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Mathematical English (a brief summary)
Jan Nekov´ar
Universit´e Paris 6c
?Jan Nekov´ar 2011 1Arithmetic
Integers
0zero10ten20twenty
1one11eleven30thirty
2two12twelve40forty
3three13thirteen50fifty
4four14fourteen60sixty
5five15fifteen70seventy
6six16sixteen80eighty
7seven17seventeen90ninety
8eight18eighteen100one hundred
9nine19nineteen1000one thousand
-245minus two hundred and forty-five22 731twenty-two thousand seven hundred and thirty-one
1 000 000one million
56 000 000fifty-six million
1 000 000 000one billion [US usage, now universal]
7 000 000 000seven billion [US usage, now universal]
1 000 000 000 000one trillion [US usage, now universal]
3 000 000 000 000three trillion [US usage, now universal]
Fractions [= Rational Numbers]
12 one half38 three eighths 13 one third269 twenty-six ninths 14 one quarter [= one fourth]-534 minus five thirty-fourths 15 one fifth237 two and three sevenths 117minus one seventeenth
Real Numbers
-0.067minus nought point zero six seven81.59eighty-one point five nine
-2.3·106minus two point three times ten to the six [=-2 300 000minus two million three hundred thousand]4·10-3four times ten to the minus three
[= 0.004 = 4/1000four thousandths]π[= 3.14159...]pi [pronounced as 'pie"]
e[= 2.71828...]e [base of the natural logarithm] 2Complex Numbers
ii3 + 4ithree plus four i
1-2ione minus two i1-2i= 1 + 2ithe complex conjugate of one minus two i equals one plus two i
The real part and the imaginary part of 3 + 4iare equal, respectively, to 3 and 4.Basic arithmetic operations
Addition:3 + 5 = 8three plus five equals [= is equal to] eight Subtraction:3-5 =-2three minus five equals [=...] minus two Multiplication:3·5 = 15three times five equals [=...] fifteen Division:3/5 = 0.6three divided by five equals [=...] zero point six (2-3)·6 + 1 =-5two minus three in brackets times six plus one equals minus five1-32+4
=-1/3one minus three over two plus four equals minus one third4! [= 1·2·3·4]four factorial
Exponentiation, Roots
52[= 5·5 = 25]five squared
53[= 5·5·5 = 125]five cubed
54[= 5·5·5·5 = 625]five to the (power of) four
5 -1[= 1/5 = 0.2]five to the minus one 5 -2[= 1/52= 0.04]five to the minus two⎷3 [= 1.73205...]the square root of three3⎷64 [= 4]the cube root of sixty four
5⎷32 [= 2]the fifth root of thirty two
In the complex domain the notation
n⎷ais ambiguous, since any non-zero complex number hasndifferentn-th roots. For example,4⎷-4 has four possible values:±1±i(with all possible combinations of signs). (1 + 2)2+2one plus two, all to the power of two plus two
eπi=-1e to the (power of) pi i equals minus one
Divisibility
The multiples of a positive integeraare the numbersa,2a,3a,4a,.... Ifbis a multiple ofa, we also say thatadividesb, or thatais a divisor ofb(notation:a|b). This is equivalent to ba being an integer. 3Division with remainder
Ifa,bare arbitrary positive integers, we can dividebbya, in general, only with a remainder. For example, 7 lies between the following two consecutive multiples of 3:2·3 = 6<7<3·3 = 9,7 = 2·3 + 1?
??73 = 2 +13 In general, ifqais the largest multiple ofawhich is less than or equal tob, then b=qa+r, r= 0,1,...,a-1. The integerq(resp.,r) is thequotient(resp., theremainder) of the division ofbbya.Euclid"s algorithm
This algorithm computes thegreatest common divisor(notation: (a,b) = gcd(a,b)) of two positive integersa,b. of the division ofbbya. This procedure, which preserves the gcd, is repeated until we arrive atr= 0.Example.Compute gcd(12,44).
44 = 3·12 + 8
12 = 1·8 + 4
8 = 2·4 + 0gcd(12,44) = gcd(8,12) = gcd(4,8) = gcd(0,4) = 4.
This calculation allows us to write the fraction
4412in its lowest terms, and also as a continued fraction: 4412
=44/412/4=113 = 3 +11 + 12 If gcd(a,b) = 1, we say thataandbarerelatively prime. addadditionner algorithmalgorithme Euclid"s algorithmalgorithme de division euclidienne bracketparenth`ese left bracketparenth`ese `a gauche right bracketparenth`ese `a droite curly bracketaccolade denominatordenominateur 4 differencediff´erence dividediviser divisibilitydivisibilit´e divisordiviseur exponentexposant factorialfactoriel fractionfraction continued fractionfraction continue gcd [= greatest common divisor]pgcd [= plus grand commun diviseur] lcm [= least common multiple]ppcm [= plus petit commun multiple] infinityl"infini iterateit´erer iterationit´eration multiplemultiple multiplymultiplier numbernombre even numbernombre pair odd numbernombre impair numeratornumerateur paircouple pairwisedeux `a deux powerpuissance productproduit quotientquotient ratiorapport; raison rationalrationnel(le) irrationalirrationnel(le) relatively primepremiers entre eux remainderreste rootracine sumsomme subtractsoustraire 5