Mathematics Glossary — Key Terms and Definitions

graph TD
    A[Limits] --> B[Continuity]
    B --> C[Differentiation]
    C --> D[Integration]

Binomial: An algebraic expression with two terms, such as (a + b). The binomial theorem provides a formula for expanding (a + b)ⁿ.

Completing the Square: A method for solving quadratic equations by rewriting ax² + bx + c in the form a(x - h)² + k. Used to derive the quadratic formula.

Discriminant: For a quadratic equation ax² + bx + c = 0, the discriminant is Δ = b² - 4ac. Determines the nature of roots: positive (two real), zero (one repeated), negative (complex).

Exponent: The power to which a number is raised. In aⁿ, n is the exponent. Laws include aᵐ · aⁿ = aᵐ⁺ⁿ and (aᵐ)ⁿ = aᵐⁿ.

Function: A relation where each input has exactly one output, written as f(x). The domain is the set of inputs; the range is the set of outputs.

Inequality: A mathematical statement comparing two expressions using <, >, ≤, ≥, or ≠. Solutions are intervals or sets of values.

Linear Equation: An equation of the form y = mx + c, where m is the slope and c is the y-intercept. Graphs as a straight line.

Logarithm: The inverse of exponentiation. log_b(x) = n means bⁿ = x. Properties include log_b(xy) = log_b(x) + log_b(y).

Matrix: A rectangular array of numbers arranged in rows and columns, denoted [aᵢⱼ]. Used to represent linear transformations and solve systems of equations.

Polynomial: An expression consisting of variables and coefficients involving only addition, subtraction, multiplication, and non-negative integer exponents. Degree is the highest power.

Quadratic Formula: The solutions to ax² + bx + c = 0 are x = (-b ± √(b² - 4ac)) / (2a). Derived from completing the square.

Quadratic Function: A polynomial function of degree 2, f(x) = ax² + bx + c. Its graph is a parabola opening upward if a > 0.

Set: A well-defined collection of distinct objects, denoted {a, b, c}. Operations include union (∪), intersection (∩), and complement (A’).

System of Equations: Two or more equations with the same variables. Can be solved by substitution, elimination, or matrix methods.

Variable: A symbol representing an unknown quantity in mathematical expressions and equations.

Antiderivative: A function F(x) such that F’(x) = f(x). Also called an indefinite integral. Related: Integral.

Chain Rule: The derivative of a composite function: d/dx[f(g(x))] = f’(g(x)) · g’(x). Essential for differentiating nested functions.

Convergence: A sequence or series approaches a finite limit. The series Σaₙ converges if partial sums approach a finite value.

Critical Point: A point where f’(x) = 0 or f’(x) is undefined. May be a local maximum, local minimum, or saddle point.

Derivative: The instantaneous rate of change of a function: f’(x) = lim[h→0] (f(x+h) - f(x))/h. Represents slope of tangent line.

Differential Equation: An equation involving derivatives of a function. Solutions are functions that satisfy the equation. Types: ODE, PDE.

Differentiation: The process of finding derivatives. Rules include power rule, product rule, quotient rule, and chain rule.

Fundamental Theorem of Calculus: Links differentiation and integration: ∫ₐᵇ f(x)dx = F(b) - F(a), where F’(x) = f(x).

Improper Integral: An integral with infinite limits or discontinuous integrand. Evaluated as a limit: ∫₁^∞ f(x)dx = lim[b→∞] ∫₁ᵇ f(x)dx.

Indefinite Integral: The family of antiderivatives: ∫f(x)dx = F(x) + C. C is the constant of integration.

Inflection Point: A point where the concavity of a function changes. Occurs where f”(x) = 0 or f”(x) is undefined and changes sign.

Integral: The limit of a Riemann sum representing the area under a curve: ∫ₐᵇ f(x)dx = lim[n→∞] Σf(xᵢ)Δx. See also Antiderivative.

Integration: The process of finding integrals. Methods include substitution, integration by parts, partial fractions, and trigonometric substitution.

Limit: The value a function approaches as the input approaches a point: lim[x→c] f(x) = L. Foundation of calculus.

Local Maximum: A point where f(x) is greater than all nearby values. Occurs at critical points where f”(x) < 0.

Local Minimum: A point where f(x) is less than all nearby values. Occurs at critical points where f”(x) > 0.

Mean Value Theorem: If f is continuous on [a,b] and differentiable on (a,b), there exists c ∈ (a,b) such that f’(c) = (f(b) - f(a))/(b - a).

Partial Derivative: The derivative of a multivariable function with respect to one variable, holding others constant. Denoted ∂f/∂x.

Riemann Sum: An approximation of an integral using rectangles: Σf(xᵢ*)Δx. As n → ∞, it approaches the exact integral.

Series: The sum of a sequence of terms. A convergent series has a finite sum; a divergent series does not.

Taylor Series: An infinite series representation of a function: f(x) = Σf⁽ⁿ⁾(a)/n! · (x-a)ⁿ. Approximates functions near a point.

Basis: A set of linearly independent vectors that span a vector space. Every vector in the space can be uniquely expressed as a linear combination of basis vectors.

Column Space: The set of all linear combinations of a matrix’s columns. Equal to the range of the matrix transformation.

Determinant: A scalar value computed from a square matrix that indicates whether the matrix is invertible. det(A) = 0 means A is singular.

Eigenvalue: A scalar λ such that Av = λv for some non-zero vector v. Found by solving det(A - λI) = 0.

Eigenvector: A non-zero vector v that, when multiplied by matrix A, results in a scalar multiple of itself: Av = λv.

Linear Combination: An expression of the form c₁v₁ + c₂v₂ + … + cₙvₙ, where cᵢ are scalars and vᵢ are vectors.

Linear Independence: Vectors are linearly independent if no vector can be written as a linear combination of the others. Equivalent to the only solution of c₁v₁ + … + cₙvₙ = 0 being all cᵢ = 0.

Linear Transformation: A function T: V → W satisfying T(u + v) = T(u) + T(v) and T(cv) = cT(v). Represented by matrix multiplication.

Null Space: The set of all vectors x such that Ax = 0. A subspace of the domain.

Rank: The dimension of the column space (or row space) of a matrix. Equals the number of linearly independent rows or columns.

Row Space: The set of all linear combinations of a matrix’s rows. Has the same dimension as the column space.

Scalar Multiplication: Multiplying a vector by a scalar, scaling its magnitude without changing direction (unless negative).

Vector Space: A set of vectors closed under vector addition and scalar multiplication, satisfying eight axioms. Examples include ℝⁿ and function spaces.

Vector: A quantity with both magnitude and direction, represented as an ordered tuple of components. Operations include addition and scalar multiplication.

Cauchy Sequence: A sequence where terms become arbitrarily close to each other: for all ε > 0, there exists N such that |aₙ - aₘ| < ε for all n, m > N.

Compactness: A property of sets where every open cover has a finite subcover. In ℝⁿ, equivalent to being closed and bounded (Heine-Borel theorem).

Continuity: A function f is continuous at a point c if lim[x→c] f(x) = f(c). Intuitively, the function has no jumps or breaks.

Convergence: A sequence {aₙ} converges to L if for all ε > 0, there exists N such that |aₙ - L| < ε for all n > N.

Divergence: A sequence or series that does not approach a finite limit. May diverge to ±∞ or oscillate.

Domain: The set of inputs for which a function is defined.

Extreme Value Theorem: A continuous function on a closed interval [a,b] attains both a maximum and minimum value.

Intermediate Value Theorem: If f is continuous on [a,b] and k is between f(a) and f(b), there exists c ∈ [a,b] such that f(c) = k.

L’Hôpital’s Rule: For indeterminate forms 0/0 or ∞/∞, lim[f(x)/g(x)] = lim[f’(x)/g’(x)], provided the limit on the right exists.

Limit Superior: The largest limit point of a sequence. Also called lim sup. Measures the “long-term” behavior of oscillating sequences.

Monotonic Sequence: A sequence that is either entirely non-increasing or non-decreasing. Monotonic bounded sequences converge.

One-to-One (Injection): A function where each output corresponds to exactly one input. f(x₁) = f(x₂) implies x₁ = x₂.

Onto (Surjection): A function where every element in the codomain is the image of at least one element in the domain.

Sequence: An ordered list of numbers a₁, a₂, a₃, … indexed by natural numbers.

Series: The sum of a sequence: Σₙ₌₁^∞ aₙ = a₁ + a₂ + a₃ + …

Topology: The study of properties preserved under continuous deformations. Deals with open sets, continuity, and connectedness.

Bayes’ Theorem: P(A|B) = P(B|A)P(A)/P(B). Relates conditional probabilities and is fundamental to Bayesian inference.

Combination: Selection of items where order doesn’t matter: C(n,r) = n! / (r!(n-r)!). Also written as “n choose r.”

Conditional Probability: The probability of event A given event B has occurred: P(A|B) = P(A ∩ B) / P(B).

Continuous Distribution: A probability distribution with a continuous random variable. Defined by a probability density function (PDF).

Correlation: A measure of the linear relationship between two variables, ranging from -1 to 1. Correlation does not imply causation.

Discrete Distribution: A probability distribution with a discrete random variable. Defined by a probability mass function (PMF).

Expected Value: The average value of a random variable: E[X] = ΣxᵢP(xᵢ) for discrete variables.

Mean: The arithmetic average of a set of values. For a sample: x̄ = (1/n)Σxᵢ.

Median: The middle value when data is ordered. For even sample sizes, the average of the two middle values.

Normal Distribution: A bell-shaped continuous distribution: f(x) = (1/σ√(2π))e^(-(x-μ)²/2σ²). Characterized by mean μ and standard deviation σ.

Permutation: Arrangement of items where order matters: P(n,r) = n! / (n-r)!.

Probability: The likelihood of an event occurring, ranging from 0 to 1. P(A) = (favorable outcomes) / (total outcomes).

Random Variable: A variable whose value is determined by the outcome of a random experiment. Can be discrete or continuous.

Sample Space: The set of all possible outcomes of a random experiment, denoted S.

Standard Deviation: The square root of variance: σ = √(Σ(xᵢ - μ)²/N). Measures the spread of data.

Variance: The average of squared deviations from the mean: Var(X) = E[(X - μ)²]. Measures dispersion.

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