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MATHEMATICAL SYMBOLS


=

Equal sign: equality


Not equal sign: inequality


Approximately equal: approximation


>

Strict inequality: greater than


<

Strict inequality: less than


Inequality: greater than or equal to


Inequality: less than or equal to


( )

Parentheses: calculate expression inside first


[ ]

Brackets: calculate expression inside first


+

Plus sign: addition


Minus sign: subtraction


±

Plus / minus: both plus and minus operations


*

Asterisk: multiplication


×

Times sign: multiplication


Multiplication dot: multiplication


÷

Division sign / obelus: division


/

Division slash: division


Horizontal line: division / fraction


mod

Modulo: remainder calculation


.

Period: decimal point / decimal separator


ab

Power: exponent


a^b

Caret: exponent


a

Square root: √a ⋅ √a = a


3a

Cube root: 3a3a3a = a


4a

Fourth root: 4a4a4a4a = a


na

N-th root (radical)


%

Percent: 1% = 1/100


Per-mille: 1‰ = 1/1000 = 0.1%


ppm

Per-million: 1ppm = 1/1000000


ppb

Per-billion: 1ppb = 1/1000000000


ppt

Per-trillion: 1ppt = 10^-12




GEOMETRY SYMBOLS


Angle: formed by two rays


Right angle: 90°


°

Degree: 1 turn = 360°


deg

Degree: 1 turn = 360deg


Prime: arcminute, 1° = 60′


Double prime: arcsecond, 1′ = 60″


AB

Line segment: line from point A to point B


Perpendicular: perpendicular lines (90° angle)


Parallel: parallel lines


Congruent to: equivalence of geometric shapes and size


~

Similarity: same shapes, not same size


Δ

Triangle: triangle shape


|x-y|

Distance: distance between points x and y


π

Pi constant: π = 3.141592654... is the ratio between the circumference and diameter of a circle


rad

Radians: radians angle unit


c

Radians: radians angle unit


grad

Gradians / gons: grads angle unit


g

Gradians / gons: grads angle unit




ALGEBRA SYMBOLS


x

Variable x: unknown value to find


Equivalence: identical to


Equal by definition


:=

Equal by definition


~

Approximately equal: weak approximation


Proportional to


Lemniscate: infinity symbol


Much less than


Much greater than


{ }

Braces: set


⌊x⌋

Floor brackets: rounds number to lower integer


⌈x⌉

Ceiling brackets: rounds number to upper integer


x!

Exclamation mark: factorial


|x|

Vertical bars: absolute value


f(x)

Function of x: maps values of x to f(x)


(f∘g)

Function composition: (f∘g) (x) = f(g(x))


(a,b)

Open interval: (a,b) = {x | a < x < b}


[a,b]

Closed interval: [a,b] = {x | a ≤ x ≤ b}


Delta: change / difference


Discriminant: Δ = b2 - 4ac


Σ

Sigma: summation / sum of all values in range of series


ΣΣ

Double sigma: double summation


Capital pi: product / product of all values in range of series


e

Constant e / Euler's number: e = 2.718281828...


γ

Euler-Mascheroni constant: γ = 0.5772156649...


φ

Golden ratio constant




LINEAR ALGEBRA SYMBOLS


·

Dot: scalar product


×

Cross: vector product


A⊗B

Tensor product: tensor product of A and B


[ ]

Brackets: matrix of numbers


( )

Parentheses: matrix of numbers


| A |

Determinant: determinant of matrix A


det(A)

Determinant: determinant of matrix A


|| x ||

Double vertical bars: norm


AT

Transpose: matrix transpose


A†

Hermitian matrix: matrix conjugate transpose


A*

Hermitian matrix: matrix conjugate transpose


A-1

Inverse matrix: A A-1 = I


rank(A)

Matrix rank: rank of matrix A


dim(U)

Dimension: dimension of matrix A




STATISTICS SYMBOLS


P(A)

Probability function: probability of event A


P(A ⋂ B)

Probability of events intersection: probability of events A and B


P(A ⋃ B)

Probability of events union: probability of events A or B


P(A | B)

Conditional probability function: probability of event A given event B occured


f(x)

Probability density function (pdf): P(a ≤ x ≤ b) = ∫ f(x) dx


F(x)

Cumulative distribution function (cdf): F(x) = P(X ≤ x)


μ

Population mean: mean of population values


E(X)

Expectation value: expected value of random variable X


E(X | Y)

Conditional expectation: expected value of random variable X given Y


var(X)

Variance: variance of random variable X


σ2

Variance: variance of population values


std(X)

Standard deviation: standard deviation of random variable X


σX

Standard deviation: standard deviation of random variable X


cov(X,Y)

Covariance: covariance of random variables X and Y


corr(X,Y)

Correlation: correlation of random variables X and Y


ρX,Y

Correlation: correlation of random variables X and Y


Summation: sum of all values in range of series


∑∑

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


s2

Sample variance: population samples variance estimator


s

Sample standard deviation: population samples standard deviation estimator


zx

Standard score: zx = (x-x) / sx


X~

Distribution of X: distribution of random variable X


N(μ,σ2)

Normal distribution: gaussian distribution


U(a,b)

Uniform distribution: equal probability in range a,b


exp(λ)

Exponential distribution: f(x) = λe-λx , x ≥ 0


gamma(c,λ)

Gamma distribution: f(x) = λcxc-1e-λx / Γ(c), x ≥ 0


χ2(k)

Chi-square distribution: f(x) = xk/2-1e-x/2 / (2k/2 Γ(k/2))


F(k1, k2)

F distribution


Bin(n,p)

Binomial distribution: f(k) = nCk pk(1-p)n-k


Poisson(λ)

Poisson distribution: f(k) = λke / k!


Geom(p)

Geometric distribution: f(k) = p(1-p)k


HG(N,K,n)

Hyper-geometric distribution


Bern(p)

Bernoulli distribution




COMBINATORICS SYMBOLS


n!

Factorial - n! = 1⋅2⋅3⋅...⋅n


nPk

Permutation: nPk = n! / (n-k)!


nCk

Combination: nCk = n! / k!(n-k)!




SET THEORY SYMBOLS


{ }

Set: a collection of elements


A ∩ B

Intersection: objects that belong to set A and set B


A ∪ B

Union: objects that belong to set A or set B


A ⊆ B

Subset: A is a subset of B; set A is included in set B


A ⊂ B

Proper subset / strict subset: set A is a subset of set B, and set A is not equal to set B


A ⊄ B

Not subset: set A is not a subset of set B


A ⊇ B

Superset: A is a superset of B; set A includes set B


A ⊃ B

Proper superset / strict superset: A is a superset of B, and A is not equal to B


A ⊅ B

Not superset: set A is not a superset of set B


A ⊅ B

Not superset: set A is not a superset of set B


2A

Power set: all subsets of A


P(A)

Power set: all subsets of A


A = B

Equality: both sets have the same members


Ac

Complement: all the objects that do not belong to set A


A \ B

Relative complement: objects that belong to A and not to B


A - B

Relative complement: objects that belong to A and not to B


A ∆ B

Symmetric difference: objects that belong to A or B but not to their intersection


A ⊖ B

Symmetric difference: objects that belong to A or B but not to their intersection


a ∈ A

Element of, belongs to: set membership


x ∉ A

Not element of: no set membership


(a,b)

Ordered pair: collection of 2 elements


A×B

Cartesian product: set of all ordered pairs from A and B


|A|

Cardinality: the number of elements of set A


#A

Cardinality: the number of elements of set A


|

Vertical bar: such that


N0

Aleph-null: infinite cardinality of natural numbers set


N1

Aleph-one: cardinality of countable ordinal numbers set


Ø

Empty set: Ø = { }


U

Universal set: set of all possible values


0

Natural numbers / whole numbers set (with zero): ℕ0 = {0, 1, 2, 3, 4,...}


1

Natural numbers / whole numbers set (without zero): ℕ1 = {1, 2, 3, 4, 5,...}


*

Natural numbers / whole numbers set (without zero): ℕ* = {1, 2, 3, 4, 5,...}


Integer numbers set: ℤ = {...-3, -2, -1, 0, 1, 2, 3,...}


Rational numbers set: ℚ = {x | x = a/b, a,b ∈ ℤ}


Real numbers set: ℝ = {x | -∞ < x < +∞}


Complex numbers set: ℂ = {z | z = a+bi, -∞ < a < +∞, -∞ < b < +∞}




LOGIC SYMBOLS


And


^

Caret / circumflex: and


&

Ampersand: and


+

Plus: or


Reversed caret: or


|

Vertical line: or


x'

Single quote: not / negation


x

Bar: not / negation


¬

Not / negation


!

Exclamation mark: not / negation


Circled plus / oplus: exclusive or / xor


~

Tilde: negation


Implies


Equivalent: if and only if (iff)


Equivalent: if and only if (iff)


For all


There exists


There does not exist


Therefore


Because / since


Because / since




CALCULUS & ANALYSIS SYMBOLS


limx→af(x)

Limit value of a function


ε

Epsilon: represents a very small number, near zero


y'

Derivative: Lagrange's notation


y''

Second derivative: derivative of derivative


y''

Second derivative: derivative of derivative


y(n)

Nth derivative: n times derivation


dy

——

dx

Derivative: Leibniz's notation


d2y

——

dx2

Derivative: Leibniz's notation


dny

——

dxn

Nth derivative: n times derivation


Dxy

Derivative: Euler's notation


Dx2y

Second derivative: derivative of derivative


∂f(x,y)

———

∂x

Partial derivative


Integral: opposite to derivation


∫∫

Double integral: integration of function of 2 variables


∫∫∫

Triple integral: integration of function of 3 variables


Closed contour / line integral


Closed surface integral


Closed volume integral


i

Imaginary unit: i ≡ √-1


z*

Complex conjugate: z = a+bi → z* = a-bi


z

Complex conjugate: z = a+bi → z = a-bi


Re(z)

Real part of a complex number: z = a+bi → Re(z) = a


Im(z)

Imaginary part of a complex number: z = a+bi → Im(z) = b


| z |

Absolute value / magnitude of a complex number: |z| = |a+bi| = √(a2+b2)


arg(z)

Argument of a complex number: the angle of the radius in the complex plane


Nabla / del: gradient / divergence operator


x * y

Convolution: y(t) = x(t) * h(t)


δ

Delta function




GREEK ALPHABET LETTERS


Α - α - Alpha


Β - β - Beta


Γ - γ - Gamma


Δ - δ - Delta


Ε - ε - Epsilon


Ζ - ζ - Zeta


Η - η - Eta


Θ - θ - Theta


Ι - ι - Iota


Κ - κ - Kappa


Λ - λ - Lambda


Μ - μ - Mu


Ν - ν - Nu


Ξ - ξ - Xi


Ο - ο - Omicron


Π - π - Pi


Ρ - ρ - Rho


Σ - σ - Sigma


Τ - τ - Tau


Υ - υ - Upsilon


Φ - φ - Phi


Χ - χ - Chi


Ψ - ψ - Psi


Ω - ω - Omega




OTHER SYMBOLS


Rightwards arrow: limx→a


Rightwards arrow from bar: associate


|

Vertical bar: such that


:

Colon: such that