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Mathematical English (a brief summary)

Jan Nekov´ar

Universit´e Paris 6c

?Jan Nekov´ar 2011 1

Arithmetic

Integers

0zero10ten20twenty

1one11eleven30thirty

2two12twelve40forty

3three13thirteen50fifty

4four14fourteen60sixty

5five15fifteen70seventy

6six16sixteen80eighty

7seven17seventeen90ninety

8eight18eighteen100one hundred

9nine19nineteen1000one thousand

-245minus two hundred and forty-five

22 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 117
minus one seventeenth

Real Numbers

-0.067minus nought point zero six seven

81.59eighty-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] 2

Complex Numbers

ii

3 + 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 five

1-32+4

=-1/3one minus three over two plus four equals minus one third

4! [= 1·2·3·4]four factorial

Exponentiation, Roots

5

2[= 5·5 = 25]five squared

5

3[= 5·5·5 = 125]five cubed

5

4[= 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 three

3⎷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

e

πi=-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. 3

Division 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

4412
in 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

Algebra

Algebraic Expressions

A=a2capital a equals small a squared

a=x+ya equals x plus y b=x-yb equals x minus y c=x·y·zc equals x times y times z c=xyzc equals x y z (x+y)z+xyx plus y in brackets times z plus x y x

2+y3+z5x squared plus y cubed plus z to the (power of) five

x n+yn=znx to the n plus y to the n equals z to the n (x-y)3mx minus y in brackets to the (power of) three m x minus y, all to the (power of) three m 2 x3ytwo to the x times three to the y ax

2+bx+ca x squared plus b x plus c⎷x+3⎷ythe square root of x plus the cube root of y

n⎷x+ythe n-th root of x plus y a+bc-da plus b over c minus d?n m?(the binomial coefficient) n over m

Indices

x

0x zero; x nought

x

1+yix one plus y i

R ij(capital) R (subscript) i j; (capital) R lower i j M kij(capital) M upper k lower i j; (capital) M superscript k subscript i j?n i=0aixisum of a i x to the i for i from nought [= zero] to n; sum over i (ranging) from zero to n of a i (times) x to the i?∞ m=1bmproduct of b m for m from one to infinity; product over m (ranging) from one to infinity of b m?n j=1aijbjksum of a i j times b j k for j from one to n; sum over j (ranging) from one to n of a i j times b j k?n i=0? n i?xiyn-isum of n over i x to the i y to the n minus i for i from nought [= zero] to n 6

Matrices

columncolonne column vectorvecteur colonne determinantd´eterminant index (pl. indices)indice matrixmatrice matrix entry (pl. entries)coefficient d"une matrice m×nmatrix [mbynmatrix]matrice `amlignes etncolonnes multi-indexmultiindice rowligne row vectorvecteur ligne squarecarr´e square matrixmatrice carr´ee

Inequalities

x > yx is greater than y x≥yx is greater (than) or equal to y x < yx is smaller than y x >0x is positive x≥0x is positive or zero; x is non-negative x <0x is negative ?The French terminology is different! x > yx est strictement plus grand que y x≥yx est sup´erieur ou ´egal `a y x < yx est strictement plus petit que y x >0x est strictement positif x≥0x est positif ou nul x <0x est strictement n´egatif

Polynomial equations

A polynomial equation of degreen≥1 with complex coefficients 7 f(x) =a0xn+a1xn-1+···+an= 0 (a0?= 0) hasncomplex solutions (= roots), provided that they are counted with multiplicities.

For example, a quadratic equation

ax

2+bx+c= 0 (a?= 0)

can be solved by completing the square,i.e., by rewriting the L.H.S. as a(x+ constant)2+ another constant.

This leads to an equivalent equation

a x+b2a? 2 =b2-4ac4a, whose solutions are x

1,2=-b±⎷Δ

2a, where Δ =b2-4ac(=a2(x1-x2)2) is the discriminant of the original equation. More precisely, ax

2+bx+c=a(x-x1)(x-x2).

If all coefficientsa,b,care real, then the sign of Δ plays a crucial rˆole: if Δ = 0, thenx1=x2(=-b/2a) is a double root; if Δ>0, thenx1?=x2are both real; if Δ<0, thenx1=x

2are complex conjugates of each other (and non-real).

coefficientcoefficient degreedegr´e discriminantdiscriminant equation´equation

L.H.S. [= left hand side]terme de gauche

R.H.S. [= right hand side]terme de droite

polynomialadj.polynomial(e) polynomialn.polynˆome provided that`a condition que rootracine simple rootracine simple double rootracine double triple rootracine triple multiple rootracine multiple root of multiplicity mracine de multiplicit´e m 8 solutionsolution solver´esoudre

Congruences

Two integersa,barecongruentmodulo a positive integermif they have the same remainder when divided bym(equivalently, if their differencea-bis a multiple ofm). a≡b(modm)a is congruent to b modulo m a≡b(m) ?Some people use the following, slightly horrible, notation:a=b[m]. Fermat"s Little Theorem.Ifpis a prime number andais an integer, then a p≡a(modp). In other words,ap-ais always divisible byp. Chinese Remainder Theorem.Ifm1,...,mkare pairwise relatively prime integers, then the system of congruences has a unique solution modulom1···mk, for any integersa1,...,ak. ?The definite article (and its absence) measure theoryth´eorie de la mesure number theoryth´eorie des nombres

Chapter onele chapitre un

Equation (7)l"´equation (7)

Harnack"s inequalityl"in´egalit´e de Harnack the Harnack inequality the Riemann hypothesisl"hypoth`ese de Riemann the Poincar´e conjecturela conjecture de Poincar´e

Minkowski"s theoremle th´eor`eme de Minkowski

the Minkowski theorem the Dirac delta functionla fonction delta de Dirac

Dirac"s delta function

the delta functionla fonction delta 9

GeometryA BCD

ELetEbe the intersection of the diagonals of the rectangleABCD. The lines (AB) and (CD) are parallel to each other (and similarly for (BC) and (DA)). We can see on this picture severalacute angles:?EAD,?EAB,?EBA,?AED,?BEC ...;right angles: ?ABC,?BCD,?CDA,?DABandobtuse angles:?AEB,?CED. Pe Q R rLetPandQbe two points lying on an ellipsee. Denote byRthe intersection point of the respective tangent lines toeatPandQ. The linerpassing throughPandQis called the polar of the pointRw.r.t. the ellipsee. 10 Here we see three concentric circles with respective radii equal to 1, 2 and 3. If we draw a line through each vertex of a given triangle and the midpoint of the opposite side, we obtain three lines which intersect at the barycentre (= the centre of gravity) of the triangle.Above, three circles have a common tangent at their (unique) intersection point. 11

Euler"s Formula

LetPbe a convex polyhedron. Euler"s formula asserts that

V-E+F= 2,

V= the number of vertices ofP,

E= the number of edges ofP,

F= the number of faces ofP.

Exercise.Use this formula to classify regular polyhedra (there are precisely five of them: tetrahedron, cube, octahedron, dodecahedron and icosahedron). For example, an icosahedron has 20 faces, 30 edges and 12 vertices. Each face is an isosceles triangle, each edge belongs to two faces and there are 5 faces meeting at each vertex. The midpoints of its faces form a dual regular polyhedron, in this case a dodecahedron, which has 12 faces (regular pentagons), 30 edges and 20 vertices (each of them belonging to 3 faces). angleangle acute angleangle aigu obtuse angleangle obtus right angleangle droit areaaire axis (pl. axes)axe coordinate axisaxe de coordonn´ees horizontal axisaxe horisontal vertical axisaxe vertical centre [US: center]centre circlecercle colinear (points)(points) align´es conic (section)(section) conique conecˆone convexconvexe cubecube curvecourbe dimensiondimension distancedistance dodecahedrondodeca`edre edgearˆete ellipseellipse ellipsoidellipso¨ıde faceface hexagonhexagone hyperbolahyperbole hyperboloidhyperbolo¨ıde 12 one-sheet (two-sheet) hyperboloidhyperbolo¨ıde `a une nappe (`a deux nappes) icosahedronicosa`edre intersectintersecter intersectionintersection latticer´eseau lettucelaitue lengthlongeur linedroite midpoint ofmilieu de octahedronocta`edre orthogonal; perpendicularorthogonal(e); perpendiculaire parabolaparabole parallelparall`el(e) parallelogramparall´elogramme pass throughpasser par pentagonpentagone planeplan pointpoint (regular) polygonpolygone (r´egulier) (regular) polyhedron (pl. polyhedra)poly`edre (r´egulier) projectionprojection central projectionprojection conique; projection centrale orthogonal projectionprojection orthogonale parallel projectionprojection parall`ele quadrilateralquadrilat`ere radius (pl. radii)rayon rectanglerectangle rectangularrectangulaire rotationrotation sidecˆot´e slopepente spheresph`ere squarecarr´e square latticer´eseau carr´e surfacesurface tangent totangent(e) `a tangent linedroite tangente tangent hyper(plane)(hyper)plan tangent tetrahedrontetra`edre triangletriangle equilateral triangletriangle ´equilat´eral isosceles triangletriangle isoc`ele right-angled triangletriangle rectangle vertexsommet 13

Linear Algebra

basis (pl. bases)base change of basischangement de base bilinear formforme bilin´eaire coordinatecoordonn´ee (non-)degenerate(non) d´eg´en´er´e(e) dimensiondimension codimensioncodimension finite dimensiondimension finie infinite dimensiondimension infinie dual spaceespace dual eigenvaluevaleur propre eigenvectorvecteur propre (hyper)plane(hyper)plan imageimage isometryisom´etrie kernelnoyau linearlin´eaire linear formforme lin´eaire linear mapapplication lin´eaire linearly dependentli´es; lin´eairement d´ependants linearly independentlibres; lin´eairement ind´ependants multi-linear formforme multilin´eaire originorigine orthogonal; perpendicularorthogonal(e); perpendiculaire orthogonal complementsuppl´ementaire orthogonal orthogonal matrixmatrice orthogonale (orthogonal) projectionprojection (orthogonale) quadratic formforme quadratique reflectionr´eflexion representrepr´esenter rotationrotation scalarscalaire scalar productproduit scalaire subspacesous-espace (direct) sumsomme (directe) skew-symmetricanti-sym´etrique symmetricsym´etrique trilinear formforme trilin´eaire vectorvecteur vector spaceespace vectoriel vector subspacesous-espace vectoriel vector space of dimensionnespace vectoriel de dimensionn 14

Mathematical arguments

Set theory

x?Ax is an element of A; x lies in A; x belongs to A; x is in A x??Ax is not an element of A; x does not lie in A; x does not belong to A; x is not in A x,y?A(both) x and y are elements of A;...lie in A; ...belong to A;...are in A x,y??A(neither) x nor y is an element of A;...lies in A; ...belongs to A;...is in A ∅the empty set (= set with no elements)

A=∅A is an empty set

A?=∅A is non-empty

A?Bthe union of (the sets) A and B; A union B

A∩Bthe intersection of (the sets) A and B; A intersection B A×Bthe product of (the sets) A and B; A times B A∩B=∅Ais disjoint fromB; the intersection of A and B is empty {x|...}the set of all x such that...

Cthe set of all complex numbers

Qthe set of all rational numbers

Rthe set of all real numbers

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