Introduction
Mathematical notation is a language of its own, often appearing cryptic to the uninitiated. This cheat sheet serves as a decoder for the symbols, abbreviations, and conventions used across various mathematical fields. Understanding these notations is crucial for anyone studying or working with mathematics, computer science, physics, engineering, and many other technical disciplines.
Core Mathematical Symbols
Basic Operators
| Symbol | Name | Meaning | Example |
|---|---|---|---|
| + | Plus | Addition | 3 + 2 = 5 |
| − | Minus | Subtraction | 7 − 4 = 3 |
| × or · or * | Times | Multiplication | 5 × 2 = 10 |
| ÷ or / | Divided by | Division | 8 ÷ 2 = 4 |
| ^ or ** | Caret/power | Exponentiation | 2^3 = 8 |
| √ | Radical | Square root | √9 = 3 |
| ∛ | Cube root | Cube root | ∛27 = 3 |
| % | Percent | Per hundred | 15% = 0.15 |
| ≡ | Identical to | Equivalence | x ≡ y (mod n) |
| ≈ | Approximately equal | Approximation | π ≈ 3.14 |
| ≠ | Not equal | Inequality | 5 ≠ 7 |
| ± | Plus-minus | Both operations | x = 3 ± 2 |
Comparison Operators
| Symbol | Name | Meaning | Example |
|---|---|---|---|
| = | Equals | Equality | 2 + 2 = 4 |
| < | Less than | Smaller value | 3 < 5 |
| > | Greater than | Larger value | 7 > 4 |
| ≤ | Less than or equal to | Not greater than | x ≤ 10 |
| ≥ | Greater than or equal to | Not less than | y ≥ 5 |
| ≪ | Much less than | Significantly smaller | 1 ≪ 1000 |
| ≫ | Much greater than | Significantly larger | 1000 ≫ 1 |
Set Notation
| Symbol | Name | Meaning | Example |
|---|---|---|---|
| ∈ | Element of | Membership | 3 ∈ {1,2,3,4} |
| ∉ | Not element of | Non-membership | 5 ∉ {1,2,3,4} |
| ∋ | Contains as member | Membership (reverse) | {1,2,3} ∋ 2 |
| ⊂ | Subset of | Proper containment | {1,2} ⊂ {1,2,3} |
| ⊆ | Subset or equal | Containment | {1,2} ⊆ {1,2} |
| ⊃ | Superset of | Proper containment | {1,2,3} ⊃ {1,2} |
| ⊇ | Superset or equal | Containment | {1,2} ⊇ {1,2} |
| ∪ | Union | Combination | {1,2} ∪ {2,3} = {1,2,3} |
| ∩ | Intersection | Common elements | {1,2,3} ∩ {2,3,4} = {2,3} |
| \ or − | Set difference | Removal | {1,2,3} \ {2} = {1,3} |
| ∅ or {} | Empty set | No elements | ∅ |
| ℕ | Natural numbers | {1, 2, 3, …} | n ∈ ℕ |
| ℤ | Integers | {…, -2, -1, 0, 1, 2, …} | z ∈ ℤ |
| ℚ | Rational numbers | Fractions p/q | q ∈ ℚ |
| ℝ | Real numbers | All points on number line | r ∈ ℝ |
| ℂ | Complex numbers | a + bi form | c ∈ ℂ |
Logic Notation
| Symbol | Name | Meaning | Example |
|---|---|---|---|
| ∧ | And | Logical conjunction | A ∧ B |
| ∨ | Or | Logical disjunction | A ∨ B |
| ¬ or ~ | Not | Logical negation | ¬A |
| ⊕ | Exclusive or (XOR) | Either but not both | A ⊕ B |
| → | Implies | If-then | A → B |
| ↔ | If and only if | Biconditional | A ↔ B |
| ∀ | For all | Universal quantifier | ∀x P(x) |
| ∃ | There exists | Existential quantifier | ∃x P(x) |
| ∄ | There does not exist | Negated existential | ∄x P(x) |
| ⊨ | Models | Semantic consequence | A ⊨ B |
| ⊢ | Proves | Syntactic consequence | A ⊢ B |
Calculus Notation
Derivatives and Integrals
| Symbol | Name | Meaning | Example |
|---|---|---|---|
| dx/dt | Derivative | Rate of change | dy/dx = 2x |
| f'(x) | Derivative (prime) | Rate of change | f'(x) = 2x |
| f”(x) | Second derivative | Change of rate of change | f”(x) = 2 |
| ∂f/∂x | Partial derivative | Derivative with respect to one variable | ∂z/∂x |
| ∫ | Integral | Antiderivative | ∫ x dx = x²/2 + C |
| ∮ | Closed curve integral | Integration around closed path | ∮ F·dr |
| ∫_a^b | Definite integral | Area under curve from a to b | ∫_0^1 x² dx = 1/3 |
| ∑ | Summation | Sum of sequence | ∑_{i=1}^n i = n(n+1)/2 |
| ∏ | Product | Product of sequence | ∏_{i=1}^n i = n! |
| lim | Limit | Approaching value | lim_{x→0} sin(x)/x = 1 |
| ∇ | Nabla/Del | Vector differential operator | ∇f = (∂f/∂x, ∂f/∂y, ∂f/∂z) |
| ∆ | Delta | Change in variable | ∆x = x₂ – x₁ |
Series and Sequences
| Symbol | Name | Meaning | Example |
|---|---|---|---|
| {a_n} | Sequence | Ordered list of terms | {1, 1/2, 1/3, …} |
| a_n | nth term | Specific element of sequence | a_3 = 1/3 |
| ∑_{n=1}^∞ a_n | Infinite series | Sum of infinite sequence | ∑_{n=1}^∞ 1/n² = π²/6 |
| a_n → L | Convergence | Sequence approaches limit | a_n = 1/n → 0 |
Linear Algebra Notation
Vectors and Matrices
| Symbol | Name | Meaning | Example |
|---|---|---|---|
| →v or v̄ or v | Vector | Quantity with magnitude and direction | v = (3, 4, 5) |
| |v| or ‖v‖ | Vector norm | Length/magnitude of vector | |(3,4)| = 5 |
| v · w | Dot product | Scalar product of vectors | (1,2) · (3,4) = 11 |
| v × w | Cross product | Vector product | (1,0,0) × (0,1,0) = (0,0,1) |
| A | Matrix | Rectangular array | A = [1 2; 3 4] |
| A^T | Transpose | Flipped matrix | [1 2; 3 4]^T = [1 3; 2 4] |
| A^{-1} | Inverse | Matrix inverse | AA^{-1} = I |
| det(A) or |A| | Determinant | Scalar value of square matrix | |[1 2; 3 4]| = -2 |
| tr(A) | Trace | Sum of diagonal elements | tr([1 2; 3 4]) = 5 |
| λ | Eigenvalue | Scalar in Av = λv | det(A-λI) = 0 |
| v | Eigenvector | Vector in Av = λv | Av = λv |
Probability and Statistics
| Symbol | Name | Meaning | Example |
|---|---|---|---|
| P(A) | Probability | Likelihood of event A | P(heads) = 0.5 |
| P(A|B) | Conditional probability | Probability of A given B | P(A|B) = P(A∩B)/P(B) |
| P(A∩B) | Joint probability | Probability of both A and B | P(A∩B) = P(A)P(B) (if independent) |
| P(A∪B) | Union probability | Probability of either A or B | P(A∪B) = P(A) + P(B) – P(A∩B) |
| E[X] or μ | Expected value | Average of random variable | E[X] = ∑x·P(X=x) |
| Var(X) or σ² | Variance | Spread of distribution | Var(X) = E[(X-μ)²] |
| σ | Standard deviation | Square root of variance | σ = √Var(X) |
| ρ | Correlation coefficient | Linear relationship | -1 ≤ ρ ≤ 1 |
| X ~ D | Distribution | Random variable follows distribution | X ~ N(μ,σ²) |
| H₀ | Null hypothesis | Statement being tested | H₀: μ = 0 |
| H₁ or H_a | Alternative hypothesis | Alternative statement | H₁: μ ≠ 0 |
Number Theory Notation
| Symbol | Name | Meaning | Example |
|---|---|---|---|
| a|b | Divides | a divides b evenly | 3|6 (true) |
| a mod n | Modulo | Remainder after division | 7 mod 3 = 1 |
| gcd(a,b) | Greatest common divisor | Largest common factor | gcd(12,18) = 6 |
| lcm(a,b) | Least common multiple | Smallest common multiple | lcm(4,6) = 12 |
| a ≡ b (mod n) | Congruence | a and b have same remainder mod n | 17 ≡ 2 (mod 5) |
| φ(n) | Euler’s totient | Count of numbers coprime to n | φ(10) = 4 |
| p | Prime number | Only divisible by 1 and itself | 2, 3, 5, 7, 11, … |
Abstract Algebra Notation
| Symbol | Name | Meaning | Example |
|---|---|---|---|
| (G,∗) | Group | Set with binary operation | (ℤ,+) |
| e or 1 | Identity element | Element where a∗e = a | 0 in (ℤ,+) |
| a^{-1} | Inverse | Element where a∗a^{-1} = e | -a in (ℤ,+) |
| Ker(f) | Kernel | Elements mapping to identity | Ker(f) = {x ∈ G | f(x) = e} |
| Im(f) | Image | Range of homomorphism | Im(f) = {f(x) | x ∈ G} |
| G/H | Quotient group | Cosets of normal subgroup | ℤ/nℤ |
Common Functions and Constants
| Symbol | Name | Meaning | Example |
|---|---|---|---|
| sin, cos, tan | Trigonometric functions | Ratios in right triangles | sin(π/2) = 1 |
| ln | Natural logarithm | Log with base e | ln(e) = 1 |
| log | Logarithm | Log with specified base | log₁₀(100) = 2 |
| e | Euler’s number | Base of natural logarithm | e ≈ 2.71828 |
| π | Pi | Ratio of circumference to diameter | π ≈ 3.14159 |
| i | Imaginary unit | Square root of -1 | i² = -1 |
| ∞ | Infinity | Unbounded quantity | lim_{x→0} 1/x = ∞ |
| ! | Factorial | Product of integers | 5! = 120 |
| n! | n factorial | Product 1×2×…×n | 5! = 120 |
| (n k) or C(n,k) or ₙCₖ | Binomial coefficient | Ways to choose k from n | (5 2) = 10 |
| [x] | Floor function | Greatest integer ≤ x | [3.7] = 3 |
| ⌈x⌉ | Ceiling function | Least integer ≥ x | ⌈3.2⌉ = 4 |
| |x| | Absolute value | Distance from zero | |-5| = 5 |
| arg(z) | Argument | Angle of complex number | arg(1+i) = π/4 |
Step-by-Step Approach to Understanding New Notation
Identify the context
- Determine the mathematical field (calculus, algebra, statistics, etc.)
- Look for surrounding familiar symbols or keywords
Look for sub-components
- Break complex notations into smaller parts
- Recognize standard modifiers (subscripts, superscripts, etc.)
Check for standard patterns
- Many notations follow conventional patterns
- Recognize function notation, operators, and special symbols
Refer to definitions
- Mathematical papers/texts typically define notations
- Check the “Notation” or “Preliminaries” sections
Translate to words
- Practice expressing the notation in plain language
- Understanding verbal equivalents aids comprehension
Common Challenges and Solutions
| Challenge | Solution |
|---|---|
| Overloaded symbols | Pay attention to context; the same symbol can have different meanings in different fields |
| Dense notation | Break into smaller parts and understand each component separately |
| Implied operations | Look for conventional omissions (e.g., multiplication signs often omitted) |
| Unfamiliar subscripts/superscripts | These often indicate specific instances, dimensions, or iterations |
| Abstract notation | Focus on the properties and relationships being described rather than concrete values |
| Multiple notation styles | Recognize equivalent notations (e.g., f'(x), df/dx, and Df all represent derivatives) |
Best Practices for Working with Mathematical Notation
Create a personal notation dictionary
- Keep a running list of notations you encounter
- Include context and examples for each
Practice translation
- Convert between different notation styles
- Express mathematical statements in words and vice versa
Draw diagrams
- Visual representations help understand abstract concepts
- Use graphs, trees, or other visualizations when appropriate
Work with examples
- Apply notation to concrete examples
- Verify understanding through specific cases
Maintain consistency
- When creating your own notation, keep it consistent
- Follow established conventions when possible
Explain notation in your work
- Define symbols clearly when writing mathematics
- Don’t assume readers know specialized notation
Resources for Further Learning
Books
- “Mathematical Notation: A Guide for Engineers and Scientists” by Edward R. Scheinerman
- “The Princeton Companion to Mathematics” edited by Timothy Gowers
- “Handbook of Mathematical Functions” by Abramowitz and Stegun
Online Resources
- MathWorld – Comprehensive encyclopedia of mathematical terms and notation
- Detexify – Draw a symbol to find its LaTeX code
- The Comprehensive LaTeX Symbol List
- Mathematical Notation: Past and Future
Communities
- Mathematics Stack Exchange – Q&A forum for mathematics
- MathOverflow – For research-level mathematics questions
- r/math on Reddit – Discussion forum for mathematical topics
Tips for Typesetting Mathematics
LaTeX Commands for Common Symbols
- Basic symbols:
\alpha,\beta,\gamma,\sum,\prod,\int - Operators:
\times,\div,\pm,\leq,\geq - Sets:
\in,\subset,\cup,\cap,\emptyset - Special sets:
\mathbb{R},\mathbb{Z},\mathbb{N},\mathbb{Q},\mathbb{C} - Greek letters:
\alpha,\beta,\gamma,\delta,\epsilon,\pi - Functions:
\sin,\cos,\tan,\log,\exp,\lim
Online Equation Editors
- Overleaf – Online LaTeX editor
- MathJax – JavaScript display engine for mathematics
- Codecogs Equation Editor – Generate LaTeX equations as images