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 × (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	3 ∓ 5 = -2 and 8
*	asterisk	multiplication	2 * 3 = 6
×	times sign	multiplication	2 × 3 = 6
∙	multiplication dot	multiplication	2 ∙ 3 = 6
÷	division sign / obelus	division	6 ÷ 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
³√a	cube root		³√8 = 2
⁴√a	forth root		⁴√16 = ±2
ⁿ√a	n-th root (radical)		for n=3, ⁿ√8 = 2
%	percent	1% = 1/100	10% × 30 = 3
‰	per-mille	1‰ = 1/1000 = 0.1%	10‰ × 30 = 0.3
ppm	per-million	1ppm = 1/1000000	10ppm × 30 = 0.0003
ppb	per-billion	1ppb = 1/1000000000	10ppb × 30 = 3×10^-7
ppt	per-trillion	1ppb = 10^-12	10ppb × 30 = 3×10^-10

Geometry symbols

Symbol	Symbol Name	Meaning / definition	Example
∠	angle	formed by two rays	∠ABC = 30º
∡	measured angle		∡ABC = 30º
∢	spherical angle		∢AOB = 30º
∟	right angle	= 90º	α = 90º
º	degree	1 turn = 360º	α = 60º
´	arcminute	1º = 60´	α = 60º59'
´´	arcsecond	1´ = 60´´	α = 60º59'59''
AB	line	line from point A to point B
	ray	line that start from point A
\|	perpendicular	perpendicular lines (90ºangle)	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	360º = 2π rad
grad	grads	grads angle unit	360º = 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! = 123*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 \| a ≤ x ≤ b}	x ∈ [2,6]
∆	delta	change / difference	∆t = t₁- t₀
∆	discriminant	Δ = b² - 4ac
∑	sigma	summation - sum of all values in range of series	∑ x_i= x₁+x₂+...+x_n
∑∑	sigma	double summation
∏	capital pi	product - product of all values in range of series	∏ x_i=x₁∙x₂∙...∙x_n
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
×	cross	vector product	a × b
A⊗B	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	(A^T)_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(A ∩ B)	probability of events intersection	probability that of events A and B	P(A∩B) = 0.5
P(A ∪ B)	probability of events union	probability that of events A or B	P(A∪B) = 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
σ_X	standard deviation	standard deviation value of random variable X	σ_X= 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
ρ_X,Y	correlation	correlation of random variables X and Y	ρ_X,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 = (x_max+x_min)/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
s²	sample variance	population samples variance estimator	s² = 4
s	sample standard deviation	population samples standard deviation estimator	s = 2
z_x	standard score	z_x = (x-x) / s_x
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 x^c-1e^-λx / Γ(c),x≥0
χ²(k)	chi-square distribution	f (x) = x^k^/2-1e^-x/2 / ( 2^k/2Γ(k/2) )
F (k₁, k₂)	F distribution
Bin(n,p)	binomial distribution	f (k) = _nC_k p^k(1-p)^n-k
Poisson(λ)	Poisson distribution	f (k) = λ^ke^-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
_nP_k	permutation	$_{n}P_{k}=\frac{n!}{(n-k)!}$	₅P₃ = 5! / (5-3)! = 60
_nC_k	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}
2^A	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
A^c	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}
a∈A	element of	set membership	A={3,9,14}, 3 ∈ A
x∉A	not element of	no set membership	A={3,9,14}, 1 ∉ A
(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={3,9,14}, \|A\|=3
#A	cardinality	the number of elements of set A	A={3,9,14}, #A=3
א	aleph	infinite cardinality
Ø	empty set	Ø = { }	C = {Ø}
U	universal set	set of all possible values
ℕ₀	natural numbers set (with zero)	ℕ₀ = {0,1,2,3,4,...}	0 ∈ ℕ₀
ℕ₁	natural numbers set (without zero)	ℕ₁ = {1,2,3,4,5,...}	6 ∈ ℕ₁
ℤ	integer numbers set	ℤ = {...-3,-2,-1,0,1,2,3,...}	-6 ∈ ℤ
ℚ	rational numbers set	ℚ = {x \| x=a/b, a,b∈ℕ}	2/6 ∈ ℚ
ℝ	real numbers set	ℝ = {x \| -∞ < x <∞}	6.343434 ∈ ℝ
ℂ	complex numbers set	ℂ = {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

Calculus & analysis symbols

Symbol	Symbol Name	Meaning / definition	Example
$\lim_{x\to x0}f(x)$	limit	limit value of a function
ε	epsilon	represents a very small number, near zero	ε → 0
e	e constant / Euler's number	e = 2.718281828...	e = lim (1+1/x)^x , x→∞
y '	derivative	derivative - Leibniz's notation	(3x³)' = 9x²
y ''	second derivative	derivative of derivative	(3x³)'' = 18x
y⁽ⁿ⁾	nth derivative	n times derivation	(3x³)⁽³⁾ = 18
$\frac{dy}{dx}$	derivative	derivative - Lagrange's notation	d(3x³)/dx = 9x²
$\frac{d^2y}{dx^2}$	second derivative	derivative of derivative	d²(3x³)/dx² = 18x
$\frac{d^ny}{dx^n}$	nth derivative	n times derivation
$\dot{y}$	time derivative	derivative by time - Newton notation
	time second derivative	derivative of derivative
$\frac{\partial f(x,y)}{\partial x}$	partial derivative		∂(x²+y²)/∂x = 2x
∫	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
[a,b]	closed interval	[a,b] = {x \| a ≤ x ≤ b}
(a,b)	open interval	(a,b) = {x \| a < x < b}
i	imaginary unit	i ≡ √-1	z = 3 + 2i
z*	complex conjugate	z = a+bi → z*=a-bi	z* = 3 + 2i
z	complex conjugate	z = a+bi → z = a-bi	z = 3 + 2i
∇	nabla / del	gradient / divergence operator	∇f (x,y,z)
	vector
	unit vector
x * y	convolution	y(t) = x(t) * h(t)
ℒ	Laplace transform	F(s) = ℒ{f (t)}
ℱ	Fourier transform	X(ω) = ℱ{f (t)}
δ	delta function

Numeral symbols

Name	European	Roman	Hindu Arabic	Hebrew
zero	0		٠
one	1	I	١	א
two	2	II	٢	ב
three	3	III	٣	ג
four	4	IV	٤	ד
five	5	V	٥	ה
six	6	VI	٦	ו
seven	7	VII	٧	ז
eight	8	VIII	٨	ח
nine	9	IX	٩	ט
ten	10	X	١٠	י
eleven	11	XI	١١	יא
twelve	12	XII	١٢	יב
thirteen	13	XIII	١٣	יג
fourteen	14	XIV	١٤	יד
fifteen	15	XV	١٥	טו
sixteen	16	XVI	١٦	טז
seventeen	17	XVII	١٧	יז
eighteen	18	XVIII	١٨	יח
nineteen	19	XIX	١٩	יט
twenty	20	XX	٢٠	כ
thirty	30	XXX	٣٠	ל
fourty	40	XL	٤٠	מ
fifty	50	L	٥٠	נ
sixty	60	LX	٦٠	ס
seventy	70	LXX	٧٠	ע
eighty	80	LXXX	٨٠	פ
ninety	90	XC	٩٠	צ
one hundred	100	C	١٠٠	ק

Greek alphabet letters

Greek Symbol		Greek Letter Name	English Equivalent	Pronunciation
Upper Case	Lower Case	Greek Letter Name	English Equivalent	Pronunciation
Α	α	Alpha	a	al-fa
Β	β	Beta	b	be-ta
Γ	γ	Gamma	g	ga-ma
Δ	δ	Delta	d	del-ta
Ε	ε	Epsilon	e	ep-si-lon
Ζ	ζ	Zeta	z	ze-ta
Η	η	Eta	h	eh-ta
Θ	θ	Theta	th	te-ta
Ι	ι	Iota	i	io-ta
Κ	κ	Kappa	k	ka-pa
Λ	λ	Lambda	l	lam-da
Μ	μ	Mu	m	m-yoo
Ν	ν	Nu	n	noo
Ξ	ξ	Xi	x	x-ee
Ο	ο	Omicron	o	o-mee-c-ron
Π	π	Pi	p	pa-yee
Ρ	ρ	Rho	r	row
Σ	σ	Sigma	s	sig-ma
Τ	τ	Tau	t	ta-oo
Υ	υ	Upsilon	u	oo-psi-lon
Φ	φ	Phi	ph	f-ee
Χ	χ	Chi	ch	kh-ee
Ψ	ψ	Psi	ps	p-see
Ω	ω	Omega	o	o-me-ga

Roman numerals

Number	Roman numeral
1	I
2	II
3	III
4	IV
5	V
6	VI
7	VII
8	VIII
9	IX
10	X
11	XI
12	XII
13	XIII
14	XIV
15	XV
16	XVI
17	XVII
18	XVIII
19	XIX
20	XX
30	XXX
40	XL
50	L
60	LX
70	LXX
80	LXXX
90	XC
100	C
200	CC
300	CCC
400	CD
500	D
600	DC
700	DCC
800	DCCC
900	CM
1000	M
5000	V
10000	X
50000	L
100000	C
500000	D
1000000	M

1- Mathematic Symbols

Symbols

For all	∀	Partial differential	∂	Plus	+
There exists	∃	Differential operator (nabla)	∇	Minus	−
Such that		Infinity	∞	Multiply	×
Element of	∈	Angle	∠	Divide	÷
Not an element of	∉	Integral	∫	Fraction slash	⁄
Contains as member	∋	Function	ƒ	Plus or minus	±
Logical and	∧	Radical	√	Asterisk operator	∗
Logical or	∨	Direct sum	⊕	Proportional to	∝
Intersection (cap)	∩	Vector product	⊗	Tilde operator	∼
Union (cup)	∪	Perpendicular	⊥	Approximately equal to	≅
Subset of or implied by	⊂	Dot operator	⋅	Almost equal to	≈
Superset of or implies	⊃	Middle dot	·	Not equal to	≠
Not a subset of	⊄	Imaginary part	ℑ	Identical to	≡
Subset of or equal to	⊆	Real part	ℜ	Less than or equal to	≤
Superset of or equal to	⊇	Aleph	ℵ	Greater than or equal to	≥
Empty set	∅	h/2π		Less than	<>
Not	¬	Lagrangian		Greater than;	>
Therefore	∴	Hamiltonian		N-ary summation	∑
				N-ary product	∏

Greek Letters used in Math

Table of Formulas For Geometry

A table of formulas for geometry, related to area and perimeter of triangles, rectangles, cercles, sectors, and volume of sphere, cone, cylinder are presented.

Right Triangle and Pythagora's theorem

Pythagora's theorem: The two sides a and b of a right triangle and the hypotenuse c are related by

a² + b² = c²

Area and Perimeter of Triangle

Perimeter = a + b + c

There are several formulas for the area.

If the base b and the corresponding height h are known, we use the formula

Area = (1 / 2) * b * h.If two sides and the angle between them are known, we use one of the formulas, depending on which side and which angle are known

Area = (1 / 2)* b * c sin A

Area = (1 / 2)* a * c sin B

Area = (1 / 2)* a * b sin C .If all three sides are known, we may use Heron's formula for the area.

Area = sqrt [ s(s - a)(s - b)(s - c) ] , where s = (a + b + c)/2.

Area and Perimeter of Rectangle

Perimeter = 2L + 2W

Area = L * W

Circumference of a Circle and Area of a Circular Region

Circumference = 2*Pi*r

Area = Pi*r²

Arclength and Area of a Circular Sector

Arclength: s = r*t

Area = (1/2) *r² * t

where t is the central angle in RADIANS.

Volume and Surface Area of a Rectangular Solid

Volume = L*W*H

Surface Area = 2(L*W + H*W + H*L)

Volume and Surface Area of a Sphere

Volume = (4/3)* Pi * r³

Surface Area = 4 * Pi * r²

Volume and Surface Area of a Right Circular Cylinder

Volume = Pi * r² * h

Surface Area = 2 * Pi * r * h

Volume and Surface Area of a Right Circular Cone

Volume = (1/3)* Pi * r² * h

Surface Area = Pi * r * sqrt (r² + h²)

Regular Polygons

A tutorial to explore the formulas and other properties of regular polygons, inscribed and circumscribed circles. An applet is also used for an interactive tutorial.

Regular polygons have all sides equal and all angles equal. Below is an example of a 5 sided regular polygon also called a pentagon.

where x is the side of the pentagon, r is the radius of the inscribed circle and R is the radius of the circumscribed circle.
Let us develop formulas to find the area of an n sided regular polygons as a function of x, r and R. We shall follow the following route: Find the area of one triangle, such as triangle OAB, and multiply it by n ,the number of sides of the polygon, to find the total area of the polygon.

Relationship between x, r and R.

Let t be angle AOB.

t = 360^o / n

From trigonometry of right triangles, we have

tan(t / 2) = (x / 2) / r and sin (t / 2) = (x / 2) / R

which gives r and R in term of x as follows

r = (x / 2) cot (180^o / n)

and

R = (x / 2) csc (180^o / n)
Formula 1

The area of triangle AOB = (1/2) x r

Area of polygon = n * area of triangle AOB

= (1/2) n x r
Formula 2

Another possible formula for the area of triangle AOB in terms of R is

Area of triangle AOB = (1/2) sin ( t ) R ²

= (1/2) R ² sin (360^o / n)
Area of polygon = n * area of triangle AOB

= (1/2) n R ² sin (360^o / n)Formula 3

Another formula may be obtained if r found above is substituted in formula 1.
Area of polygon = (1/2) n x r

= (1/2) n x [ (x / 2) cot (180^o / n) ]

= (1 / 4) n x² cot (180^o / n)
Formula 4

Another formula may be obtained if x in r = (x / 2) cot (180^o / n) is substituted in formula 1.

Area of polygon = (1/2) n x r

= (1/2) n [ 2 r tan (180^o / n) ] r

= n r² tan (180^o / n)
names of polygons according to the number of sides

number of sides		name
3		equilateral triangle
4		square
5		pentagon
6		hexagon
7		heptagon
8		octagon
9		nonagon
10		decagon
11		undecagon
12		dodecagon

Interactive Tutorial

1 - Press the button above to start the applet.

2 - In this applet, the radius of the circumscribed circle is constant and equal to 3. The number n of the sides of the polygon may be changed using the slider.

3 - Use the formulas found above to check the area of the polygon for different values of n.

4 - When n increases, what happens to the three areas: that of the circumscribed circle, the polygon and the inscribed circle? See problem 4 in polygons problems.

Polygons Problems

Regular polygons problems with detailed solutions.

Problem 1: A 6 sided regular polygon (hexagon) is inscribed in a circle of radius 10 cm, find the length of one side of the hexagon.

Solution to Problem 1:

Angle AOB is given by

angle (AOB) = 360^o / 6 = 60^o
Since OA = OB = 10 cm, triangle OAB is isosceles which gives

angle (OAB) = angle (OBA)
So all three angles of the triangle are equal and therefore it is an equilateral triangle. Hence

AB = OA = OB = 10 cm.

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