Mathematical Notations
Mathematical notation is the language of mathematics. Learn it, and you can read any math textbook, paper, or formula. Struggle with it, and every equation becomes a puzzle.
This is my comprehensive reference guide. I’ve organized it by category, explained what each symbol means, and included examples. Bookmark this page. You’ll need it.
Essential Notations
These are the workhorses of mathematics. You’ll see them everywhere: calculus, statistics, physics, computer science, economics. Master these first.
Summation (Sigma Notation)
$$\sum_{i=1}^{n} a_i = a_1 + a_2 + a_3 + \cdots + a_n$$
The capital Greek letter sigma (Σ) means “add up.” The subscript tells you where to start, the superscript tells you where to stop, and the expression after tells you what to add.
Examples:
- \( \sum_{i=1}^{5} i = 1 + 2 + 3 + 4 + 5 = 15 \)
- \( \sum_{k=0}^{3} k^2 = 0 + 1 + 4 + 9 = 14 \)
- \( \sum_{j=1}^{n} 1 = n \) (adds 1 a total of n times)
Pro Tip: The index variable (i, j, k, n) is a “dummy variable.” It doesn’t matter what letter you use: \( \sum_{i=1}^{5} i = \sum_{k=1}^{5} k = \sum_{🐱=1}^{5} 🐱 \). Same answer.
Product (Pi Notation)
$$\prod_{i=1}^{n} a_i = a_1 \cdot a_2 \cdot a_3 \cdot \cdots \cdot a_n$$
The capital Greek letter pi (Π) means “multiply together.” Same indexing convention as summation.
Examples:
- \( \prod_{i=1}^{4} i = 1 \cdot 2 \cdot 3 \cdot 4 = 24 = 4! \)
- \( \prod_{k=1}^{n} k = n! \) (this is how factorial is defined)
- \( \prod_{i=1}^{3} 2 = 2 \cdot 2 \cdot 2 = 8 = 2^3 \)
Integral Notation
$$\int_a^b f(x) \, dx$$
The elongated S (∫) stands for “sum” (it’s a stylized S from the Latin “summa”). Integration is continuous summation—adding up infinitely many infinitesimally small pieces.
Components:
- \( a \) = lower limit (where to start)
- \( b \) = upper limit (where to stop)
- \( f(x) \) = integrand (what you’re integrating)
- \( dx \) = differential (indicates integration with respect to x)
Variations:
| Notation | Name | Meaning |
|---|---|---|
| \( \int f(x) \, dx \) | Indefinite integral | Antiderivative (no limits) |
| \( \int_a^b f(x) \, dx \) | Definite integral | Area under curve from a to b |
| \( \iint f(x,y) \, dx \, dy \) | Double integral | Integration over 2D region |
| \( \iiint f \, dV \) | Triple integral | Integration over 3D region |
| \( \oint f \, ds \) | Contour/line integral | Integration around closed curve |
Limit Notation
$$\lim_{x \to a} f(x) = L$$
Read as: “The limit of f(x) as x approaches a equals L.”
This means: as x gets arbitrarily close to a, f(x) gets arbitrarily close to L.
Variations:
- \( \lim_{x \to a^+} \) = limit from the right (x approaches a from above)
- \( \lim_{x \to a^-} \) = limit from the left (x approaches a from below)
- \( \lim_{x \to \infty} \) = limit as x goes to infinity
- \( \lim_{n \to \infty} a_n \) = limit of a sequence
Derivative Notations
Multiple notations exist for the same concept. Different fields prefer different styles.
| Notation | Name | Read as | Common in |
|---|---|---|---|
| \( \frac{dy}{dx} \) | Leibniz | “dy dx” or “derivative of y with respect to x” | Physics, engineering |
| \( f'(x) \) | Lagrange (prime) | “f prime of x” | Pure math |
| \( \dot{y} \) | Newton (dot) | “y dot” | Physics (time derivatives) |
| \( D_x f \) | Euler | “D sub x of f” | Differential equations |
Higher derivatives:
- Second derivative: \( \frac{d^2y}{dx^2} \) or \( f”(x) \) or \( \ddot{y} \)
- Third derivative: \( \frac{d^3y}{dx^3} \) or \( f”'(x) \)
- nth derivative: \( \frac{d^n y}{dx^n} \) or \( f^{(n)}(x) \)
Partial Derivative
$$\frac{\partial f}{\partial x} \quad \text{or} \quad f_x \quad \text{or} \quad \partial_x f$$
The curly ∂ (called “del” or “partial”) indicates a partial derivative—differentiation with respect to one variable while treating others as constants.
Example: If \( f(x, y) = x^2 y + y^3 \), then:
- \( \frac{\partial f}{\partial x} = 2xy \) (treat y as constant)
- \( \frac{\partial f}{\partial y} = x^2 + 3y^2 \) (treat x as constant)
Mixed partials: \( \frac{\partial^2 f}{\partial x \partial y} = \frac{\partial}{\partial x}\left( \frac{\partial f}{\partial y} \right) \)
Factorial
$$n! = n \cdot (n-1) \cdot (n-2) \cdot \cdots \cdot 2 \cdot 1$$
The exclamation mark means “factorial”—multiply all positive integers from 1 to n.
Key values:
| \( n \) | \( n! \) |
|---|---|
| 0 | 1 (by definition) |
| 1 | 1 |
| 2 | 2 |
| 3 | 6 |
| 4 | 24 |
| 5 | 120 |
| 6 | 720 |
| 10 | 3,628,800 |
Why 0! = 1: By convention, and because it makes formulas work. The empty product (multiplying zero things together) equals 1.
Binomial Coefficient
$$\binom{n}{k} = \frac{n!}{k!(n-k)!} = C(n,k) = {}_nC_k$$
Read as “n choose k.” Counts the number of ways to choose k items from n items without regard to order.
Examples:
- \( \binom{5}{2} = \frac{5!}{2! \cdot 3!} = \frac{120}{2 \cdot 6} = 10 \)
- \( \binom{n}{0} = \binom{n}{n} = 1 \)
- \( \binom{n}{1} = n \)
- \( \binom{n}{k} = \binom{n}{n-k} \) (symmetry)
Vectors and Matrices
Vector notation:
$$\vec{v} = \begin{bmatrix} v_1 \\ v_2 \\ \vdots \\ v_n \end{bmatrix} \quad \text{or} \quad \mathbf{v} = (v_1, v_2, \ldots, v_n)$$
Vectors can be written as columns (common in linear algebra) or as tuples (common in physics). The arrow \( \vec{v} \) or bold \( \mathbf{v} \) indicates “this is a vector, not a scalar.”
Matrix notation:
$$A = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{bmatrix}$$
An \( m \times n \) matrix has m rows and n columns. Element \( a_{ij} \) is in row i, column j.
Greek Letters in Mathematics
Greek letters are everywhere in math. Here’s what they typically represent:
Lowercase Greek Letters
| Letter | Name | Common Uses |
|---|---|---|
| \( \alpha \) | alpha | Angles, coefficients, significance level |
| \( \beta \) | beta | Angles, coefficients, Type II error |
| \( \gamma \) | gamma | Euler-Mascheroni constant, angles |
| \( \delta \) | delta | Small change, Kronecker delta |
| \( \epsilon, \varepsilon \) | epsilon | Arbitrarily small positive number |
| \( \zeta \) | zeta | Riemann zeta function |
| \( \eta \) | eta | Efficiency, viscosity |
| \( \theta, \vartheta \) | theta | Angles (especially in trig) |
| \( \iota \) | iota | Index, imaginary unit (rarely) |
| \( \kappa \) | kappa | Curvature, condition number |
| \( \lambda \) | lambda | Eigenvalues, wavelength, rate parameter |
| \( \mu \) | mu | Mean, micro- prefix, measure |
| \( \nu \) | nu | Frequency, degrees of freedom |
| \( \xi \) | xi | Random variable, dummy variable |
| \( \pi \) | pi | 3.14159…, projection |
| \( \rho, \varrho \) | rho | Density, correlation, radius |
| \( \sigma \) | sigma | Standard deviation, sum (lowercase) |
| \( \tau \) | tau | Time constant, torque, 2π |
| \( \upsilon \) | upsilon | Rarely used (looks like v) |
| \( \phi, \varphi \) | phi | Golden ratio, angles, functions |
| \( \chi \) | chi | Chi-squared distribution |
| \( \psi \) | psi | Wave function, angles |
| \( \omega \) | omega | Angular frequency, last element |
Uppercase Greek Letters
| Letter | Name | Common Uses |
|---|---|---|
| \( \Gamma \) | Gamma | Gamma function, graphs |
| \( \Delta \) | Delta | Change/difference, Laplacian |
| \( \Theta \) | Theta | Big-Theta (complexity) |
| \( \Lambda \) | Lambda | Diagonal matrix of eigenvalues |
| \( \Xi \) | Xi | Cascade particle, grand canonical |
| \( \Pi \) | Pi | Product operator |
| \( \Sigma \) | Sigma | Summation, covariance matrix |
| \( \Phi \) | Phi | CDF of normal distribution, flux |
| \( \Psi \) | Psi | Wave function |
| \( \Omega \) | Omega | Sample space, ohms, Big-O variant |
Note: Some uppercase Greek letters look identical to Latin letters (A, B, E, Z, H, I, K, M, N, O, P, T, X) and are rarely used in math to avoid confusion.
Set Theory Notation
Defining Sets
| Notation | Meaning | Example |
|---|---|---|
| \( \{a, b, c\} \) | Set with elements a, b, c | \( \{1, 2, 3\} \) |
| \( \{a_1, a_2, \ldots, a_n\} \) | Finite set with n elements | \( \{1, 2, \ldots, 100\} \) |
| \( \{a_1, a_2, \ldots\} \) | Infinite set | \( \{1, 2, 3, \ldots\} \) |
| \( \{x : P(x)\} \) | Set of all x satisfying property P | \( \{x : x > 0\} \) |
| \( \{x \in A : P(x)\} \) | Elements of A satisfying P | \( \{x \in \mathbb{R} : x^2 < 4\} \) |
| \( \emptyset \) or \( \{\} \) | Empty set (no elements) | — |
Pro Tip: The colon “:” in set-builder notation can also be written as “|”. Both \( \{x : x > 0\} \) and \( \{x \mid x > 0\} \) mean the same thing.
Membership and Subsets
| Notation | Meaning | Example |
|---|---|---|
| \( a \in A \) | a is an element of A | \( 3 \in \{1, 2, 3\} \) |
| \( a \notin A \) | a is not an element of A | \( 4 \notin \{1, 2, 3\} \) |
| \( A \subset B \) | A is a proper subset of B | \( \{1\} \subset \{1, 2\} \) |
| \( A \subseteq B \) | A is a subset of B (possibly equal) | \( \{1, 2\} \subseteq \{1, 2\} \) |
| \( A \supset B \) | A is a proper superset of B | \( \{1, 2\} \supset \{1\} \) |
| \( A \supseteq B \) | A is a superset of B | \( \{1, 2\} \supseteq \{1, 2\} \) |
Set Operations
| Notation | Name | Meaning |
|---|---|---|
| \( A \cup B \) | Union | Elements in A or B (or both) |
| \( A \cap B \) | Intersection | Elements in both A and B |
| \( A \setminus B \) or \( A – B \) | Set difference | Elements in A but not in B |
| \( A^c \) or \( \bar{A} \) | Complement | Elements not in A |
| \( A \times B \) | Cartesian product | All ordered pairs (a, b) |
| \( \mathcal{P}(A) \) or \( 2^A \) | Power set | Set of all subsets of A |
| \( |A| \) or \( \#A \) | Cardinality | Number of elements in A |
Standard Number Sets
These are so common they get their own symbols (blackboard bold):
| Symbol | Name | Definition |
|---|---|---|
| \( \mathbb{N} \) | Natural numbers | \( \{0, 1, 2, 3, \ldots\} \) (sometimes excludes 0) |
| \( \mathbb{N}^+ \) or \( \mathbb{Z}^+ \) | Positive integers | \( \{1, 2, 3, \ldots\} \) |
| \( \mathbb{Z} \) | Integers | \( \{\ldots, -2, -1, 0, 1, 2, \ldots\} \) |
| \( \mathbb{Q} \) | Rational numbers | \( \{p/q : p \in \mathbb{Z}, q \in \mathbb{Z}^+\} \) |
| \( \mathbb{R} \) | Real numbers | All points on the number line |
| \( \mathbb{R}^+ \) | Positive reals | \( \{x \in \mathbb{R} : x > 0\} \) |
| \( \mathbb{C} \) | Complex numbers | \( \{a + bi : a, b \in \mathbb{R}\} \) |
| \( \mathbb{P} \) | Prime numbers | \( \{2, 3, 5, 7, 11, 13, \ldots\} \) |
| \( \mathbb{R}^n \) | n-dimensional real space | All n-tuples of real numbers |
The hierarchy: \( \mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R} \subset \mathbb{C} \)
Logic Symbols
Propositional Logic
| Symbol | Name | Meaning | Example |
|---|---|---|---|
| \( \land \) or \( \wedge \) | And (conjunction) | Both true | \( P \land Q \) |
| \( \lor \) or \( \vee \) | Or (disjunction) | At least one true | \( P \lor Q \) |
| \( \neg \) or \( \sim \) | Not (negation) | Opposite truth value | \( \neg P \) |
| \( \Rightarrow \) or \( \to \) | Implies | If P then Q | \( P \Rightarrow Q \) |
| \( \Leftrightarrow \) or \( \iff \) | If and only if | P and Q have same truth value | \( P \iff Q \) |
| \( \therefore \) | Therefore | Conclusion follows | \( \therefore Q \) |
| \( \because \) | Because | Reason given | \( \because P \) |
Quantifiers
| Symbol | Name | Read as | Example |
|---|---|---|---|
| \( \forall \) | Universal quantifier | “For all” or “for every” | \( \forall x \in \mathbb{R}, x^2 \geq 0 \) |
| \( \exists \) | Existential quantifier | “There exists” | \( \exists x : x^2 = 2 \) |
| \( \exists! \) | Unique existence | “There exists exactly one” | \( \exists! x : x + 1 = 2 \) |
| \( \nexists \) | Non-existence | “There does not exist” | \( \nexists x \in \mathbb{Q} : x^2 = 2 \) |
Relations and Operators
Equality and Inequality
| Symbol | Meaning |
|---|---|
| \( a = b \) | a equals b (identical values) |
| \( a \neq b \) | a is not equal to b |
| \( a \equiv b \) | a is identically equal to b (true for all values) |
| \( a \approx b \) | a is approximately equal to b |
| \( a \sim b \) | a is similar to b, or a is distributed as b |
| \( a \propto b \) | a is proportional to b |
| \( a := b \) or \( a \triangleq b \) | a is defined as b |
Ordering
| Symbol | Meaning |
|---|---|
| \( a < b \) | a is less than b |
| \( a > b \) | a is greater than b |
| \( a \leq b \) or \( a \le b \) | a is less than or equal to b |
| \( a \geq b \) or \( a \ge b \) | a is greater than or equal to b |
| \( a \ll b \) | a is much less than b |
| \( a \gg b \) | a is much greater than b |
Arithmetic Operators
| Symbol | Operation | Notes |
|---|---|---|
| \( + \) | Addition | — |
| \( – \) | Subtraction | — |
| \( \times \) or \( \cdot \) | Multiplication | \( \cdot \) preferred in algebra |
| \( \div \) or \( / \) | Division | \( / \) or fraction bar preferred |
| \( \pm \) | Plus or minus | Two possible values |
| \( \mp \) | Minus or plus | Opposite of \( \pm \) |
| \( a^n \) | Exponentiation | a to the power n |
| \( \sqrt{a} \) or \( a^{1/2} \) | Square root | — |
| \( \sqrt[n]{a} \) or \( a^{1/n} \) | nth root | — |
| \( |a| \) | Absolute value | Distance from zero |
| \( \lfloor x \rfloor \) | Floor function | Greatest integer ≤ x |
| \( \lceil x \rceil \) | Ceiling function | Smallest integer ≥ x |
Interval Notation
| Notation | Set-Builder | Description |
|---|---|---|
| \( (a, b) \) | \( \{x : a < x < b\} \) | Open interval (excludes endpoints) |
| \( [a, b] \) | \( \{x : a \leq x \leq b\} \) | Closed interval (includes endpoints) |
| \( [a, b) \) | \( \{x : a \leq x < b\} \) | Half-open (includes a, excludes b) |
| \( (a, b] \) | \( \{x : a < x \leq b\} \) | Half-open (excludes a, includes b) |
| \( (-\infty, b) \) | \( \{x : x < b\} \) | All values less than b |
| \( (a, \infty) \) | \( \{x : x > a\} \) | All values greater than a |
| \( (-\infty, \infty) \) | \( \mathbb{R} \) | All real numbers |
Warning: Infinity is never included in an interval. Always use \( ( \) with \( \infty \), never \( [ \).
Calculus Notation
Special Limits and Values
| Notation | Meaning |
|---|---|
| \( f(x) \to L \) as \( x \to a \) | f(x) approaches L as x approaches a |
| \( f(x) \to \infty \) | f(x) increases without bound |
| \( f(x) \to -\infty \) | f(x) decreases without bound |
| \( O(g(x)) \) | Big-O: bounded above by g(x) asymptotically |
| \( o(g(x)) \) | Little-o: dominated by g(x) asymptotically |
| \( f \sim g \) | f is asymptotically equivalent to g |
Vector Calculus Operators
| Notation | Name | Result Type |
|---|---|---|
| \( \nabla f \) or \( \text{grad } f \) | Gradient | Vector field |
| \( \nabla \cdot \mathbf{F} \) or \( \text{div } \mathbf{F} \) | Divergence | Scalar field |
| \( \nabla \times \mathbf{F} \) or \( \text{curl } \mathbf{F} \) | Curl | Vector field |
| \( \nabla^2 f \) or \( \Delta f \) | Laplacian | Scalar field |
The symbol \( \nabla \) (nabla or del) is defined as \( \nabla = \left( \frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z} \right) \).
Linear Algebra Notation
| Notation | Name | Meaning |
|---|---|---|
| \( A^T \) or \( A’ \) | Transpose | Swap rows and columns |
| \( A^{-1} \) | Inverse | Matrix such that \( AA^{-1} = I \) |
| \( A^* \) or \( A^\dagger \) | Conjugate transpose | Transpose + complex conjugate |
| \( \det(A) \) or \( |A| \) | Determinant | Scalar value from square matrix |
| \( \text{tr}(A) \) | Trace | Sum of diagonal elements |
| \( \text{rank}(A) \) | Rank | Dimension of column space |
| \( \ker(A) \) or \( \text{null}(A) \) | Kernel/null space | Solutions to Ax = 0 |
| \( I \) or \( I_n \) | Identity matrix | 1s on diagonal, 0s elsewhere |
| \( \mathbf{0} \) | Zero matrix/vector | All entries are zero |
| \( \mathbf{u} \cdot \mathbf{v} \) | Dot product | \( \sum u_i v_i \) |
| \( \mathbf{u} \times \mathbf{v} \) | Cross product | Vector perpendicular to both |
| \( \|\mathbf{v}\| \) | Norm (length) | \( \sqrt{\mathbf{v} \cdot \mathbf{v}} \) |
| \( \langle \mathbf{u}, \mathbf{v} \rangle \) | Inner product | Generalized dot product |
Probability and Statistics Notation
| Notation | Meaning |
|---|---|
| \( P(A) \) | Probability of event A |
| \( P(A|B) \) | Probability of A given B |
| \( P(A \cap B) \) | Probability of A and B |
| \( P(A \cup B) \) | Probability of A or B |
| \( E[X] \) or \( \mu \) | Expected value (mean) |
| \( \text{Var}(X) \) or \( \sigma^2 \) | Variance |
| \( \text{SD}(X) \) or \( \sigma \) | Standard deviation |
| \( \text{Cov}(X, Y) \) | Covariance |
| \( \rho_{XY} \) or \( \text{Corr}(X,Y) \) | Correlation coefficient |
| \( X \sim \text{Dist} \) | X is distributed as Dist |
| \( X \sim N(\mu, \sigma^2) \) | X is normally distributed |
| \( \bar{x} \) | Sample mean |
| \( s^2 \) | Sample variance |
| \( \hat{\theta} \) | Estimator of parameter θ |
Number Theory Notation
| Notation | Name | Meaning |
|---|---|---|
| \( a \mid b \) | Divides | a divides b evenly |
| \( a \nmid b \) | Does not divide | a does not divide b evenly |
| \( a \equiv b \pmod{n} \) | Congruence | a and b have same remainder mod n |
| \( \gcd(a, b) \) | Greatest common divisor | Largest number dividing both |
| \( \text{lcm}(a, b) \) | Least common multiple | Smallest number both divide |
| \( \phi(n) \) | Euler’s totient | Count of integers coprime to n |
| \( \lfloor x \rfloor \) | Floor | Greatest integer ≤ x |
| \( \lceil x \rceil \) | Ceiling | Least integer ≥ x |
| \( \{x\} \) | Fractional part | \( x – \lfloor x \rfloor \) |
Geometry Notation
| Notation | Meaning |
|---|---|
| \( \overline{AB} \) | Line segment from A to B |
| \( \overleftrightarrow{AB} \) | Line through A and B |
| \( \overrightarrow{AB} \) | Ray from A through B |
| \( |AB| \) or \( d(A,B) \) | Distance from A to B |
| \( \angle ABC \) | Angle at vertex B |
| \( \triangle ABC \) | Triangle with vertices A, B, C |
| \( \parallel \) | Is parallel to |
| \( \perp \) | Is perpendicular to |
| \( \cong \) | Is congruent to |
| \( \sim \) | Is similar to |
Special Constants
| Symbol | Name | Value |
|---|---|---|
| \( \pi \) | Pi | 3.14159265358979… |
| \( e \) | Euler’s number | 2.71828182845904… |
| \( i \) | Imaginary unit | \( \sqrt{-1} \) |
| \( \varphi \) or \( \phi \) | Golden ratio | \( \frac{1 + \sqrt{5}}{2} \approx 1.618 \) |
| \( \gamma \) | Euler-Mascheroni constant | 0.57721566490153… |
| \( \infty \) | Infinity | Unbounded growth |
Special Functions
| Notation | Name | Definition/Use |
|---|---|---|
| \( \sin, \cos, \tan \) | Trigonometric functions | Ratios in right triangles |
| \( \arcsin, \arccos, \arctan \) | Inverse trig functions | Also written \( \sin^{-1} \), etc. |
| \( \sinh, \cosh, \tanh \) | Hyperbolic functions | Exponential-based analogs |
| \( \ln x \) or \( \log x \) | Natural logarithm | Base e |
| \( \log_b x \) | Logarithm base b | \( \log_b x = \frac{\ln x}{\ln b} \) |
| \( \exp(x) \) | Exponential function | \( e^x \) |
| \( \Gamma(n) \) | Gamma function | Extension of factorial: \( \Gamma(n) = (n-1)! \) |
| \( \zeta(s) \) | Riemann zeta function | \( \sum_{n=1}^{\infty} n^{-s} \) |
| \( \delta_{ij} \) | Kronecker delta | 1 if i = j, else 0 |
| \( \delta(x) \) | Dirac delta function | Infinite at 0, integral = 1 |
Typography Conventions
How something is written often carries meaning:
| Style | Typical Use | Example |
|---|---|---|
| Italic lowercase | Variables, parameters | \( x, y, a, b \) |
| Italic uppercase | Sets, random variables | \( A, B, X, Y \) |
| Bold | Vectors, matrices | \( \mathbf{v}, \mathbf{A} \) |
| Blackboard bold | Number sets | \( \mathbb{R}, \mathbb{Z}, \mathbb{C} \) |
| Calligraphic | Special sets, spaces | \( \mathcal{L}, \mathcal{H} \) |
| Fraktur | Ideals, Lie algebras | \( \mathfrak{a}, \mathfrak{g} \) |
| Roman (upright) | Functions, operators, units | \( \sin, \exp, \text{kg} \) |
| Hat/caret | Unit vectors, estimators | \( \hat{x}, \hat{\theta} \) |
| Bar/overline | Averages, closures, conjugates | \( \bar{x}, \overline{A} \) |
| Tilde | Approximations, transforms | \( \tilde{f}, \tilde{x} \) |
Common Abbreviations
| Abbreviation | Meaning | Usage |
|---|---|---|
| iff | if and only if | Logical equivalence |
| w.l.o.g. | without loss of generality | Simplifying assumption |
| s.t. | such that | Conditions |
| w.r.t. | with respect to | Derivatives, relations |
| a.e. | almost everywhere | Measure theory |
| a.s. | almost surely | Probability |
| i.i.d. | independent and identically distributed | Statistics |
| QED | quod erat demonstrandum | End of proof |
| \( \square \) or \( \blacksquare \) | End of proof marker | Tombstone/halmos |
Frequently Asked Questions
What’s the difference between = and ≡?
The equals sign (=) means two expressions have the same value for the current context. The identity symbol (≡) means two expressions are identical for ALL possible values of variables. For example, x + x = 6 is true only when x = 3, but x + x ≡ 2x is true for every x. In some contexts, ≡ also denotes congruence modulo n.
Why are there so many ways to write derivatives?
Different notations serve different purposes. Leibniz notation (dy/dx) emphasizes that derivative is a ratio of differentials and makes chain rule transparent. Lagrange notation (f'(x)) is compact for higher-order derivatives. Newton’s dot notation (ẏ) is used in physics for time derivatives. Each field developed its own conventions before standardization.
Does ℕ include 0 or not?
This varies by convention and drives mathematicians crazy. In logic, set theory, and computer science, ℕ typically includes 0. In analysis and algebra, it often starts at 1. When in doubt, check how the author defines it. To be explicit, use ℕ₀ for {0, 1, 2, …} or ℕ⁺ or ℤ⁺ for {1, 2, 3, …}.
What’s the difference between ∂ and d?
The straight d is used for ordinary derivatives of single-variable functions. The curly ∂ is used for partial derivatives of multi-variable functions. If f depends only on x, use df/dx. If f depends on x, y, z, use ∂f/∂x to mean “differentiate with respect to x while holding y and z constant.”
What does the colon in set notation mean?
In set-builder notation {x : P(x)}, the colon means “such that.” It separates the variable from the condition it must satisfy. Some authors use a vertical bar | instead: {x | P(x)}. Both mean the same thing: “the set of all x such that property P holds.”
Why is factorial of 0 equal to 1?
By convention, the empty product (multiplying zero things together) equals 1. This makes formulas work correctly. For example, the binomial coefficient C(n,0) = n!/(0! × n!) should equal 1 (there’s one way to choose nothing). That requires 0! = 1. It also preserves the recursion n! = n × (n-1)! when n = 1.
What’s the difference between ⊂ and ⊆?
⊆ means “subset or equal” (A is contained in B, possibly equal to B). ⊂ traditionally means “proper subset” (A is contained in B and A ≠ B). However, some authors use ⊂ to mean ⊆. To avoid ambiguity, use ⊊ for strict/proper subset. Always check the author’s conventions.
What do the different brackets mean?
Parentheses () are for function arguments, grouping, and open intervals. Square brackets [] are for closed intervals, floor/ceiling, and sometimes matrices. Curly braces {} define sets. Angle brackets ⟨⟩ denote inner products or expected values. Vertical bars |x| mean absolute value or cardinality. Double bars ‖v‖ mean vector norm.
What’s the difference between log and ln?
ln always means natural logarithm (base e). log is ambiguous: in pure math and most of Europe, log usually means natural log (base e). In engineering and American textbooks, log often means base 10. In computer science, log often means base 2. When precision matters, write log_e, log₁₀, or log₂ explicitly.
How do I type mathematical symbols?
For documents: use LaTeX (\sum, \int, \alpha). For quick typing: learn Unicode shortcuts or use symbol palettes. On Mac, use the Character Viewer. On Windows, use Win+. for emoji/symbol picker. Online tools like Detexify let you draw symbols to find LaTeX commands. Many text editors have math input modes or plugins.
