie provine de la alchimie, vechiul nume al Egiptului. Civilizacia Egiptului Antic a atins un nalt nivel nc din cele mai vechi timpuri. Datorit Nilului _i climei, Egiptul avea tot ce-i necesar dezvoltrii unei civilizacii nfloritoare. Egiptul era _i u_or de aprat avnd o lung granic cu de_ertul Sahara, a_a c s beneficiat de perioade lungi de pace, perioade n care societatea s-a dezvoltat rapid. Cu 3.000 de ani .C., n Egipt era dezvoltat puternic agricultura pe baza inundaciilor bianuale ale Nilului. Apa revrsat aducea aluviuni care mbogceau solul; surplusul de ap era dirijat printr-un sistem complicat de canale _i ecluze, astfel ca ea s fie folosit _i n perioadele secetoase. Construirea _i ntrecinerea unui astfel de sistem de irigacii a necesitat importante cuno_tince de geometrie, mecanic, hidraulic. Cunoa_terea cu precizie a perioadelor din an n care se produceau inundaciile era de maxim importanc. Problema a fost rezolvat de cuno_tincele avansate de astronomie care le-a permis realizarea unui calendar foarte precis. Teritoriul pe care se ntindea Egiptul fiind vast, era nevoie de un sistem administrativ eficient. Pentru calcularea taxelor _i repartizarea sumelor colectate pentru construccii, armat _.a. era nevoie de cuno_tince de aritmetic. Din 3.000 .C. a nceput construccia piramidelor; astfel marea piramid de la Ghiza a fost construit prin 2.650 .C. Construccia piramidelor necesita vaste cuno_tince _i imense resurse materiale. n acea perioad, Egiptenii aveau pus la punct sistemul de scriere hieroglific. Sistemul de numeracie folosit nu era foarte bun pentru realizarea calculelor aritmetice. Operaciile aritmetice, a_a cum le cunoa_tem azi, erau foarte greu de realizat: adunarea _i scderea se puteau efectua relativ u_or; nmulcirea _i mprcirea erau de-a dreptul imposibile. Totu_i, egiptenii au dezvoltat metode remarcabile pentru a trece peste acest neajuns. La nceput, numerele erau sculptate n piatr pentru a comunica diferite mrimi. Deoarece nu era nevoie s se opereze mult cu ele, pentru cifre nu existau hieroglife speciale. Din momentul n care s-a trecut la utilizarea papirusului pentru scriere, a aprut necesitatea dezvoltrii unor mijloace mai rapide de scriere, a aprut necesitatea crerii unor hieroglife pentru scrierea numerelor. Papirusurile descoperite arat c egiptenii, spre deosebire de greci care s-au preocupat de studiul matematicii abstracte, erau legaci de rezolvarea unor probleme de aritmetic legate exclusiv de practic. Sistemul de numeracie folosit de ei era zecimal _i pozicional, dar nu n accepcia actual. "Cifrele" folosite se obcineau prin compunerea a _apte simboluri de baz:  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/egi-1.gif" \* MERGEFORMATINET 1un bc de msurat INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/egi-10.gif" \* MERGEFORMATINET 10un val INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/egi-100.gif" \* MERGEFORMATINET 100sfoara de msurat INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/egi-1000.gif" \* MERGEFORMATINET 1.000floarea de lotus INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/egi-10000.gif" \* MERGEFORMATINET 10.000degetul arttor INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/egi-100000.gif" \* MERGEFORMATINET 100.000o broasc INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/egi-1000000.gif" \* MERGEFORMATINET 1.000.000un zeu cu minile ridicate deasupra capuluiScrierea se fcea n ordinea cresctoare a valorii. Iat cteva exemple: 3.244 = INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/egi-1.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/egi-1.gif" \* MERGEFORMATINET   INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/egi-1.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/egi-1.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/egi-10.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/egi-10.gif" \* MERGEFORMATINET   INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/egi-10.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/egi-10.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/egi-100.gif" \* MERGEFORMATINET   INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/egi-100.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/egi-1000.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/egi-1000.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/egi-1000.gif" \* MERGEFORMATINET 4402003.00021.237 = INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/egi-1.gif" \* MERGEFORMATINET   INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/egi-1.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/egi-1.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/egi-1.gif" \* MERGEFORMATINET   INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/egi-1.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/egi-1.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/egi-1.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/egi-10.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/egi-10.gif" \* MERGEFORMATINET   INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/egi-10.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/egi-100.gif" \* MERGEFORMATINET   INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/egi-100.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/egi-1000.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/egi-10000.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/egi-10000.gif" \* MERGEFORMATINET 7302001.00020.000dar se putea scrie _i pe vertical: 200 INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/egi-100.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/egi-100.gif" \* MERGEFORMATINET 4.000 INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/egi-1000.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/egi-1000.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/egi-1000.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/egi-1000.gif" \* MERGEFORMATINET 70 INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/egi-10.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/egi-10.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/egi-10.gif" \* MERGEFORMATINET   INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/egi-10.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/egi-10.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/egi-10.gif" \* MERGEFORMATINET 600 INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/egi-100.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/egi-100.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/egi-100.gif" \* MERGEFORMATINET   INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/egi-100.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/egi-100.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/egi-100.gif" \* MERGEFORMATINET 6 INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/egi-1.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/egi-1.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/egi-1.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/egi-1.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/egi-1.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/egi-1.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/egi-10.gif" \* MERGEFORMATINET 20 INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/egi-10.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/egi-10.gif" \* MERGEFORMATINET 2764 INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/egi-1.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/egi-1.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/egi-1.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/egi-1.gif" \* MERGEFORMATINET 4.624Deoarece se foloseau semne diferite pentru unitci, zeci, sute, mii, ..., nu are importanc ordinea scrierii. Nu era nevoie nici de simbol pentru zero.  INCLUDEPICTURE "http://math4all.home.ro/images/separator.gif" \* MERGEFORMATINET  Efectuarea unei nmulciri era destul de complicat. S considerm produsul 41 59. Construim o tabl astfel: rndul 1 al doilea factor, 59, pe rndurile urmtoare se scrie dublul rndului precedent pn cnd multiplicatorul devine mai mare ca primul factor, n cazul nostru pn la 32 < 41 < 64: multiplicatorvaloaremultiplicatorvaloare159141211828242364164847283281694416656321.888321.3122.4192.419Apoi efectum o serie de scderi: 41  32 = 9; 9  8 = 1; 1  1 = 0 _i scriem 41 = 32 + 8 + 1. Selectm multiplii corespunztori _i summ. Putem s schimbm ordinea factorilor, 59 41. Avem 59  32 = 27; 27  16 = 11; 11  8 = 3; 3  2 = 1; 1 1=0. _i scriem suma multiplilor 59 = 32 + 16 + 8 + 2 + 1. Metoda folosit se bazeaz pe teorema care spune c orice numr poate fi scris ca o sum a puterilor lui 2. Egiptenii nu aveau o dovad n acest sens _i nici nu-i interesa s-o obcin. ^tiau c metoda este bun _i o aplicau. Pur _i simplu! Totu_i, noi ne putem permite s scriem: 41 = 120 + 021 + 022 + 123 + 024 + 125, respectiv: 59 = 120 + 121 + 022 + 123 + 124 + 125.  INCLUDEPICTURE "http://math4all.home.ro/images/separator.gif" \* MERGEFORMATINET  mprcirea se realiza tot prin dublare. S lum, de exemplu, numrul 1.495 _i s-l mprcim la 65. Construim un tabel ca la nmulcire: multiplicatorvaloare165213042608520161.0401.495_i ne oprim n momentul n care valoarea din tabel devine mai mare dect demprcitul, adic la 1.040<1.495<2.080. Avem: 1.495  1.040 = 455; 455  260 = 195; 195  130 = 65, 65  65 = 0, deci: 1.495= 1.040 + 260 + 130 + 65. Adunm multiplicatori corespunztori: 1 + 2 + 4 + 16 = 23. Acesta este ctul mprcirii 1.495 : 65. n exemplul de mai sus, 1.495 se divide cu 65. Cum se calculeaz n cazul n care demprcitul nu se divide cu mprcitorul? S considerm mprcirea 1.500 : 65. Construim tabelul: multiplicatorvaloare165213042608520161.0401.495^i de data aceasta ne oprim n momentul n care valoarea din tabel devine mai mare dect demprcitul, adic la 1.040<1.500<2.080. Adunm valorile n pentru care avem: 1.500  65<n INCLUDEPICTURE "http://math4all.home.ro/simboluri/less&equ.gif" \* MERGEFORMATINET 1.500: 1.040 + 260 + 130 + 65 = 1.465 Diferenca 1.500  1.465 = 5 reprezint restul mprcirii. Summ multiplicatorii corespunztori: 1 + 2 + 4 + 16 = 23. Acesta este ctul mprcirii. Atunci se poate scrie: 1.500 : 65 = 23 + 5/65 = 23 1/13  INCLUDEPICTURE "http://math4all.home.ro/images/separator.gif" \* MERGEFORMATINET  Egiptenii foloseau numai fraccii cu numrtorul 1, cu excepcia a dou fraccii mai des folosite: 2/3 _i 3/4. Iat cteva exemple:  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/egi-frac.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/egi-frac.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/egi-1.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/egi-1.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/egi-1.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/egi-10.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/egi-10.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/egi-frac.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/egi-1.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/egi-1.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/egi-1.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/egi-10.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/egi-10.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/egi-1.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/egi-1.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/egi-1.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/egi-1.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/egi-1.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/egi-1.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/egi-1.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/egi-1.gif" \* MERGEFORMATINET   INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/egi-1.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/egi-1.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/egi-1.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/egi-10.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/egi-10.gif" \* MERGEFORMATINET   INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/egi-10.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/egi-10.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/egi-100.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/egi-100.gif" \* MERGEFORMATINET 1/31/251/269Urmtoarea problem pe care ne-o punem este cum se efectueaz nmulcirea _i mprcirea cu fraccii. S lum ca mprcitor fraccia 1/5. Am fi tentaci s procedm ca mai sus, prin dublarea acesteia: 1/5 + 1/5. Din motive pe care nu le discutm, egiptenii, n loc s efectueze acest calcul ar fi adunat 1/3 + 1/15. Papirusul Rhind concine o tabl care permitea dublarea unor fraccii de tipul 1/n, pentru 5 < n < 101 impar, cu numrtorul 1. Iat nceputul acestei table: Fraccia de dublatFracciile care dubleaz1/51/3 + 1/151/71/4 + 1/281/91/6 + 1/181/111/6 + 1/661/131/10 + 1/26 + 1/651/151/10 + 1/301/171/12 + 1/51 + 1/68. . . .. . . . . . . . . . . . . .Este remarcabil de observat c papirusul nu concine erori (apr cteva din copiere), c termenii descompunerii sunt fraccii cu numitori apropiaci ca valoare _i c niciodat nu sunt mai mulci de 4. Cum rezolvau egiptenii ecuacia: 2/3 + 1/15 + x = 1 ? Se multiplic cu 15: 10 + 1 + y = 15. Aceasta era numit auxiliar ro_u, deoarece scribul folosea cerneal ro_ie la scrierea ei. Solucia ei este, evident, 4. Pentru a obcine solucia ecuaciei iniciale scriem: dublu (dublu 1/15) Din tabla de mai sus observm c dublu 1/15 este suma 1/10 + 1/30, pe care dublnd-o se obcine 1/5 + 1/15, care este solucia ecuaciei date. Iat _i o problem: O cantitate adugat la un sfert din cantitate d 15. Ct este cantitatea ? Problema se transcrie n limbaj modern astfel: x + x / 4 = 15 Presupunem c x ar fi egal cu 4. Atunci x + x / 4 = 5, ceea ce nu este corect. Dar 15 este de 3 ori 5. A_a c presupunerea trebuie multiplicat cu 3. Deci, rspunsul corect este x = 12. Mai multe probleme din papirusul Rhind folosesc n rezolvare metoda falsei ipoteze (aplicat mai sus). Cum procedau egiptenii pentru a rezolva calculul: (1 + 1/3 + 1/5) (30 + 1/3) ? Foloseau metoda dublrii: 11 + 1/3 + 1/522 + 2/3 + 1/3 + 1/15 = 3 + 1/1546 + 1/10 + 1/30812 + 1/5 + 1/151624 + 1/3 + 1/15 + 1/10 + 1/302/32/3 + 1/6 + 1/18 + 1/10 + 1/301/31/3 + 1/12 + 1/36 + 1/20 + 1/60Penultima linie din tabel s-a obcinut astfel: 2/3 din 1 este 2/3; 2/3 din 1/3 este dublul lui 1/9 care este 1/6 + 1/18; 2/3 din 1/5 este dublul lui 1/15 care este 1/10 + 1/30. Acum trebuie gsite numerele din prima coloan care nsumate dau 30+1/3. Rezultatul se obcine sumnd valorile din a doua coloan. Acesta este: 46 + 1/5 + 1/10 + 1/12 + 1/15 + 1/30 + 1/36. O alt problem din papirusul Rhind: Un teren rotund are diametrul de 9 khet. Ce arie are ? Solucia prezentat n papirus este urmtoarea: Se afl 1/9 din diametru, adic 1; restul este 8. nmulcind 8 cu 8 ne d 64. A_a c terenul are 64 setat. 191/91 19216432864De observat c solucia este echivalent cu p = 4(8/9)2 = 3.1605. Calculnd acum, obcinem 3.160493 care difer de rezultatul obcinut de egipteni dect la a 4-a zecimal. Este un lucru remarcabil dac cinem cont de perioada n care a fost obcinut.  INCLUDEPICTURE "http://math4all.home.ro/images/separator.gif" \* MERGEFORMATINET  n papirusul din Moscova este prezentat urmtoarea problem, ilustrat n figura alturat: Problema cere s se calculeze un trunchi de piramid pornind de la urmtoarele date: baza mare este un ptrat cu latura de 4 cubit, baza mic este un ptrat cu latura de 2 cubit _i distanca dintre baze este de 6 cubit. n primul rnd trebuie remarcat c prin s se calculeze un trunchi de piramid se ncelege s se calculeze volumul unui trunchi de piramid. INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/egi-mosc.gif" \* MERGEFORMATINET Calculul ncepe cu aflarea ariei bazei mari: 4 4 = 16. Se calculeaz apoi aria bazei mici: 2 2 = 4. Se nmulcesc latura bazei mari cu latura bazei mici: 4 2 = 8. Se adun rezultatele: 16 + 4 + 8 = 28. Se calculeaz 1/3 din nlcime, adic: 2. n final, se nmulce_te ultimul rezultat cu suma calculat anterior _i se obcine 56. Aceast problem arat c egiptenii _tiau formula volumului trunchiului de piramid. Astfel, lund a latura bazei mari, b latura bazei mici _i h nlcimea, formula s-ar traduce n limbaj modern: V = h/3 (a2 + ab + b2)  INCLUDEPICTURE "http://math4all.home.ro/images/separator.gif" \* MERGEFORMATINET  Dup inventarea scrierii pe papirus, egiptenii au creat "cifrele" hieratice. Cu ajutorul lor, numerele puteau fi scrise ntr-o manier mult mai compact. n noua scriere existau simboluri pentru 1,.., 9; 10, ..., 90; 100, ..., 900; 1.000, ..., 9.000. De exemplu, numrul 9.999 se scria acum cu 4 hieroglife n loc de 36. Iat un exemplu:  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/egi-2765.gif" \* MERGEFORMATINET  Cele dou sisteme de scriere au coexistat mai bine de 2.000 de ani. Cel hieratic era folosit pentru scrierea pe papirus, cel obi_nuit continund s se utilizeze pentru inscripcii cioplite n piatr. INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/egi-hiera.gif" \* MERGEFORMATINET Grecesc Pn prin secolul 3 .C., aproape fiecare republic greceasc folosea un alt sistem de notare a numerelor. Apoi, schimburile comerciale tot mai intense au dus la adoptarea unui sistem de notare preluat de la fenicieni. Totu_i, diferencele ntre sistemele de numeracie nu erau majore, principalul lor rol fiind legat de tranzacciile economice. Primul de care ne vom ocupa este sistemul acrofonic folosit n mileniul 1 .C. (Acrofonic nseamn c notacia unui numr se face prin prima liter a numelui su). Sistemul mai este cunoscut _i sub numele de Attica, dup regiunea din jurul Atenei n care a fost folosit. Principalele simboluri folosite erau: Numele litereiSimbolul litereiNumele numruluiValoareaIotaIIota (iota)1PiGPenta (penta)5DeltaDDeka (deka)10EtaHHekaton (hekaton)100ChiCCilioi (chilioi)1.000MuMMnrioi (myrioi)10.000n afar de caracterul I - care este o abreviere - simbol ntlnit n aproape toate sistemele pentru notarea numrului (cifrei) unu, celelalte simboluri deriv din fonograme. De exemplu, simbol G este o versiune mai veche a lui P (pi), prima liter a cuvntului Penta (penta), care nseamn cinci. La fel, simbolul D este delta, prima liter a cuvntului Deka (deca), ce nseamn zece; simbolul H este eta, prima liter a cuvntului Hekaton (hecaton), ce nseamn o sut. Ca _i cel  HYPERLINK "http://math4all.home.ro/aritmetica/sist_egi.htm" egiptean sau cel  HYPERLINK "http://math4all.home.ro/aritmetica/sist_rom.htm" roman, primul sistem de numeracie grecesc este unul aditiv. De exemplu, pentru a afla numrul: DDGIII se adun valorile individuale: DDGIII = 10 + 10 + 5 + 1 + 1 + 1 = 28 De observat c nu are importanc ordinea n care sunt scrise simboluri din alctuirea numrului. Totu_i, era preferat scrierea simbolurilor n ordine descresctoare a valorii acestora. Pentru reprezentarea unor numere (mai ales a celor mari), este nevoie de foarte multe simboluri. De ex., pentru a scrie 9.999 se folosesc 32 de simboluri: CCCCCCCCCHHHHHHHHHDDDDDDDDDGIIII Nu este surprinztor c apare un simbol pentru 5 de vreme ce baza de numeracie folosit este 10, provenind de la cele 10 degete, cci la o mn avem 5 degete. Este ns interesant c vechii greci aveau simboluri pentru 50, 500, 5.000 _i 50.000, simboluri care proveneau din combinacia simbolului pentru 5 cu simbolurile pentru 10, 100, 1.000, 10.000: Simbolul litereiValoarea INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/gre-50.gif" \* MERGEFORMATINET 50 INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/gre-500.gif" \* MERGEFORMATINET 500 INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/gre-5000.gif" \* MERGEFORMATINET 5.000 INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/gre-50000.gif" \* MERGEFORMATINET 50.000A_a cum am spus _i mai nainte, existau diference ntre reprezentrile caracterelor n diferite state. Iat cteva pentru 50:  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/gre-dif5.gif" \* MERGEFORMATINET  Pentru vechii greci numrul nu era o nociune abstract, era legat de obiectele pe care le numra. Astfel, pentru a preciza o sum de 5.678 de drahme ei scriau:  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/gre-drahma.gif" \* MERGEFORMATINET  n care se observ c simbolul unitcilor este caracteristic pentru drahm. Ca s reprezinte 3.807, taleri scriau:  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/gre-taler.gif" \* MERGEFORMATINET  Se observ c pentru unitci se folose_te simbolul T de la Taler. Suma de 3.807 drahme _i 5 oboli (1 obol = 1/6 drahme) era reprezentat astfel:  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/gre-dra-obo.gif" \* MERGEFORMATINET   INCLUDEPICTURE "http://math4all.home.ro/images/separator.gif" \* MERGEFORMATINET  Cel de-al doilea sistem de care ne ocupm, folosit de vechii greci, este sistemul alfabetic. n acest sistem numerele sunt reprezentate prin literele alfabetului. Acest mod de reprezentare a fost copiat de la fenicieni. De fapt, _i alfabetul pentru scrierea cuvintelor fusese copiat de la ei. Grecii aveau 24 de litere n alfabet. Au mai adugat 3 litere scoase din uz (simbolurile pentru 6, 90 _i 900: digamma, koppa _i sampi): 1A a(alpha)10Ii(iota)100Rr(rho)2B b(beta)20Kk(kappa)200Ss(sigma)3G g(gamma)30Ll(lambda)300Tt(tau)4D d(delta)40Mm(mu)400Uu(upsilon)5E e(epsilon)50Nn(nu)500Ff(phi)6 INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/gre-digamma-m.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/gre-digamma.gif" \* MERGEFORMATINET (digamma)60Xx(xi)600Cc(chi)7Z z(zeta)70Oo(omicron)700Yy(psi)8H h(eta)80Pp(pi)800Ww(omega)9Q q(theta)90 INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/gre-koppa-m.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/gre-koppa.gif" \* MERGEFORMATINET (koppa)900 INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/gre-sampi-m.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/gre-sampi.gif" \* MERGEFORMATINET (sampi)Evident, sistemul este zecimal. Ca _i  HYPERLINK "http://math4all.home.ro/aritmetica/" \l "sistemul acrofonic" sistemul acrofonic, cel alfabetic este unul aditiv. De exemplu: FMB = 500 + 40 + 2 = 542 Uneori, pentru a le deosebi de literele alfabetului, deasupra acestor simboluri se trasa o bar. Doar cu aceste simboluri puteau fi reprezentate doar numere mai mici ca 1.000. Pentru numerele cuprinse ntre 1,000 _i 9,999 au fost create simboluri compuse din simbolurile de baz crora li se ata_a litera i (iota) fie n stnga-sus, fie n stnga-jos: 1.000iA iA 2.000iB iB 3.000iG iG 4.000iD iD 5.000iE iE 6.000i INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/gre-digamma-m.gif" \* MERGEFORMATINET i INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/gre-digamma-m.gif" \* MERGEFORMATINET 7.000iZ iZ 8.000iH iH 9.000iQ iQ Pentru numere mai mari se scria deasupra literei M (de la Miriad) un numr cuprins ntre 1 _i 9.999, valoarea acestuia multiplicndu-se cu 10.000: bslaM= 20.000M= 3.210.000De multe ori, n loc s se scrie deasupra literei M, se scria n faca acesteia: iZROEMiEWOE = 71.755.875 Acest sistem le permitea grecilor s scrie toate numerele aprute n viaca de zi cu zi. Numere ca 71.755.875 erau, evident, o raritate, iar numere mai mari nu aveau de unde s apar. O alt modalitate de a multiplica cu zece mii valoarea unui simbol era scrierea cu dou puncte deasupra simbolului respectiv. De exemplu:  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/sist_g1.gif" \* MERGEFORMATINET   INCLUDEPICTURE "http://math4all.home.ro/images/separator.gif" \* MERGEFORMATINET  Vechii greci foloseau _i fraccii. Notacia era ambigu _i determinarea valorii depindea n cea mai mare msur de context. Un accent plasat dup un grup de simboluri numerice transforma grupul n inversul acestuia: b' = 1/2, iar mg' = 1/43 dar putea s nsemne _i: mg' = 401/3. Spre deosebire de  HYPERLINK "http://math4all.home.ro/aritmetica/sist_egi.htm" egipteni, grecii foloseau _i fraccii cu numrtor mai mare ca 1: napd' = 51/84 Se observ c numrtorul este precizat cu suprabarare.  INCLUDEPICTURE "http://math4all.home.ro/images/separator.gif" \* MERGEFORMATINET  nmulcirea se realiza folosind distributivitatea: spz b = (s + p + z) b (200 + 80 + 7) 2 = 400 + 160 + 14 = 500 + 70 + 4 fod Interesant este c vechii greci efectuau mprcirea n acela_i mod cum o efectum _i noi azi. Roman naintea romanilor, cea mai dezvoltat civilizacie din peninsula Italic a fost cea a etruscilor. Etruscii au copiat  HYPERLINK "http://math4all.home.ro/aritmetica/sist_gre.htm" \l "sistemul acrofonic" sistemul grecesc de numeracie acrofonic. Romanii au copiat sistemul de la etrusci _i l-au adaptat alfabetului lor. IVXLCDM1510501005001.000Sistemul de numeracie roman nu este un simplu  HYPERLINK "http://math4all.home.ro/aritmetica/sist_num.htm" \l "Sistemul aditiv" sistem aditiv, ci unul aditiv-subtractiv. La nceput, cifrele V, L _i D lipseau. 1Iun bc vertical10Xdou bece ncruci_ate100Ciniciala cuvntului centum (100)1.000Miniciala cuvntului mille (1.000)Pentru 1.000, romanii foloseau inicial o alt notacie:  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/rom-1000.gif" \* MERGEFORMATINET . Pe 10.000 l notau  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/rom-10000.gif" \* MERGEFORMATINET , iar pe 100.000 cu  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/rom-100000.gif" \* MERGEFORMATINET . Regulile de scriere (preluate tot de la etrusci) erau: Orice semn pus la dreapta altuia de valoare mai mare sau egal dect a lui, se adun. Exemplu: XX = 10 + 10 = 20 XII = 10 + 1 + 1 = 12 Orice semn pus la stnga altuia de valoare mai mare dect a lui, se scade. Exemplu: IX = 10 - 1 = 9 IXC = 100 - (10 - 1) = 100 - 9 = 91 XCII = 100 - 10 + 1 + 1 = 100 - 10 + 1 + 1 = 92 Un punct de vedere propriu: Celelalte cifre romane cunoscute au aprut din nevoia de simplificare a scrierii. Interesant este c, spre deosebire de  HYPERLINK "http://math4all.home.ro/aritmetica/sist_gre.htm" sistemul grecesc, noile simboluri sunt _i din punct de vedere grafic jumtci ale unitcilor din care provin. Simbolul X = 10, dac este tiat n dou pe linia median orizontal genereaz dou litere V (din care una rsturnat) avnd fiecare valoarea 5; n mod analog, tindu-l pe C = 100 (s nu uitm c forma sa cioplit nu era rotunjit, artnd astfel  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/rom-100.gif" \* MERGEFORMATINET ) se obcin dou litere L (din care una este rsturnat), fiecare cu valoarea 50; tind pe linia median vertical pe  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/rom-1000.gif" \* MERGEFORMATINET , vechiul simbol pentru 1.000, se obcin dou simboluri  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/rom-500.gif" \* MERGEFORMATINET , foarte apropiat de litera D (din care una n oglind), fiecare avnd valoarea 500. Sistemul de numeracie roman este foarte greu de folosit, n special la scrierea numerelor mari. Ca fapt divers: consemnarea numrului de prizonieri n urma luptei cu cartaginezii, estimat la 2.300.000, s-a fcut prin repetarea semnului  INCLUDEPICTURE "http://math4all.home.ro/aritmetica/images/rom-100000.gif" \* MERGEFORMATINET de 23 de ori ! Dar chiar n cazul numerelor mici scrierea poate fi destul de complicat. De exemplu, numrul 879: 879 = 800 + 70 + 9 DCCCLXXIX O alt lacun a acestui sistem de numeracie este ambiguitatea regulilor de scriere. Astfel, numrul 8 poate fi scris fie ca VIII, fie ca IIX. n vechiul sistem de numeracie roman (1.200 .C.), cnd au fost introduse formele substractive (ca IV = 4 sau IX = 9), era posibil s se reprezinte orice numr mai mic dect 5.000 cu ajutorul unei serii de simboluri n care oricare nu aprea mai mult de 4 ori. De exemplu, 2.976 = MMDCCCCLXXVI. Sistemul aditiv este foarte u_or de folosit n calcule simple. Adunarea se face n doi pa_i: mai nti se scriu mpreun toate simbolurile din care sunt alctuite cele dou numere, apoi se "colecteaz" simbolurile de cea mai mic valoare _i se nlocuiesc, dac este cazul, cu simboluri de valoare superioar: 2319+MMCCCXIX821DCCCXXI3140MMDCCCCCCXXXIXIMMDCCCCCCXXXXMMDDCXLMMMCXLnmulcirea se face ceva mai greu: 28XXVIII12XII56XXVIIIoriIegalXXVIII28XXVIIIoriIegalXXVIII336XXVIIIoriXegalCCLXXXCCLXXXXXXXVVIIIIIICCLXXXXXXXVVVICCLXXXXXXXXVICCLLXXXVICCCXXXVIScderea _i mprcirea sunt ceva mai complicate, dar pot fi efectuate. Zecimal Este cel mai rspndit sistem de numeracie. Cifrele folosite n sistemul zecimal se numesc cifre zecimale. A_adar, cifrele sistemului zecimal sunt numerele naturale mai mici ca 10 _i se noteaz n ordine, respectiv cu: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 Secvenca de cifre zecimale 192544, sau mai precis 192544(10) reprezint numrul natural: 1105 + 9104 + 2103 + 5102 + 4101 + 4100 Abacul  Originea abacului se pierde n negura timpului. La nceput, tabla abacului era o suprafac plan pe care erau trasate linii de-a lungul crora se puteau mi_ca o serie de pietre pentru a se executa operacii aritmetice. Foarte multe civilizacii au folosit abacul pentru realizarea calculelor aritmetice. n Europa, odat cu rspndirea cifrelor arabe _i cu aparicia hrtiei, majoritatea au renuncat la utilizarea lui. Se _tie c n Italia abacul a disprut n sec.16. n partea de nord a Europei, din cauza nivelului sczut de trai _i a conservatorismului, abacul a rmas n uz pn n sec.18. Avem impresia gre_it c numai cei din orient au folosit abacul, asta pentru c n Europa a disprut de foarte mult vreme, pe cnd n orient mai este folosit _i azi. Dup nfrngerea lui Napoleon n campania de invadare a Rusiei (1812), unii dintre soldaci s-au ntors n Franca aducnd ca pe o curiozitate abace ruse_ti. Nu _tiau c bunicii lor le folosiser n n treburile lor zilnice! Dar nu numai europenii _i asiaticii au utilizat abacul, conchistadorii au gsit abace la populaciile de pe actualele teritorii ale Mexicului _i Peruului n forma sa modern, abacul a fost inventat de chinezi prin secolul 13, de la care a fost preluat de coreeni (sec.15) _i de japonezi (sec.17). Se _tie c abacul mai era nc folosit acum 250 de ani n unele zone ale Europei. Mulci dintre termenii moderni din matematic _i comerc provin din denumirile legate de abac. De exemplu, romanii denumeau pietrele abacului calculi; de aici provin termenul a calcula _i derivatele sale calcul, calculator. n Anglia, tabla abacului era denumit, n general, counting board sau, mai simplu, counter. Fire_te, fiecare comerciant avea n magazin un counter cu care fcea calculul valorii mrfurilor vndute. De aici provine termenul modern de contor (pl. contoare). n Europa, abacul a fost standardizat n sec.13: o tabl pe care erau trasate linii care indicau locul de pozicionare a pietrelor de socotit. Linia de jos era linia unitcilor, urmtoarele linii aveau valoarea de 10 ori mai mare dect a liniei de dedesubt; fiecare spaciu dintre linii avea valoarea de 5 ori mai mare dect a liniei de dedesubt. Nu puteau fi puse mai mult de 4 pietre pe o linie _i cel mult una pe un spaciu. Atunci cnd numrul de pietre de pe o linie ar fi devenit 5, erau scoase de pe linie _i pus o piatr n spaciul de deasupra acesteia. Dac numrul de pietre de pe un spaciu ar fi devenit 2, erau luate de pe spaciu _i pus o piatr pe linia de deasupra spaciului. Prin secolul 13 pietrele folosite pentru calcul la abac au cptat forma de moned. Aceste piese erau cast, aruncate sau apsate pe tabla abacului. n Franca piesele acestea erau numite jetoane din verbul jeter (a arunca). Abacul chinezesc Prin anii 1.500 .C., chinezii au inventat abacul. Acesta era format dintr-un cadru de lemn cu o bar orizontal care poate culisa pe vertical, mprcindu-l n dou compartimente. Pe vertical sunt fixate mai multe bare (de obicei 17) pe care pot culisa 7 bile: dou n partea de sus _i cinci n cea de jos. Fiecare bil de jos are valoarea 1; fiecare bil de sus are valoarea de 5. Astfel, pe o bar poate fi reprezentat orice numr cuprins ntre 1 _i 15. Orice valoare mai mare ca 9 este stocat temporar, ca rezultat intermediar. n imagine este prezentat un abac chinezesc n care este stocat valoarea 279. Abacul japonez  INCLUDEPICTURE "C:\\My Documents\\Mate\\abac_files\\sist_aba_files\\abac_japonez.gif" \* MERGEFORMATINET  La abacul japonez, pe fiecare bar vertical erau 4 bile n partea de jos _i una n partea de sus. Probabil c japonezii au adus abacului ultima perfeccionare nainte de desfiincarea sa. (Abacul din imagine stocheaz numrul 279) Poate c nu-i lipsit de importanc c n 1946 (!), un funccionar din administracia japonez folosea pentru calcule aritmetice un abac. S-a organizat un concurs ntre acesta _i un calculator electric cu probe constnd n adunarea sau scderea unor coloane lungi de numere, nmulcirea unor ntregi cu 5-12 cifre, mprcirea unor ntregi cu 5-12 cifre, rezolvarea unor probleme complexe concinnd toate aceste operacii. A fost btut de ma_in doar la unele nmulciri. Rezumat 40.000 .C.nsemnri pe lemne _i pe oase despre numrul de piei decinute, de animale prinse. Obiceiul s-a pstrat nc mult vreme, astfel c ciobanii fceau _i n secolul 20 astfel de nsemnri pe rboj. 20.000 .C.n pe_teri s-au descoperit desenate sau cioplite pe pereci linii folosite probabil la numrare 9.000 .C.Oamenii au folosit pentru socotit pietricele _i beciga_e 4.000 .C. HYPERLINK "http://math4all.home.ro/aritmetica/sist_sum.htm" Primul sistem de numeracie zecimal creat de sumerieni 3.500 .C. HYPERLINK "http://math4all.home.ro/aritmetica/sist_sum.htm" \l "tablite de lut" Folosirea tblicelor de lut pentru calcule 3.200 .C. HYPERLINK "http://math4all.home.ro/aritmetica/sist_sum.htm" \l "Sumerul este cucerit de akkadieni" Sistemul zecimal sumerian se une_te cu sistemul de numeracie akkadian rezultnd un un sistem de numeracie pseudosexagesimal (60 de semne dar fr scriere pozicional) 2.700 .C. HYPERLINK "http://math4all.home.ro/aritmetica/sist_egi.htm" Egiptenii creeaz un sistem de numeracie zecimal 2.500 .C. HYPERLINK "http://math4all.home.ro/aritmetica/sist_bab.htm" Babilonienii, situaci n sudul Mesopotamiei, preiau cuno_tincele de la sumerieni _i akkadieni _i realizeaz un adevrat sistem de numeracie sexagesimal. Semicii creeaz un sistem de numeracie zecimal 2.350 .C.Sistemul de numeracie zecimal se rspnde_te n toat Mesopotamia 2.300 .C.Vechii indieni creeaz un sistem de numeracie cu baza 4 2.000 .C.Invention of the Gurumikhi, an analogical calculator, Hittite and Indusian decimal numeration 1.900 .C.n Babilon se impune sistemul de numeracie zecimal 1.500 .C.Sistemul de numeracie asirian n Anatolia se folose_te un sistem de numeracie cu baza 4  HYPERLINK "http://math4all.home.ro/aritmetica/sist_aba.htm" Este inventat abacul n zona chinei de azi 1.000 .C. HYPERLINK "http://math4all.home.ro/aritmetica/sist_gre.htm" Sistemul de numeracie grecesc 650 .C.Sistemul de numeracie etrusc 650 .C.Fenicienii folosesc un sistem de numeracie cu baza 3 500 .C.Hindu_ii folosesc un sistem de numeracie zecimal preluat Zapothec and pre-Colombian numeration in base 20 Sistemul de numeracie cu baza 4 n Grecia numeration in base 5 in Yemen 200 .C.Chinese learned numeration in base 1 and 5  INCLUDEPICTURE "C:\\My Documents\\Mate\\tot_files\\sist_num_files\\separator.gif" \* MERGEFORMATINET  Sisteme de numeracie Generalitci Oamenii au _tiut s socoteasc nainte de a _ti s scrie. Se foloseau de degete de nsemnri pe bece sau pe oase, foloseau funia cu noduri, pietricele etc. Dar scrierea _i citirea numerelor permit o evaluare rapid a cantitcii de obiecte pe care o reprezint. Se nume_te sistem de numeracie totalitatea regulilor de reprezentare a numerelor folosind un anumit set de simboluri distincte, numit alfabet; simbolurile sunt numite cifre.n sistemele de numeracie primitive, pentru a reprezenta un numr de 5 oi se desenau prin repeticie 5 oi. ntr-o faz superioar - cea simbolic -, acela_i lucru era realizat desenndu-se o oaie urmat (precedat) de cinci semne (puncte, linii etc.). Aceasta a dus treptat la desprinderea de concret, la aparicia ideii de numr ca nociune abstract. Baza unui sistem de numeracie pozicional este dat de numrul de elemente care formeaz alfabetul sistemului de numeracie. De exemplu, sistemul de numeracie n baza 2 are alfabetul {0,1}; sistemul de numeracie n baza 16 are alfabetul {0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F}. n decursul timpului, n diferite zone ale globului au fost folosite sisteme de numeracie cu o varietate destul de mare de baze de numeracie: 3, 4, 5, 6, 10, 12, 20, 60. Dar cea mai des folosit a fost baza 10, probabil ca urmare a socotitului pe degete.  INCLUDEPICTURE "C:\\My Documents\\Mate\\tot_files\\sist_num_files\\separator.gif" \* MERGEFORMATINET  Semnele folosite pentru notarea cifrelor sunt destul de variate. Modul lor de grupare pentru reprezentarea unui numr calific un sistem de numeracie ca nepozicional ( HYPERLINK "http://math4all.home.ro/aritmetica/sist_num.htm" \l "Sistemul aditiv" aditiv,  HYPERLINK "http://math4all.home.ro/aritmetica/sist_num.htm" \l "Sistemul multiplicativ" multiplicativ) sau  HYPERLINK "http://math4all.home.ro/aritmetica/sist_num.htm" \l "Sistemul pozitional" pozicional. Sistemele aditive n aceste sisteme exist semne distincte (cifre) pentru fiecare grup de obiecte folosit n procesul numrrii. Sistemul de numeracie  HYPERLINK "http://math4all.home.ro/aritmetica/sist_egi.htm" egiptean este un astfel de sistem. Valoarea unui numr se obcine prin adunarea cifrelor dup anumite reguli. De exemplu: 21.237 = INCLUDEPICTURE "C:\\My Documents\\Mate\\tot_files\\sist_num_files\\egi-1.gif" \* MERGEFORMATINET   INCLUDEPICTURE "C:\\My Documents\\Mate\\tot_files\\sist_num_files\\egi-1.gif" \* MERGEFORMATINET  INCLUDEPICTURE "C:\\My Documents\\Mate\\tot_files\\sist_num_files\\egi-1.gif" \* MERGEFORMATINET  INCLUDEPICTURE "C:\\My Documents\\Mate\\tot_files\\sist_num_files\\egi-1.gif" \* MERGEFORMATINET   INCLUDEPICTURE "C:\\My Documents\\Mate\\tot_files\\sist_num_files\\egi-1.gif" \* MERGEFORMATINET  INCLUDEPICTURE "C:\\My Documents\\Mate\\tot_files\\sist_num_files\\egi-1.gif" \* MERGEFORMATINET  INCLUDEPICTURE "C:\\My Documents\\Mate\\tot_files\\sist_num_files\\egi-1.gif" \* MERGEFORMATINET  INCLUDEPICTURE "C:\\My Documents\\Mate\\tot_files\\sist_num_files\\egi-10.gif" \* MERGEFORMATINET  INCLUDEPICTURE "C:\\My Documents\\Mate\\tot_files\\sist_num_files\\egi-10.gif" \* MERGEFORMATINET   INCLUDEPICTURE "C:\\My Documents\\Mate\\tot_files\\sist_num_files\\egi-10.gif" \* MERGEFORMATINET  INCLUDEPICTURE "C:\\My Documents\\Mate\\tot_files\\sist_num_files\\egi-100.gif" \* MERGEFORMATINET   INCLUDEPICTURE "C:\\My Documents\\Mate\\tot_files\\sist_num_files\\egi-100.gif" \* MERGEFORMATINET  INCLUDEPICTURE "C:\\My Documents\\Mate\\tot_files\\sist_num_files\\egi-1000.gif" \* MERGEFORMATINET  INCLUDEPICTURE "C:\\My Documents\\Mate\\tot_files\\sist_num_files\\egi-10000.gif" \* MERGEFORMATINET  INCLUDEPICTURE "C:\\My Documents\\Mate\\tot_files\\sist_num_files\\egi-10000.gif" \* MERGEFORMATINET = 7 + 30 + 200 + 1.000 + 20.000Sistemul de numeracie  HYPERLINK "http://math4all.home.ro/aritmetica/sist_rom.htm" roman este un sistem aditiv-substractiv. Valoarea unui numr se obcine prin adunarea sau scderea cifrelor dup anumite reguli. De exemplu, XI = 11, MMCIII = 2103; IV = 4, IX = 9. Sistemele multiplicative Un sistem multiplicativ este acela n care pentru aflarea valorii unui numr este necesar s se nmulceasc anumite perechi de simboluri ntr-o manier asemntoare sistemului aditiv ( HYPERLINK "http://math4all.home.ro/aritmetica/sist_chi.htm" sistemul de numeracie chinez). Sistemele pozicionale n sistemele de numeracie pozicionale, un simbol din alctuirea unui numr (cifr) are valoare intrinsec dar _i o valoare prin pozicia pe care o ocup n numr. Aceasta implic existenca unui simbol cu valoare intrinsec nul (zero). n unele sisteme pozicionale ( HYPERLINK "http://math4all.home.ro/aritmetica/sist_bab.htm" babilonian) n care regulile o permit, este posibil s se renunce la acest simbol. Regulile folosite n aceste sisteme sunt mai complexe dect n cele aditive. Iat cum se scrie un numr n  HYPERLINK "http://math4all.home.ro/aritmetica/sist_zec.htm" sistemul zecimal pozicional: 40.323 = 4104 + 0103 + 3102 + 2101 + 3100 Trebuie remarcat c cifra 3 apare de dou ori n scrierea numrului. Cnd se afl pe ultimul loc, reprezint trei obiecte; cnd se afl pe antepenultimul loc, reprezint trei sute de obiecte. Sistemele de numeracie pozicionale folosesc acela_i sistem de reguli de reprezentare a numerelor; ele difer doar prin alfabetul pe care l utilizeaz _i, implicit, prin baz. Sisteme de numeracie pozicionale Din cele mai vechi timpuri, s-a pus problema gsirii unor procedee de scriere a numerelor naturale care s permit o rapid a ordinului lor de mrime _i elaborarea unor reguli eficiente de efectuare a operaciilor cu acestea. Adoptarea sistemului de numeracie zecimal s-a ncheiat abia n secolele 16-17 _i reprezint o etap important n dezvoltarea matematicii. Un sistem de numeracie este un ansamblu de reguli prin care valorile numerice pot fi scrise prin intermediul simbolurilor, denumite numere. Relacia de ordine introdus pe mulcimea INCLUDEPICTURE "C:\\My Documents\\Mate\\tot_files\\sist_num_files\\mult_n.gif" \* MERGEFORMATINET permite aranjarea numerelor naturale ntr-un _ir cresctor: 0 < 1 < 2 < ... < n < n  1 < ... Fie u un numr natural mai mare ca 1. Reprezentarea numerelor naturale n sistemul de numeracie de baz u se fundamenteaz pe cteva rezultate care sunt prezentate n continuare: Lema 1 Fie u > 1 un numr natural. Oricare ar fi numrul natural a > 0, exist numerele naturale n, q0,q1,...,qn  INCLUDEPICTURE "C:\\My Documents\\Mate\\tot_files\\sist_num_files\\mult_n.gif" \* MERGEFORMATINET astfel nct: a = uq0 + a0, 0 INCLUDEPICTURE "C:\\My Documents\\Mate\\tot_files\\sist_num_files\\less&equ.gif" \* MERGEFORMATINET a0 < u, q0 = uq1 + a1, 0 INCLUDEPICTURE "C:\\My Documents\\Mate\\tot_files\\sist_num_files\\less&equ.gif" \* MERGEFORMATINET a1 < u, . . . . . . . . . . . . . . . . . . . . . . . qn2 = uqn1 + an-1, 0 INCLUDEPICTURE "C:\\My Documents\\Mate\\tot_files\\sist_num_files\\less&equ.gif" \* MERGEFORMATINET an1 < u, qn1 = qn 0 INCLUDEPICTURE "C:\\My Documents\\Mate\\tot_files\\sist_num_files\\less&equ.gif" \* MERGEFORMATINET an < u, unde s-a notat cu n  1 numrul v  INCLUDEPICTURE "C:\\My Documents\\Mate\\tot_files\\sist_num_files\\mult_n.gif" \* MERGEFORMATINET , astfel nct v  1 = n _i cu n  2 numrul w  INCLUDEPICTURE "C:\\My Documents\\Mate\\tot_files\\sist_num_files\\mult_n.gif" \* MERGEFORMATINET , astfel nct w 1=v. Demonstracie Dac a < u, lum n = 0, q0 = 0, a0 = a _i lema este adevrat. Dac a INCLUDEPICTURE "C:\\My Documents\\Mate\\tot_files\\sist_num_files\\grate&equ-1.gif" \* MERGEFORMATINET 0, fie q0, a0  INCLUDEPICTURE "C:\\My Documents\\Mate\\tot_files\\sist_num_files\\mult_n-1.gif" \* MERGEFORMATINET , astfel nct: a = uq0 + a0,0 INCLUDEPICTURE "C:\\My Documents\\Mate\\tot_files\\sist_num_files\\less&equ-1.gif" \* MERGEFORMATINET a0 < u,Cum a INCLUDEPICTURE "C:\\My Documents\\Mate\\tot_files\\sist_num_files\\grate&equ-1.gif" \* MERGEFORMATINET 0, avem q0 > 0. Exist q1, a1  INCLUDEPICTURE "C:\\My Documents\\Mate\\tot_files\\sist_num_files\\mult_n-1.gif" \* MERGEFORMATINET , astfel nct: a0 = uq1 + a1,0 INCLUDEPICTURE "C:\\My Documents\\Mate\\tot_files\\sist_num_files\\less&equ-1.gif" \* MERGEFORMATINET a1 < u,Dac q1 = 0, lema este adevrat, cu n = 1. Astfel, exist q2, a2  INCLUDEPICTURE "C:\\My Documents\\Mate\\tot_files\\sist_num_files\\mult_n-1.gif" \* MERGEFORMATINET , astfel nct: a1 = uq2 + a2,0 INCLUDEPICTURE "C:\\My Documents\\Mate\\tot_files\\sist_num_files\\less&equ-1.gif" \* MERGEFORMATINET a2 < u,_i a_a mai departe. Dac qi  0, din 1 < u deducem qi < uqi INCLUDEPICTURE "C:\\My Documents\\Mate\\tot_files\\sist_num_files\\less&equ-1.gif" \* MERGEFORMATINET uqi + ai = qi 1, deci: a > q0 > q1 > ... > qi 1 > qi > ... Exist n astfel nct qn 1  0 _i qn = 0. ntr-adevr, fie A = {qi|qi  0} _i b cel mai mic numr din A. Exist n astfel nct b = qn 1. Cum qn<qn 1= b, rezult c qn 1 A, deci qn = 0. A_adar, 0 < qn1 = an < u. Lema 2 Fie u, a0, a1, ..., an  INCLUDEPICTURE "C:\\My Documents\\Mate\\tot_files\\sist_num_files\\mult_n.gif" \* MERGEFORMATINET astfel nct u > 1, 0  INCLUDEPICTURE "C:\\My Documents\\Mate\\tot_files\\sist_num_files\\less&equ.gif" \* MERGEFORMATINET ai < u pentru 0  INCLUDEPICTURE "C:\\My Documents\\Mate\\tot_files\\sist_num_files\\less&equ.gif" \* MERGEFORMATINET i  INCLUDEPICTURE "C:\\My Documents\\Mate\\tot_files\\sist_num_files\\less&equ.gif" \* MERGEFORMATINET n. Atunci:  INCLUDEPICTURE "C:\\My Documents\\Mate\\tot_files\\sist_num_files\\sist_n0.gif" \* MERGEFORMATINET Demonstracie Induccie dup n. Dac n = 0, atuncia0 < u. Presupunnd afirmacia adevrat pentru n, atunci:  INCLUDEPICTURE "C:\\My Documents\\Mate\\tot_files\\sist_num_files\\sist_n1.gif" \* MERGEFORMATINET  Teorem Fieu > 1 un numr natural. Atunci oricare ar fi numrul natural a > 0, exist n, a0,a1,...,an  INCLUDEPICTURE "C:\\My Documents\\Mate\\tot_files\\sist_num_files\\mult_n.gif" \* MERGEFORMATINET  unic determinate, astfel nct: a = anun + an 1un 1 + ... + a1u1 + a0u0, unde0 < an < u_i 0  INCLUDEPICTURE "C:\\My Documents\\Mate\\tot_files\\sist_num_files\\less&equ.gif" \* MERGEFORMATINET ai < u pentru 0  INCLUDEPICTURE "C:\\My Documents\\Mate\\tot_files\\sist_num_files\\less&equ.gif" \* MERGEFORMATINET i  INCLUDEPICTURE "C:\\My Documents\\Mate\\tot_files\\sist_num_files\\less&equ.gif" \* MERGEFORMATINET n  1.Demonstracie Summ egalitcile din enuncul  HYPERLINK "http://math4all.home.ro/aritmetica/sist_num.htm" \l "Lema 1" lemei 1 nmulcite, respectiv, cu 1, u, u2, ..., un: a = anun + an 1un 1 + ... + a1u1 + a0u0, Fie, de asemenea, numerele naturale m, bm,bm 1,...,b0 astfel nct a = bmum + mm 1um 1 + ... + b1u1 + b0u0, unde: 0 < bm < u_i 0  INCLUDEPICTURE "C:\\My Documents\\Mate\\tot_files\\sist_num_files\\less&equ.gif" \* MERGEFORMATINET bi < u pentru 0  INCLUDEPICTURE "C:\\My Documents\\Mate\\tot_files\\sist_num_files\\less&equ.gif" \* MERGEFORMATINET i  INCLUDEPICTURE "C:\\My Documents\\Mate\\tot_files\\sist_num_files\\less&equ.gif" \* MERGEFORMATINET m. Dac n < m, atunci n + 1  INCLUDEPICTURE "C:\\My Documents\\Mate\\tot_files\\sist_num_files\\less&equ-1.gif" \* MERGEFORMATINET m, de unde:  INCLUDEPICTURE "C:\\My Documents\\Mate\\tot_files\\sist_num_files\\sist_n2.gif" \* MERGEFORMATINET  Contradiccie. Analog se exclude cazul m < n. Rmne deci adevrat cazul n = m. S artm c ai = bi, 0  INCLUDEPICTURE "C:\\My Documents\\Mate\\tot_files\\sist_num_files\\less&equ.gif" \* MERGEFORMATINET i  INCLUDEPICTURE "C:\\My Documents\\Mate\\tot_files\\sist_num_files\\less&equ.gif" \* MERGEFORMATINET n. Dac n = 0, atunci a0 = a = b0. Presupunem c n > 0 _i c afirmacia este adevrat pentru n  1. Cum a = a0 + u(anun 1 + ... + a1) = b0 + u(bnun 1 + ... + b1), din teorema mprcirii cu rest rezult a0 = b0 _i anun 1 + ... + a1 = bnun 1 + ... + b1. Folosind ipoteza inducciei, din ultima egalitate deducem _i c a1 = b1, a2 = b2, ..., an = bn. La fiecare numr natural a > 0, asociem secvenca finit unic determinat de numere naturale anan 1 ... a1a0, unde ai < u, 0  INCLUDEPICTURE "C:\\My Documents\\Mate\\tot_files\\sist_num_files\\less&equ.gif" \* MERGEFORMATINET i  INCLUDEPICTURE "C:\\My Documents\\Mate\\tot_files\\sist_num_files\\less&equ.gif" \* MERGEFORMATINET n, an0 _i:  INCLUDEPICTURE "C:\\My Documents\\Mate\\tot_files\\sist_num_files\\sist_n3.gif" \* MERGEFORMATINET  A_adar, anan 1 ... a1a0  INCLUDEPICTURE "C:\\My Documents\\Mate\\tot_files\\sist_num_files\\=def.gif" \* MERGEFORMATINET  anun + an 1un 1 + ... + a1u1 + a0u0. Spunem c prin corespondenca de mai sus se realizeaz scrierea numerelor naturale n sistemul de numeracie cu baza u. Dac u este baza sistemului de numeracie, atunci numerele naturale c < u se numesc cifrele sistemului de numeracie cu baza u. Cel mai rspndit sistem de numeracie este cel cu baza 10, numit  HYPERLINK "http://math4all.home.ro/aritmetica/sist_zec.htm" sistemul zecimal. Sistemul de numeracie cu baza 2 se nume_te  HYPERLINK "http://math4all.home.ro/aritmetica/sist_bin.htm" binar. Este limbajul circuitelor electronice. Sistemul de numeracie cu baza 16 este numit  HYPERLINK "http://math4all.home.ro/aritmetica/sist_hex.htm" hexazecimal _i este folosit n programarea calculatoarelor. Are avantajul c este relativ apropiat de sistemul zecimal _i c numerele scrise n baza 16 pot fi foarte u_or convertite n baza de numeracie 2. Propozicia 1 Fie a _i b dou numere naturale scrise n baza u: a = amam 1 ... a1a0 _i b = bnbn 1 ... b1b0. Atunci a < b dac _i numai dac m < n sau m = n _i at < bt, unde t este cel mai mare indice i pentru care ai bi.Demonstracie: Dac m < n, atunci din  HYPERLINK "http://math4all.home.ro/aritmetica/sist_num.htm" \l "Lema 2" lema 2 rezult c a < um+1 INCLUDEPICTURE "C:\\My Documents\\Mate\\tot_files\\sist_num_files\\less&equ-1.gif" \* MERGEFORMATINET  un  INCLUDEPICTURE "C:\\My Documents\\Mate\\tot_files\\sist_num_files\\less&equ-1.gif" \* MERGEFORMATINET b, deci a < b. Presupunem c m = n _i at < bt, unde t = max{ i | ai bi }. Avem at+1 INCLUDEPICTURE "C:\\My Documents\\Mate\\tot_files\\sist_num_files\\less&equ-1.gif" \* MERGEFORMATINET b, deci at ut + ut  INCLUDEPICTURE "C:\\My Documents\\Mate\\tot_files\\sist_num_files\\less&equ-1.gif" \* MERGEFORMATINET btut. Rezult c:  INCLUDEPICTURE "C:\\My Documents\\Mate\\tot_files\\sist_num_files\\sist_n7.gif" \* MERGEFORMATINET  deci a < b. Reciproc, dac a < b, atunci din prima parte a demonstraciei deducem c m < n sau m = n _i at < bt, unde t = max{ i | ai bi }. Exemple: Dac a = 122311(10), b = 92197(10), deoarece m > n, a > b. Dac a = 1011000(2), b = 1010111(2), deoarece m = n _i pentru t = 3, at > bt, rezult c a > b. Propozicia 2 Numrul maxim care poate fi reprezentat n baza u cu m cifre este um  1.Demonstracie: Fie a = cm 1cm 2 ... c0 cel mai mare numr cu m cifre, reprezentat n baza u. Atunci: ci = u  1 pentru i = 0, ..., m  1; a + 1 = 10 ... 0(u) , unde numrul are m + 1 cifre. a + 1 = 1 um, Conform definiciei ********************** a = um  1 Exemplu: Numrul maxim care se poate reprezenta cu 4 cifre, n baza 10, este 9999, adic 104  1. Conversia dintr-o baz de numeracie oarecare n baza 10 Considerm sistemul de numeracie cu baza u. Un numr racional pozitiv a reprezentat n baza u prin _irul a(u)=an 1un 2 ... u0, u 1 ... u m are, prin definicie, valoarea n baza 10:  INCLUDEPICTURE "C:\\My Documents\\Mate\\tot_files\\sist_num_files\\sist_n4.gif" \* MERGEFORMATINET = cmum + ... + c1u1 + c0u0 + c1u1 + ... + an1un1 == an1un1 + ... + c1u1 + c0u0 + c1u1 + ... + cmumExemple: 11021434(5) = 157 + 256 + 055 + 254 + 153 + 452 + 351 + 450 = = 78125 + 31250 +1250 + 125 +100 + 15 + 4 = 110869 Putem s mai scriem _i sub forma: 11021434(5) = 5(5(5(5(5(5(51 +2) + 0) + 2) + 1) + 4) + 3) + 4 = = 5(5(5(5(5(57 + 0) + 2) + 1) + 4) + 3) + 4 = = 5(5(5(5(535 + 2) + 1) + 4) + 3) + 4 = = 5(5(5(5177 + 1) + 4) + 3) + 4 = = 5(5(5886 + 4) + 3) + 4 = = 5(54434 + 3) + 4 = = 522173 + 4 = = 110869 De aici se poate desprinde u_or un algoritm de conversie: cifr2021434numr1735177886443422173110869Se nmulce_te baza cu numrul din dreapta, se adun rezultatul cu numrul din dreapta-sus _i se scrie rezultatul pe rndul de jos:  INCLUDEPICTURE "C:\\My Documents\\Mate\\tot_files\\sist_num_files\\num_alg.gif" \* MERGEFORMATINET  40A(16) = 4162 0161 + A160 + = 1024 + 10 = 1034 10,011(2) = 121 + 020 + 02 1 + 12 2 + 12 3 = 2 + 0,25 + 0,125 = 2,375. Conversia din baza 10 ntr-o baz de numeracie oarecare B Pentru a trece un numr din baza 10 ntr-o alt baza de numeracie u INCLUDEPICTURE "C:\\My Documents\\Mate\\tot_files\\sist_num_files\\mult_n.gif" \* MERGEFORMATINET , u > 1 se aplic algoritmul: A.1. Se converte_te partea ntreag a numrului, conform procedurii A.2. _i partea fraccionar a numrului conform procedurii A.3. A.2. Se fac mprciri ntregi, succesive la baza u, pornind de la numrul ntreg care se converte_te; n urma fiecrei mprciri se obcine un ct _i un rest; noul ct este demprcitul urmtoarei mprciri ntregi; algoritmul se ncheie cnd se obcine ctul 0; resturile obcinute, ncepnd cu ultimul _i pn la primul, reprezint cifrele numrului, de la cea mai semnificativ la cea mai pucin semnificativ. A.3. Se fac nmulciri succesive, cu baza u, ncepnd cu partea fraccionar a numrului care se converte_te partea fraccionar a fiecrui produs constituie denmulcitul pentru produsul urmtor partea fraccionar a numrului convertit n baza u este reprezentat de succesiunea obcinut din prcile ntregi ale tuturor produselor obcinute, ncepnd cu primul produs, care furnizeaz cifra cea mai semnificativ a rezultatului algoritmul se ncheie cu un rezultat exact atunci cnd: 1. se obcine ca produs parcial un ntreg; 2. se obcine aproximarea dorit a numrului fraccionar dup un anumit numr de pa_i. Demonstracie A.2. Fie a numrul ntreg n baza 10 care se converte_te n baza u, conform algoritmului A.2. _i fie reprezentarea n baza u obcinut prin conversie, de forma cn 1cn 2 ... c0. Algoritmul este corect dac se termin ntr-un numr finit de pa_i _i dac:  INCLUDEPICTURE "C:\\My Documents\\Mate\\tot_files\\sist_num_files\\sist_n5.gif" \* MERGEFORMATINET  Notm cu a1 ctul obcinut dup prima mprcire ntreag _i cu c0 restul acestei mprciri; au loc relaciile: c0 = a0  a1u, a1 < a0 Analog, pentru orice i = 1, ..., n au loc relaciile: ci 1 = ai -1  aiu, ai < ai 1 Din _irul de inegalitci ai < ai 1 (i = 1, ..., n) rezult finitudinea algoritmului. Fie ultimul rest, cn 1 = an 1  anu, unde an = 0. Rezult: a0 = c0 + a1u = c0 + c1u + a2u2 = ... = c0u0 + c1u1 + c2u2 + ... + an 1un 1 = c0u0 + c1u1 + ... + an 1un 1 + anun A.3. Fie a numrul subunitar n baza 10 care se converte_te n baza u conform algoritmului A.3 _i fie reprezentarea n baza u, obcinut prin conversie, de forma 0,c 1c 2 ... c m. Algoritmul este corect dac:  INCLUDEPICTURE "C:\\My Documents\\Mate\\tot_files\\sist_num_files\\sist_n6.gif" \* MERGEFORMATINET  Notm cu a1 partea fraccionar a primului produs _i cu c 1 partea ntreag a acestuia. Au loc relaciile: a1u = c 1 + a1 a1 = c 1u 1 + a1u 1 Analog, pentru orice i =1, ..., m au loc relaciile: ai 1u = c i + ai ai 1 = c iu i + aiu i Din _irul de egalitci ai 1 = c iu i + aiu i (i =1, ..., m), rezult: a0 = c 1u 1 + c 2u 2 + a2u 2 = c 1u 1 + c 2u 2 + ... + am 1u (m 1) = ... = c 1u 1 + c 2u 2 + ... + cm 1u (m 1) + c mu m Cazul 1: Dac exist un m astfel nct am = 0, atunci algoritmul este finit _i rezultatul conversiei este exact. Cazul 2: Este posibil ca numrului racional a0, reprezentat n baza 10 s i corespund un numr racional 0,c 1c 2 ... c m reprezentat in baza u printr-o fraccie periodic; n acest caz, algoritmul se ncheie atunci cnd se determin perioada. Exemple: A.2. Conversia numrului 95.244 n baza 5 se obcine astfel: numr bazactrest95244 :519048419048:5380933809:57614761:51521152:530230:5606:5111:501Rezultatul obcinut este 95.244 = 11021434(5). A.3. Conversia numrului 0,7109375 n baza 2 se obcine astfel: numr bazapartea fraccionalpartea ntreag0,7109375  21,42187510,421875 20,8437500,84375 21,687510,6875 21,37510,375 20,7500,75 21,510,5 21,01Rezultatul conversiei este 0,1011011(2). Conversia din baza u n baza uk Pentru conversia unui numr din baza u n baza uk se aplic algoritmul urmtor: Se grupeaz cifrele numrului n baza u n grupe de cte k elemente, ncepnd de la marca zecimal la dreapta _i la stnga; valoarea fiecrei grupe, n baza 10, corespunde unei cifre n baza uk; numrul n baza uk este format din cifrele obcinute prin conversiile de la pasul precedent. Demonstracie: Fie a numrul ntreg, reprezentat n baza u cu m cifre de forma: a = cm 1cm-2 ... c0; Se completeaz lungimea lui a la n cifre, cu n = k p, _i cu aj = 0 pentru m  1< j < n. Rezult: a = (c0 u0 + ... + ck 1 uk 1) + uk(ck u0 + ... + c2k 1 uk 1) + ... + uk(p 1) (ck(p-1) u0 + ... + ckp 1 uk 1) Notm: b0 = c0 u0 + ... + ck 1 uk 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . bp 1 = ck(p 1) u0 + ... + ckp 1 uk 1 Pentru orice i = 0, ..., p  1, bi este valoarea zecimal corespunztoare grupei i, de cte k cifre din alfabetul bazei u, de forma: ck(i 1)...cki 2cki 1(u). Din  HYPERLINK "http://math4all.home.ro/aritmetica/sist_num.htm" \l "Propozitia 2" propozicia 2 rezult c bi < uk pentru orice i = 0, ... , p  1, deci bi < u pentru orice i = 0, ..., p  1. Rezult egalitatea: a = b0 + b1 uk + b2 u2k + ... + bp 1 uk(p 1), care confirm corectitudinea algoritmului. Exemple: 1) Se cere s se converteasc numrul 102001121(3) n baza 9. Cum 9 = 32, mprcim numrul n grupe de cte dou cifre: 102001121(3) 1'02'00'11'21(3) 12047(9) Convertim fiecare grup n baza 9: 21(3) = 231 + 130 = 7(9) 11(3) = 131 + 130 = 4(9) 00(3) = 031 + 030 = 0(9) 02(3) = 031 + 230 = 2(9) 01(3) = 031 + 130 = 1(9) Obcinem: 102001121(3) = 12047(9) 2) S se converteasc numrul 1010,01101(2) n baza 16. Deoarece 16 = 24, mprcim numrul n grupe de cte 4 cifre: 1010,01101(2) 1010,0110'1000(2) 1010(2) = 123 + 022 + 121 + 020 = 10 = A(16) 0110(2) = 023 + 122 + 121 + 020 = 6(16) 1000(2) = 123 + 022 + 021 + 020 = 8(16) Obcinem: 1010,01101(2) = A,68(16) Conversia din baza uk n baza u Pentru conversia unui numr din baza uk n baza u se aplic algoritmul urmtor: se reprezint fiecare cifr a numrului din baza uk n baza u, pe o lungime de k cifre. reprezentarea numrului n baza u se obcine prin concatenarea grupelor de cte k cifre obcinute la pasul precedent. Demonstracie: Fie a = cn 1 ... c0 un numr ntreg n baza uk. Rezult: a = c0 + c1uk + c2u2k + ... + cn 1uk(n-1); Cum, pentru orice i = 0, ..., n  1, ci < uk, rezult c reprezentarea numrului ci n baza u are cel mult k cifre, deci exist bi, k 1, bi, k 2, ..., bi, 0, mai mici uk, astfel nct ci = bi,0u0 + ... + bi,k 1uk 1. Obcinem succesiv: a = (b0,0u0 + ... + b0,k 1uk 1) + (b1,0u0 + ... + b1,k 1uk 1)uk + ... + (bn 1,0u0 + ... + bn 1,k 1uk 1)uk(n 1) a = (b0,0u0 + ... + b0,k 1uk 1) + (b1,0uk + ... + b1,k 1u2k 1) + ... + (bn 1,0ukn k + ... + bn 1,k 1ukn 1) Notnd cij cu dki+j, obcinem: a = d0 u0 + d1 u1 + ... + dkn 1 ukn 1, reprezentarea numrului a n baza u. Exemplu: S convertim numrul A,68(16) n baza 2. 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