Chandan Mandal 11.April-2016. Mathematical Symbols

[Pages:14]Chandan Mandal 11.April-2016.

Mathematical Symbols

List of all mathematical symbols and signs - meaning and examples.

Basic math symbols Geometry symbols Algebra symbols Probability & statistics symbols Set theory symbols Logic symbols Calculus & analysis symbols Number symbols Greek symbols Roman numerals

Basic math symbols

Symbol Symbol Name

=

equals sign

not equal sign

>

strict inequality

<

strict inequality

inequality

inequality

( ) parentheses

[ ] brackets

+

plus sign

-

minus sign

?

plus - minus

minus - plus

*

asterisk

?

times sign

multiplication dot

Meaning / definition

Example

equality

5 = 2+3

inequality

5 4

greater than

5 > 4

less than

4 < 5

greater than or equal to

5 4

less than or equal to

4 5

calculate expression inside first 2 ? (3+5) = 16

calculate expression inside first [(1+2)*(1+5)] = 18

addition

1 + 1 = 2

subtraction

2 - 1 = 1

both plus and minus operations 3 ? 5 = 8 and -2

both minus and plus operations 3 5 = -2 and 8

multiplication

2 * 3 = 6

multiplication

2 ? 3 = 6

multiplication

2 3 = 6

?

division sign / obelus division

/

division slash

division

6 ? 2 = 3 6 / 2 = 3

?

horizontal line

division / fraction

mod modulo

remainder calculation

7 mod 2 = 1

.

period

decimal point, decimal separator 2.56 = 2+56/100

ab

power

exponent

23 = 8

a^b caret

exponent

2 ^ 3= 8

a square root

a ? a = a

9 = ?3

3a cube root

3a ? 3a ? 3a = a

38 = 2

4a fourth root

4a ? 4a ? 4a ? 4a = a

416 = ?2

na n-th root (radical)

for n=3, n8 = 2

%

percent

1% = 1/100

10% ? 30 = 3

per-mille

1 = 1/1000 = 0.1%

10 ? 30 = 0.3

ppm per-million ppb per-billion ppt per-trillion

1ppm = 1/1000000

1ppb = 1/1000000000 1ppt = 10-12

10ppm ? 30 = 0.0003 10ppb ? 30 = 3?10-7 10ppt ? 30 = 3?10-10

Geometry symbols

Symbol Symbol Name

Meaning / definition

angle

formed by two rays

measured angle

spherical angle

right angle = 90?

?

degree

1 turn = 360?

?

arcminute 1? = 60?

??

arcsecond 1? = 60??

line

infinite line

AB

line segment line from point A to point B

ray

line that start from point A

arc

arc from point A to point B

|

perpendicular perpendicular lines (90? angle)

||

parallel

parallel lines

Example ABC = 30?

ABC = 30?

AOB = 30? = 90? = 60? = 60?59' = 60?59'59''

= 60? AC |BC AB || CD

congruent to equivalence of geometric shapes and size

ABC XYZ

~

similarity

same shapes, not same size

ABC ~ XYZ

triangle

triangle shape

ABC BCD

|x-y| distance

distance between points x and y

| x-y | = 5

= 3.141592654...

pi constant

c = ?d = 2??r

is the ratio between the circumference and diameter of a circle

rad

radians

radians angle unit

360? = 2 rad

grad grads

grads angle unit

360? = 400 grad

Algebra symbols

Symbol Symbol Name x x variable equivalence equal by definition := equal by definition ~ approximately equal approximately equal proportional to lemniscate much less than much greater than

( ) parentheses

[ ] brackets

{ } braces

x floor brackets

x ceiling brackets x! exclamation mark | x | single vertical bar f (x) function of x

(f g) function composition

Meaning / definition unknown value to find identical to equal by definition

equal by definition weak approximation approximation proportional to infinity symbol much less than

much greater than calculate expression inside first calculate expression inside first set rounds number to lower integer rounds number to upper integer factorial absolute value maps values of x to f(x)

(f g) (x) = f (g(x))

Example when 2x = 4, then x = 2

11 ~ 10 sin(0.01) 0.01 y x when y = kx, k constant

1 1000000 1000000 1 2 * (3+5) = 16 [(1+2)*(1+5)] = 18

4.34 4.35 4! = 1*2*3*4 = 24 | -5 | = 5 f (x) = 3x+5 f (x)=3x, g(x)=x-1 (f g)(x)=3(x1)

(a,b) open interval [a,b] closed interval delta discriminant

sigma

(a,b) = {x | a < x < b}

[a,b] = {x | a x b}

change / difference = b2 - 4ac summation - sum of all values in range of series

sigma

double summation

capital pi

product - product of all values in range of series

e

e constant / Euler's number

e = 2.718281828...

EulerMascheroni constant

= 0.527721566...

golden ratio

golden ratio constant

= 3.141592654...

pi constant

is the ratio between the circumference and diameter of a circle

x (2,6) x [2,6] t = t1 - t0 xi= x1+x2+...+xn

xi=x1x2...xn e = lim (1+1/x)x , x

c = ?d = 2??r

Linear Algebra Symbols

Symbol

? AB

[ ] ( ) | A | det(A) || x || A T A A * A -1 rank(A) dim(U)

Symbol Name

dot cross

Meaning / definition

scalar product vector product

a b a ? b

Example

tensor product tensor product of A and B A B

inner product

brackets

matrix of numbers

parentheses

matrix of numbers

determinant

determinant of matrix A

determinant

determinant of matrix A

double vertical bars norm

transpose

matrix transpose

(AT)ij = (A)ji

Hermitian matrix matrix conjugate transpose (A)ij = (A)ji

Hermitian matrix matrix conjugate transpose (A*)ij = (A)ji

inverse matrix

A A-1 = I

matrix rank

rank of matrix A

rank(A) = 3

dimension

dimension of matrix A rank(U) = 3

Probability and statistics symbols

Symbol P(A) P(A B) P(A B) P(A | B) f (x)

F(x) E(X) E(X | Y) var(X) 2 std(X) X

cov(X,Y) corr(X,Y) X,Y

Symbol Name

Meaning / definition

Example

probability function probability of event A

P(A) = 0.5

probability of events probability that of events A

intersection

and B

P(AB) = 0.5

probability of events probability that of events A or

union

B

P(AB) = 0.5

conditional

probability of event A given

probability function event B occured

P(A | B) = 0.3

probability density function (pdf)

P(a x b) = f (x) dx

cumulative distribution function F(x) = P(X x) (cdf)

population mean mean of population values = 10

expectation value

expected value of random variable X

E(X) = 10

conditional expectation

expected value of random variable X given Y

E(X | Y=2) = 5

variance

variance of random variable X

var(X) = 4

variance

variance of population values 2 = 4

standard deviation

standard deviation of random variable X

std(X) = 2

standard deviation

standard deviation value of random variable X

X = 2

median

middle value of random variable x

covariance

covariance of random variables X and Y

cov(X,Y) = 4

correlation

correlation of random variables X and Y

corr(X,Y) = 0.6

correlation

correlation of random variables X and Y

X,Y = 0.6

summation

summation - sum of all values in range of series

double summation double summation

Mo

mode

value that occurs most

frequently in population

MR

mid-range

MR = (xmax+xmin)/2

Md

sample median

half the population is below this value

Q1

lower / first quartile

25% of population are below this value

Q2

median / second quartile

50% of population are below this value = median of samples

Q3

upper / third quartile

75% of population are below this value

x

sample mean

average / arithmetic mean x = (2+5+9) / 3 = 5.333

s 2

sample variance

population samples variance estimator

s2 = 4

s

sample standard deviation

population samples standard deviation estimator

s = 2

zx

standard score

zx = (x-x) / sx

X ~

distribution of X

distribution of random variable X

X ~ N(0,3)

N(,2) normal distribution gaussian distribution

X ~ N(0,3)

U(a,b) uniform distribution equal probability in range a,b X ~ U(0,3)

exp()

exponential distribution

f (x) = e-x , x0

gamma(c, )

gamma distribution

f (x) = c xc-1e-x / (c), x0

2(k)

chi-square distribution

f (x) = xk/2-1e-x/2 / ( 2k/2 (k/2) )

F (k1, k2) F distribution Bin(n,p) binomial distribution f (k) = nCk pk(1-p)n-k Poisson() Poisson distribution f (k) = ke- / k!

Geom(p)

geometric distribution

f (k) = p (1-p) k

HG(N,K,n)

hyper-geometric distribution

Bern(p)

Bernoulli distribution

Combinatorics Symbols

Symbol Symbol Name Meaning / definition

Example

n!

factorial

n! = 1?2?3?...?n

5! = 1?2?3?4?5 = 120

nPk permutation nCk

combination

5P3 = 5! / (5-3)! = 60 5C3 = 5!/[3!(5-3)!]=10

Set theory symbols

Symbol

{ } A B A B A B A B A B A B A B A B 2A

A = B Ac A \ B

Symbol Name set intersection union

subset

proper subset / strict subset

not subset

superset proper superset / strict superset not superset power set power set equality

complement relative complement

Meaning / definition

Example

a collection of elements

A = {3,7,9,14}, B = {9,14,28}

objects that belong to set A and

set B

A B = {9,14}

objects that belong to set A or A B =

set B

{3,7,9,14,28}

subset has fewer elements or {9,14,28}

equal to the set

{9,14,28}

subset has fewer elements than {9,14}

the set

{9,14,28}

left set not a subset of right set {9,66} {9,14,28}

set A has more elements or equal to the set B

{9,14,28} {9,14,28}

set A has more elements than {9,14,28}

set B

{9,14}

set A is not a superset of set B {9,14,28} {9,66}

all subsets of A

all subsets of A

both sets have the same members

A={3,9,14}, B={3,9,14}, A=B

all the objects that do not belong to set A

objects that belong to A and not A = {3,9,14},

to B

B = {1,2,3},

A - B

A B

A B aA xA (a,b) A?B |A| #A

?

0 1

relative complement

symmetric difference

symmetric difference

element of

not element of ordered pair cartesian product cardinality cardinality aleph-null aleph-one empty set universal set natural numbers / whole numbers set (with zero) natural numbers / whole numbers set (without zero) integer numbers set rational numbers set real numbers set complex numbers set

A-B = {9,14}

objects to B

that

belong to

A

and

not

A = {3,9,14}, B = {1,2,3}, A-B = {9,14}

objects that belong to A or B but not to their intersection

A = {3,9,14}, B = {1,2,3}, A B = {1,2,9,14}

objects that belong to A or B but not to their intersection

A = {3,9,14}, B = {1,2,3},

A B = {1,2,9,14}

set membership

A={3,9,14}, 3 A

no set membership

A={3,9,14}, 1 A

collection of 2 elements

set of all ordered pairs from A and B

the

number

of

elements

of

set

A

A={3,9,14}, |A|=3

the

number

of

elements

of

set

A

A={3,9,14}, #A=3

infinite cardinality of natural numbers set

cardinality of countable ordinal numbers set

? = { }

C = {?}

set of all possible values

0 = {0,1,2,3,4,...}

0 0

1 = {1,2,3,4,5,...}

= {...-3,-2,-1,0,1,2,3,...}

= {x | x=a/b, a,b } = {x | - < x ................
................

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