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ریاضی مهربان (آموزش ریاضی)
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List of all mathematical symbols and signs - meaning and examples.

Basic math symbols

Symbol Symbol Name Meaning / definition Example
= equals sign equality 5 = 2+3
not equal sign inequality 5 ≠ 4
> strict inequality greater than 5 > 4
< strict inequality less than 4 < 5
inequality greater than or equal to 5 ≥ 4
inequality less than or equal to 4 ≤ 5
( ) parentheses calculate expression inside first 2 Ч (3+5) = 16
[ ] brackets calculate expression inside first [(1+2)*(1+5)] = 18
+ plus sign addition 1 + 1 = 2
minus sign subtraction 2 − 1 = 1
± plus - minus both plus and minus operations 3 ± 5 = 8 and -2
minus - plus both minus and plus operations 35 = -2 and 8
* asterisk multiplication 2 * 3 = 6
Ч times sign multiplication 2 Ч 3 = 6
∙  multiplication dot multiplication 2 ∙ 3 = 6
ч division sign / obelus division 6 ч 2 = 3
/ division slash division 6 / 2 = 3
horizontal line division / fraction \frac{6}{2}=3
mod modulo remainder calculation 7 mod 2 = 1
. period decimal point, decimal separator 2.56 = 2+56/100
a b power exponent 23 = 8
a^b caret exponent 2 ^ 3 = 8
a square root

a · a  = a

9 = ±3
3a cube root   38 = 2
4a forth root   416 = ±2
na n-th root (radical)   for n=3, n8 = 2
% percent 1% = 1/100 10% Ч 30 = 3
per-mille 1‰ = 1/1000 = 0.1% 10‰ Ч 30 = 0.3
ppm per-million 1ppm = 1/1000000 10ppm Ч 30 = 0.0003
ppb per-billion 1ppb = 1/1000000000 10ppb Ч 30 = 3Ч10-7
ppt per-trillion 1ppb = 10-12 10ppb Ч 30 = 3Ч10-10

Geometry symbols

Symbol Symbol Name Meaning / definition Example
angle formed by two rays

ABC = 30є

measured angle   ABC = 30є
spherical angle   AOB = 30є
right angle = 90є α = 90є
є degree 1 turn = 360є α = 60є
ґ arcminute 1є = 60ґ α = 60є59'
ґґ arcsecond 1ґ = 60ґґ α = 60є59'59''
AB line line from point A to point B  
ray line that start from point A  
| perpendicular perpendicular lines (90є angle) AC | BC
|| parallel parallel lines AB || CD
congruent to equivalence of geometric shapes and size ∆ABC ∆XYZ
~ similarity same shapes, not same size ∆ABC ~ ∆XYZ
Δ triangle triangle shape ΔABC ΔBCD
| x-y | distance distance between points x and y | x-y | = 5
π pi constant π = 3.141592654...

is the ratio between the circumference and diameter of a circle

c = π·d = 2·π·r
rad radians radians angle unit 360є = 2π rad
grad grads grads angle unit 360є = 400 grad

Algebra symbols

Symbol Symbol Name Meaning / definition Example
x x variable unknown value to find when 2x = 4, then x = 2
equivalence identical to  
equal by definition equal by definition  
:= equal by definition equal by definition  
~ approximately equal weak approximation 11 ~ 10
approximately equal approximation sin(0.01) ≈ 0.01
proportional to proportional to

f(x) g(x)

lemniscate infinity symbol  
much less than much less than 1 1000000
much greater than much greater than 1000000 1
( ) parentheses calculate expression inside first 2 * (3+5) = 16
[ ] brackets calculate expression inside first [(1+2)*(1+5)] = 18
{ } braces set  
x floor brackets rounds number to lower integer 4.3= 4
x ceiling brackets rounds number to upper integer 4.3= 5
x! exclamation mark factorial 4! = 1*2*3*4 = 24
| x | single vertical bar absolute value | -5 | = 5
f (x) function of x maps values of x to f(x) f (x) = 3x+5
(f g) function composition

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

f (x)=3x, g(x)=x-1 (f g)(x)=3(x-1) 
(a,b) open interval (a,b) {x | a < x < b} x (2,6)
[a,b] closed interval [a,b] {x | axb} x [2,6]
delta change / difference t = t1 - t0
discriminant Δ = b2 - 4ac  
sigma summation - sum of all values in range of series xi= x1+x2+...+xn
∑∑ sigma double summation
capital pi product - product of all values in range of series xi=x1∙x2∙...∙xn
e e constant / Euler's number e = 2.718281828... e = lim (1+1/x)x , x→∞
γ Euler-Mascheroni  constant γ = 0.527721566...  
φ golden ratio golden ratio constant  

Linear Algebra Symbols

Symbol Symbol Name Meaning / definition Example
dot scalar product a b
Ч cross vector product a Ч b
AB tensor product tensor product of A and B A B
\langle x,y \rangle inner product    
[ ] 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  
A T transpose matrix transpose

(AT)ij = (A)ji

A Hermitian matrix matrix conjugate transpose

(A)ij = (A)ji

A * Hermitian matrix matrix conjugate transpose

(A*)ij = (A)ji

A -1 inverse matrix A A-1 = I  
rank(A) matrix rank rank of matrix A

rank(A) = 3

dim(U) dimension dimension of matrix A

rank(U) = 3

Probability and statistics symbols

Symbol Symbol Name Meaning / definition Example
P(A) probability function probability of event A P(A) = 0.5
P(AB) probability of events intersection probability that of events A and B P(AB) = 0.5
P(A B) probability of events union probability that of events A or B P(AB) = 0.5
P(A | B) conditional probability function probability of event A given event B occured P(A | B) = 0.3
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 μ = 10
E(X) expectation value expected value of random variable X E(X) = 10
E(X | Y) conditional expectation expected value of random variable X given Y E(X | Y=2) = 5
var(X) variance variance of random variable X var(X) = 4
σ2 variance variance of population values σ2 = 4
std(X) standard deviation standard deviation of random variable X std(X) = 2
σX standard deviation standard deviation value of random variable X σX  = 2
median middle value of random variable x
cov(X,Y) covariance covariance of random variables X and Y cov(X,Y) = 4
corr(X,Y) correlation correlation of random variables X and Y corr(X,Y) = 3
ρX,Y correlation correlation of random variables X and Y ρX,Y = 3
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 s 2 = 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(μ,σ2) normal distribution gaussian distribution X ~ N(0,3)
U(a,b) uniform distribution equal probability in range a,b  X ~ U(0,3)
exp(λ) exponential distribution f (x) = λe-λx , x≥0  
gamma(c, λ) gamma distribution

f (x) = λ c xc-1e-λx / Γ(c), x≥0

 
χ 2(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-λ / 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 _{n}P_{k}=\frac{n!}{(n-k)!} 5P3 = 5! / (5-3)! = 60
nCk

 

combination _{n}C_{k}=\binom{n}{k}=\frac{n!}{k!(n-k)!} 5C3 = 5!/[3!(5-3)!]=10

Set theory symbols

Symbol Symbol Name Meaning / definition Example
{ } set a collection of elements A={3,7,9,14}, B={9,14,28}
A B intersection objects that belong to set A and set B A B = {9,14}
A B union objects that belong to set A or set B A B = {3,7,9,14,28}
A B subset subset has less elements or equal to the set {9,14,28}{9,14,28}
A B proper subset / strict subset subset has less elements than the set {9,14}{9,14,28}
A B not subset left set not a subset of right set {9,66}{9,14,28}
A B superset set A has more elements or equal to the set B {9,14,28}{9,14,28}
A B proper superset / strict superset set A has more elements than set B {9,14,28}{9,14}
A B not superset set A is not a superset of set B {9,14,28}{9,66}
2A power set all subsets of A  
Ƥ (A) power set all subsets of A  
A = B equality both sets have the same members A={3,9,14}, B={3,9,14}, A=B
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={3,9,14},     B={1,2,3}, A-B={9,14}
A - B relative complement objects that belong to A and not to B A={3,9,14},     B={1,2,3}, A-B={9,14}
A ∆ B symmetric difference 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}
A B symmetric difference 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}
aA element of set membership A={3,9,14}, 3 A
xA not element of no set membership A={3,9,14}, 1 A
(a,b) ordered pair collection of 2 elements  
AЧB cartesian product set of all ordered pairs from A and B  
|A| cardinality the number of elements of set A A={3,9,14}, |A|=3
#A cardinality the number of elements of set A A={3,9,14}, #A=3
א aleph infinite cardinality  
Ш empty set Ш = { } C = {Ш}
U universal set set of all possible values  
0 natural numbers set (with zero) 0 = {0,1,2,3,4,...} 0 ∈ ℕ0
1 natural numbers set (without zero) 1 = {1,2,3,4,5,...} 6 ∈ ℕ1
integer numbers set = {...-3,-2,-1,0,1,2,3,...} -6 ∈ ℤ
rational numbers set = {x | x=a/b, a,b∈ℕ} 2/6 ∈ ℚ
real numbers set = {x | -∞ < x <∞} 6.343434 ∈ ℝ
complex numbers set = {z | z=a+bi, -∞<a<∞,      -∞<b<∞} 6+2i ∈ ℂ

Logic symbols

Symbol Symbol Name Meaning / definition Example
· and and

x · y

^ caret / circumflex and

x ^ y

& ampersand and

x & y

+ plus or

x + y

reversed caret or

x y

| vertical line or

x | y

x' single quote not - negation

x'

x bar not - negation

x

¬ not not - negation

¬ x

! exclamation mark not - negation

! x

circled plus / oplus exclusive or - xor

x y

~ tilde negation

~ x

implies    
equivalent if and only if  
for all    
there exists    
there does not exists    
therefore    
because / since    




طبقه بندی: آموزشی، دانستنی ها، دبیرستان،
[ شنبه 23 مهر 1390 ] [ 09:16 ب.ظ ] [ عبداللهی ]
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