MATH SYMBOLS: DECODED

Your 8-Bit Glossary for Computer Science Mathematics

Calculus

$\lim_{x \to a}$

Limit

Describes the value that a function approaches as the input approaches some value 'a'.

$\frac{df}{dx}$ or $f'(x)$

Derivative

Represents the instantaneous rate of change of a function, or the slope of the tangent line.

$\int_{a}^{b} f(x) \,dx$

Definite Integral

Calculates the net area under the curve of a function $f(x)$ from $x=a$ to $x=b$.

$\sum_{i=1}^{n}$

Summation (Sigma)

Represents the sum of a sequence of terms, from index $i=1$ to $n$.

$\varepsilon, \delta$

Epsilon, Delta

Used in the formal definition of limits to represent arbitrarily small positive numbers.

$\infty$

Infinity

A concept representing a quantity without bound or end.

Multivariable Calculus

$\frac{\partial f}{\partial x}$

Partial Derivative

The derivative of a multivariable function with respect to one variable, holding others constant.

$\nabla f$

Gradient (Nabla)

A vector of all partial derivatives of a function. It points in the direction of the steepest ascent.

$\iint_R f(x,y) \,dA$

Double Integral

Calculates the volume under the surface $z=f(x,y)$ over a region R in the xy-plane.

$\oint_C$

Line Integral over a Closed Path

An integral evaluated along a closed curve or path, often used in vector calculus theorems.

$\nabla \cdot \mathbf{F}$ (div)

Divergence

Measures the magnitude of a vector field's source or sink at a given point.

$\nabla \times \mathbf{F}$ (curl)

Curl

Measures the rotation or circulation of a vector field at a given point.

Differential Equations

$y', y''$

First & Second Derivatives

Represent the first and second derivatives of a function y with respect to its variable (e.g., x or t).

$\mu(x)$

Mu / Integrating Factor

A special function used to solve linear first-order differential equations by making them integrable.

$\mathcal{L}\{f(t)\}$

Laplace Transform

Transforms a function of time $f(t)$ into a function of complex frequency $F(s)$. Used to simplify solving ODEs.

Linear Algebra

$A^T, A^{-1}$

Transpose, Inverse

Transpose ($A^T$) swaps rows and columns. Inverse ($A^{-1}$) is a matrix that, when multiplied by A, yields the identity matrix.

$\det(A)$

Determinant

A scalar value that provides information about a square matrix, such as whether it's invertible.

$\mathbf{v}$

Vector

An object with both magnitude and direction, often represented as an array of numbers.

$\lambda, \mathbf{v}$ in $A\mathbf{v} = \lambda\mathbf{v}$

Eigenvalue & Eigenvector

An eigenvector $\mathbf{v}$ of a matrix A is a vector that is only scaled by a factor $\lambda$ (the eigenvalue) when A is applied to it.

$\langle \mathbf{u}, \mathbf{v} \rangle$

Inner Product (Dot Product)

A way to multiply two vectors to get a scalar, related to the projection of one vector onto another.

$\|\mathbf{v}\|$

Norm (Magnitude)

The length or size of a vector, calculated as the square root of the inner product of the vector with itself.

Discrete Math

$\neg, \land, \lor, \rightarrow, \leftrightarrow$

Logical Operators

NOT, AND, OR, IMPLIES, and IF AND ONLY IF (IFF). Used to build logical statements.

$\forall, \exists$

Quantifiers

"For All" ($\forall$) and "There Exists" ($\exists$). Used to specify the scope of a logical statement.

$\in, \cup, \cap, \subseteq$

Set Operators

"Is an element of", Union, Intersection, and "Is a subset of".

$\binom{n}{k}$

Binomial Coefficient

"n choose k". Represents the number of ways to choose k items from a set of n items without regard to order.

Probability & Stats

$P(A)$

Probability of an Event

A number between 0 and 1 representing the likelihood that event A will occur.

$P(A|B)$

Conditional Probability

The probability of event A occurring, given that event B has already occurred.

$E[X]$

Expected Value

The long-run average value of a random variable X; its probability-weighted average.

$\text{Var}(X)$

Variance

Measures the spread or dispersion of a random variable around its expected value.

$\mu, \sigma$

Mu, Sigma

Typically represent the mean ($\mu$) and standard deviation ($\sigma$) of a population or probability distribution.

$\bar{x}, s$

Sample Mean & St. Dev.

Represent the mean ($\bar{x}$) and standard deviation ($s$) calculated from a sample of data.

Number Theory & Crypto

$a \equiv b \pmod{n}$

Congruence Modulo n

"a is congruent to b modulo n". Means a and b have the same remainder when divided by n.

$\gcd(a, b)$

Greatest Common Divisor

The largest positive integer that divides both a and b without leaving a remainder.

$\varphi(n)$

Euler's Totient Function

Counts the positive integers up to a given integer n that are relatively prime to n.

Algorithm Analysis

$O(g(n))$

Big O Notation

Describes the upper bound on the time or space complexity of an algorithm as the input size n grows.

$\Omega(g(n))$

Big Omega Notation

Describes the lower bound on the complexity of an algorithm.

$\Theta(g(n))$

Big Theta Notation

Describes a tight bound on the complexity, where the algorithm's growth is bounded both above and below.