1、Mathematics Symbols and Their English Explanations1/2 a half or one half1/3 a third or one third2/3 two third1/4 a quarter or one quarter; a fourth or one fourth1/10 a tenth or one tenth1/100 a one hundredth1/1000 a one thousandth1/1234 one over a thousand two hundred and thirty-four3/4 three fourth
2、s or three quarters4/5 three fifths or four over five113/300 one hundred and thirteen over three hundredtwo and a half 21two and seven over eight or two and seven eighth87three and one eighth3four and a third14a one hundred twenty-five and three fourth quarters250.1(or .1) point one or zero point on
3、e or nought point one0.01(or .01) point 0 one or zero point zero one or naught point naught one0.25(or .25) naught point two five or point two five0.045 decimal point naught four five2.35 two point three five4. four point nine recurring93.03 three point naught three two six , two six recurring6245.6
4、7 four five forty-five point six seven38.72 three eight point seven two or thirty-eight decimal seven two0.001(or .001) 0 point 0 0 one naught point naught naught one or zero pointzero zero one or point naught naught one+ plus; positive_ minus; negativeplus or minus(or.) multiplied by ; times(or/) d
5、ivided by= is equal to ; equalsis identically equal to(or ; ;=) is approximately equal to ; approximately equalsidentical withnot equal toor approachesgreater than much grater thanequal to or less thanequal to or greater than( ) round brackets; parentheses square bracketsangular brackets bracesInter
6、sectionUnionis a member of set is a subset of similar to ; difference* denotes an operationis equivalent toImplies or empty setMaps intoThereforeBecause: ratio sign, divided by, is to: equals, as (proportion)varies asinfinitysquare root ofcube root of3 parallel to degrees minutes” secondsanglelength
7、 of line from A to BABvector product of A and B; magnitude of A times magnitude of B times sine of the angle from A to B; AB sin Bscalar product of A to B ; magnitude of A times magnitude AB:of B times cosine of the angle form A to B ;AB cos I or j imaginary or square root of -1or the imaginary cube
8、 roots of 12pi ; the ratio of the circumference of a circle to its diameter, approx.3.14159e or the base of natural logarithms, approx.2.71828 the eccentricity of a conic sectionx! factorial of xlog x to the base nlognxor log x to the base 10 (common logarithm) 10lor log x to the base (natural logar
9、ithm or Naperian logarithm )lexM modulus of common logarithms e = 0.4343, ( x = 10log10logx 0.4343)elog10 = 2.3026, ( x = x 2.3026)1 elog10lx x.xto n factors, the nth power of x, x to the power nnxor the nth root of x,x to the power one over n thn1X a x approaches the limit a(the angle between the r
10、adius vector and the polar axis) the angle thetasigmathe product of the terms indicatedthe absolute value of x xthe mean value of x ; x barb primebb double prime ; b second primeb triple primeb sub one1bb sub two2b sub mmx dot ,first derivative of x with respect to timexx two dot , second derivative
11、 of x with respect to timeexp x = (e = naperian log base )(abbreviation for )e xeb double prime sub mmbf or F functionf(x);F(x); (x) function f(or ) of xy = f(x) y is a function of x( finite difference or increment) deltaor x (the increment of x) delta x(an increment of x considered as tending to ze
12、ro )ddee of x ; dee x ; differential of xor the differential coefficient of y with respect to xyxdxyDthe first derivative of y with respect to x/ the second derivative of y with respect x;2yx/ the nth derivative of y with respect xnd/ the partial derivative of y with respect u ,uyuwhere y is a funct
13、ion of u and another variable (or variables)(x) the first derivative of function F of x with respect to xFi / +j / +k / del ; nabla; vector differential xyzoperatorn nth del (nabla)Laplacian operator 4! Factorial =1 432line integral around a closed path integralintegral between limits a and babvecto
14、r FFx+y x plus y (a+b) bracket a plus b bracket closeda = b a equals b ; a is equal to b ; a is b a b a is not equal to b ; a is not ba b a plus or minus ba b a is approximately equal to bab a is greater than bab a is much far greater than ba b a is greater than or equal to ba is not greater than ba
15、7+3 12 is greater than 7 plus 372-16=56 72 minus 16 is equals; is equal to 56; 16 from 72 leaves 56x square ; x squared; the square of x; the second power of x; 2xx to the second power =25 the second power of 5 is 25; 5 square is 25 ; 5 to the second 25power is equal to 25y cube ; y cubed ;the cube
16、of y ; y to the third power ; y to the 3ythirdy to the minus tenth(power)10= the square root of 4 is equals plus or minus 242the cube root of a3athe fifth root of x square 52xthe square root of five hundred and eighteen18the cubic root of nine hundred and thirty3903x=5 three times x equals 5x raised
17、 to the third power divided by five equals y squared325xyx squared with y squared equals 1010a equals v sub t minus v over divided by ttVaa plus b minus c multiplied by d , all divided by e equals fbcdefeight plus six and five-eighths minus three decimal 51863.42point six eight multiplied by four, a
18、ll divided by two and a half余 13 23 into 4567 goes 198 times, and 13 remainder45672319845+70+152=267 45, 70 and 152 added together are 267% percent2% two per centper mille05 five per millethree eighths (of one) percent3%80.3% point three percenthalf ton1 ton2tow third of a ton3three quarters of a ki
19、lometer km41.75km one point seven five kilometersixty miles per hour60ihrtwenty degrees2C 6 minutes 6 feet 10 seconds 10 inches010 zero degree Centigrade CelsiusC100 one a hundred degrees centigrade32 thirty-two degrees FahrenheitF0Set theory, relations, functionsX , Y SetX x is a member of the set
20、Xx is not a member of XxSet A is contained in set X,AA is not contained in XXUnion of sets X and Y,YIntersection of sets X and Y:+,+, Symmetric difference of setsUnion of all the sets X,iiXxIntersection of all the sets Xii Null set , empty set,0Complement of the set XCXX-Y, XY Difference of sets X a
21、nd YThe set of all x with the property p(),(),:()xpPx(x , y , z), ( x, y, z) Ordered set of elements x , y, and z; to be distinguished from (x, y, z), for example( x, y, z) Unordered set, the set whose elements are x, y, z, and no others , i=1,2.,n The set whose members are , where i is any 12,.ai 1
22、nia iawhole number from 1 to n , i=1,2., The set whose members are , where i is any 12,.i 1i ipositive whole numberCartesian product, set of all (x , y) such that XY ,xXyYThe set whose elements are a, where iaIiIXry ,R x , y RelationEquivalence relations, for example, congruence ,:, Transitive relat
23、ions, for example, numerical orderFunction, mapping, transformation:,f XfXYyYf Inverse mapping11,f g。F Composite functions: (g。F) ( x )=g( f( x ) )f( X ) Image of X by fInverse-image set, counter image1(f1-1, one-one Read “one-to-one correspondence”Diagram : the diagram is commutative in case yxf f=
24、g Partial mapping, restriction of function f to set AzwgAf X , card X, Cardinal of the set A, d Denumerable infinityC , c ,2 Power of continuumOrder type of the set of positive integersRead “countably ”Number, numerical functions1( 1 ) 20( 10 )100 Read “ from 1 to 20 in intervals of 1, and from 20 t
25、o 100 in intervals of 10 ”Const constanta0 The number A is nonnegative, or, the matrix A is positive definite, or, the matrix A has nonnegative entriesxy Read “ x divides y ”xy mod p Read “ x congruent to y modulo p ”Continued fractions00121.,.aaarc sine of xxsinSinh sinus hyperbolic, the hyperbolic
26、 sine a ,b Closed interval, a ,b Half-open Interval ( open at the right )b( a ,b),a ,b Open intervala, ),a, Interval closed at the left, infinite to the rightSet of all real numbers,),(max f(x),max f(x) Maximum of f(x) when x is in the set XxXxmin Minimumsup , Supremum, least upper boundlubinf , g.1
27、.b Infimum, greatest lower boundlim sup , limit superiorlimlim inf , imit inferiorl . i . m . limit in the meanComplex variables,ireyxz ieIvuwi,Z , Complex conjugateRe, Real part Im, Imaginary partArg ArgumentJacobian, functional determinant(,)(,)uvDxyIntegral (for example , Lebesgue integral ) of f
28、unction f over ()Efdset E with respect to measure f(n)log n as n f(n)/log n approaches 1 as nf(n)=0(log n ) as n f(n)/log n is bounded as nf(n)=0(log n ) f(n)/log n approaches zerof(x)b, f(x) b f(x) increases, approaching the limit bf(x)b, f(x) b f(x) decreases, approaching the limit ba . e . ,p . p
29、. almost everywhereess sup essential superemumspace of continuous function)(,0XCthe class of functions having continuous kth derivative (on a, bakb )same as C CLipchitz class of function,aLip,Lp , Lpa , b space of functions having integrable absolute pth power ( on a, b )same as L L( C ,a ), ( C , p
30、 ) cesaro summablitySpecial functionx The integral part of xBinomial coefficient n!/k!( n-k )!(),nknKCLegendre symbolpExponential function,exsinhx, coshx, tanhx Hyperbolic functionssnx , cnx , dnx Jacobi elliptic functions( x ) Weierstrass elliptic function Gamma functionBessel function)(JvCharacter
31、istic function of the set X, X(x)=1 in case x X, xX otherwise X(x)=0sgn x Signum: sgn 0=0. while sgn x=x for x 0Dirac delta function)(Algebra, tensors, operators+, . , law of composition in algebraic systems,T。 ，e , 0 Identity, unit , neutral element ( of an additive system )e ,1, I, Identity, unit
32、, neutral element ( of a general algebraic system ) e ,e ,E ,P idempotentInverse of a1aHom ( M,N ) Group of all homomorphisms of M into NG/H Factor group, group of cosetsK : k Dimension of K over k,+ Direct sumTensor product, Kronecker product Exterior product, Grassmann productVector,XxInner produc
33、t, scalar product, dot product,()Yyy Outer product, vector product, cross productx, Norm of the vector x,xAx, xA The image of x under the transformation AKronecker delta: ij 1,0whileforijTranspose of the matrix A,tAAdjoint, Hermitian conjugate of ATr A ,Sp A Trace of the matrix ADet A, Determinant o
34、f the matrix AFinite differences)(,)(xfnxfnx0,x1, x0,x1,x2, x0,xifDivided differences0,Read “ gradient of f ”,gradfRead “ divergence of v ”.iv, rot v Read “ curl of v ”curlVdiv grad Laplacian2,X, Y Poisson bracket, or commutator, or Lie productGL( n, R ) Full linear group of degree n over field R0 (
35、 n , R ) Full orthogonal groupSO ( n, R ),O ( n, R ) Special orthogonal groupTopologyEuclidean n spacenEn spherenSMetric, distance ( between points p and q )(,),pqdClosure of the set X,cl,XFrX , frX, ,bdry X Frontier, boundary of XInt X, X Interior of Xspace Hausdorff space2TUnion of countably many
36、closed setsaFIntersection of countably many open setsGDim X dimensionality, dimension of XHimotopy groups(),)nAHomology groups,;()nHHXCohomology groups(),XGProbability and statisticsX ,Y Random variablesP(X2) ,Pr(X2) Probability that X2P(X2) (Y1) Conditional probabilityE(X), E(X) Expectation of XE(XY)1 Conditional expectation of ( X/Y )Cumulative distribution function.cdfProbability density functionpCharacteristic function.cfMean (especially, sample mean)xStandard deviation,.sdVariance2,varVMoments of a distribution123,ijCoefficient of correlationPartial correlation coefficient4.
