Comprehensive guide to all mathematical notations and symbols with explanations. Includes calculus (∫, ∑, ∞), algebra (∈, ⊂, ∀), geometry (∠, △, ∥), and logic symbols. Perfect for students, researchers, and technical writers.

mathematical-notations.md

Complete List of Mathematical Notations and Symbols

A comprehensive reference guide for mathematical notation across all major branches of mathematics

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📋 Table of Contents

• Calculus & Analysis
• Geometry & Trigonometry
• Basic Arithmetic & Algebra
• Set Theory & Logic
• Number Systems & Constants
• Greek Letters

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∫ Calculus & Analysis

Symbol	Name	Usage Example	Explanation
∞	Infinity	lim(x→∞) f(x)	Unbounded quantity, larger than any real number
∑	Summation	∑(i=1 to n) a_i	Sum of sequence: a₁ + a₂ + ... + aₙ
∏	Product	∏(i=1 to n) a_i	Product of sequence: a₁ × a₂ × ... × aₙ
∫	Integral	∫f(x)dx	Area under curve or antiderivative
∬	Double Integral	∬f(x,y)dxdy	Integration over 2D area
∭	Triple Integral	∭f(x,y,z)dxdydz	Integration over 3D volume
∮	Contour Integral	∮f(z)dz	Integral along closed complex path
lim	Limit	lim(x→a) f(x)	Value function approaches as input approaches a
′	Prime/Derivative	f'(x)	First derivative of function f
d	Differential	dy/dx	Infinitesimal change, derivative notation
∂	Partial Derivative	∂f/∂x	Derivative with respect to one variable
∇	Nabla/Gradient	∇f	Vector of partial derivatives
∇²	Laplacian	∇²f	Divergence of gradient
inf	Infimum	inf S	Greatest lower bound of set S
sup	Supremum	sup S	Least upper bound of set S
→	Tends To	x → a	x approaches a

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△ Geometry & Trigonometry

Symbol	Name	Usage Example	Explanation
π	Pi	C = 2πr	Circle circumference ratio ≈ 3.14159
∠	Angle	∠ABC	Angle at vertex B
∟	Right Angle	90°	90-degree angle
∡	Measured Angle	∡A = 45°	Angle measurement
∢	Spherical Angle	∢ABC	Angle on sphere surface
∥	Parallel	AB ∥ CD	Lines never intersect
∦	Not Parallel	AB ∦ CD	Lines are not parallel
⊥	Perpendicular	AB ⊥ CD	Lines meet at 90°
Δ	Triangle	ΔABC	Triangle with vertices A, B, C
○	Circle	○ O	Circle with center O
¯	Segment	AB̅	Line segment between A and B
⌒	Arc	A͝B	Arc between points A and B
°	Degree	45°	Angle measurement unit
sin	Sine	sin(θ)	Opposite/Hypotenuse
cos	Cosine	cos(θ)	Adjacent/Hypotenuse
tan	Tangent	tan(θ)	Opposite/Adjacent
Π	Product	Π a_i	Continuous product

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➕ Basic Arithmetic & Algebra

Symbol	Name	Usage Example	Explanation
+	Plus	a + b	Addition
-	Minus	a - b	Subtraction
×	Times	a × b	Multiplication
⋅	Dot	a ⋅ b	Multiplication (alternative)
÷	Obelus	a ÷ b	Division
/	Slash	a/b	Division (fraction)
=	Equals	a = b	Equality
≠	Not Equal	a ≠ b	Inequality
≈	Approximately	π ≈ 3.14	Approximate equality
≡	Identically	f(x) ≡ g(x)	Identity for all x
<	Less Than	a < b	Comparison
>	Greater Than	a > b	Comparison
≤	Less or Equal	a ≤ b	Comparison
≥	Greater or Equal	a ≥ b	Comparison
±	Plus-Minus	x = ±5	Positive or negative value
∓	Minus-Plus	a ± b ∓ c	Opposite of ± in expressions
√	Square Root	√4 = 2	Root of number
∛	Cube Root	∛8 = 2	Cube root
∜	Fourth Root	∜16 = 2	Fourth root
%	Percent	50%	Per hundred
‰	Per Mille	50‰	Per thousand
!	Factorial	5! = 120	Product of integers
aⁿ	Exponentiation	2³ = 8	Base to power
.	Decimal Point	3.14	Decimal separator

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∈ Set Theory & Logic

Symbol	Name	Usage Example	Explanation
∈	Element Of	a ∈ S	a is in set S
∉	Not Element Of	a ∉ S	a is not in set S
∅	Empty Set	∅ = {}	Set with no elements
⊂	Subset	A ⊂ B	All elements of A in B
⊊	Proper Subset	A ⊊ B	Subset but not equal
⊃	Superset	B ⊃ A	Contains all elements
⊋	Proper Superset	B ⊋ A	Superset but not equal
∪	Union	A ∪ B	Elements in A or B
∩	Intersection	A ∩ B	Elements in both A and B
\	Set Minus	A \ B	Elements in A but not B
Aᶜ	Complement	Aᶜ	Elements not in A
U	Universal Set	U	Set of all elements
∀	For All	∀x P(x)	Universal quantifier
∃	There Exists	∃x P(x)	Existential quantifier
∄	There Does Not Exist	∄x P(x)	Negation of existence
:	Such That	{x : x > 0}	Set builder notation
∴	Therefore	P ∴ Q	Logical consequence
∵	Because	Q ∵ P	Reason
∎	Q.E.D.	Proof ∎	End of proof
⇒	Implies	P ⇒ Q	Logical implication
⇔	If and Only If	P ⇔ Q	Logical equivalence
∧	AND	P ∧ Q	Logical conjunction
∨	OR	P ∨ Q	Logical disjunction
¬	NOT	¬P	Logical negation

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ℕ Number Systems & Constants

Symbol	Name	Elements	Description
ℕ	Natural Numbers	{1, 2, 3, ...} or {0, 1, 2, ...}	Counting numbers
ℤ	Integers	{..., -2, -1, 0, 1, 2, ...}	Whole numbers
ℚ	Rational Numbers	{p/q : p,q ∈ ℤ, q ≠ 0}	Fractions
ℝ	Real Numbers	All rational and irrational numbers	Number line
ℂ	Complex Numbers	{a + bi : a,b ∈ ℝ}	Real and imaginary parts
ℍ	Quaternions	{a + bi + cj + dk}	4D number system
i	Imaginary Unit	i² = -1	Square root of -1
e	Euler's Number	e ≈ 2.71828	Natural logarithm base

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α Greek Letters (Mathematical Usage)

Symbol	Name	Common Usage
α	Alpha	Angles, constants
β	Beta	Angles, constants
γ	Gamma	Euler-Mascheroni constant
Δ	Delta (upper)	Change, discriminant
δ	Delta (lower)	Small increment, Dirac delta
ε	Epsilon	Arbitrarily small number
ζ	Zeta	Riemann zeta function
θ	Theta	Angles
λ	Lambda	Eigenvalues, wavelength
μ	Mu	Mean, micron
Σ	Sigma (upper)	Summation
ω	Omega (lower)	Angular velocity
Ω	Omega (upper)	Ohms, sample space