Math Glossary

The absolute value |x| is the distance from x to 0 on the number line — always non-negative. |3| = 3, |-3| = 3.

Amplitude is the peak deviation of a wave from its center. For y = A sin(Bx), the amplitude is |A|. Larger amplitude = taller wave.

An angle measures the rotation between two rays sharing a common endpoint (the vertex). Common units: degrees (full circle = 360°) and radians (full circle = 2π).

Area measures the size of a 2D region — how much surface it covers. Units are squared (cm², m²). Each shape has its own area formula.

Bayes' theorem reverses conditional probabilities: P(A|B) = P(B|A)P(A)/P(B). The foundation of Bayesian inference, medical testing, and ML.

A binomial is a polynomial with exactly two terms, like x + 3 or 2x² - 5. Distinguished from monomials (1 term) and trinomials (3 terms).

The chi-square test compares observed counts to expected counts in categorical data. χ² = Σ(O−E)²/E. Used for goodness-of-fit and independence tests.

A circle is the set of all points in a plane equidistant from a center. The constant distance is the radius; the longest chord through the center is the diameter (2× radius).

A coefficient is the numerical factor in front of a variable in an algebraic expression. In 5x², the coefficient is 5.

A confidence interval gives a range of plausible values for a population parameter, with a stated confidence level (e.g. 95%) describing the procedure's long-run reliability.

Two figures are congruent if one can be transformed into the other by rigid motion (translation, rotation, reflection) — same shape AND same size.

A function is continuous at a point if its value there equals the limit of its values as inputs approach the point — no jumps, holes, or asymptotes.

A sequence or series converges if it approaches a finite limit. Otherwise it diverges. Convergence tests determine which case applies.

A coordinate system assigns numbers to points in space. Cartesian (x, y) is most common in 2D; polar (r, θ) is used for circular symmetry.

Correlation measures the strength and direction of the linear relationship between two variables. The Pearson coefficient r is in [-1, 1]: 1 = perfect positive, -1 = perfect negative, 0 = no linear relationship.

Cosecant is the reciprocal of sine: csc(θ) = 1/sin(θ). Domain excludes angles where sin = 0 (i.e. multiples of π).

Cotangent is the reciprocal of tangent: cot(θ) = cos(θ)/sin(θ). Domain excludes angles where sin = 0.

The curl of a vector field measures local rotation. ∇×F gives a vector pointing along the rotation axis with magnitude proportional to spin rate.

A derivative measures the instantaneous rate of change of a function — equivalently, the slope of the tangent line to the function's graph at a single point.

The divergence of a vector field measures the net "outflow" at each point. ∇·F > 0 means a source; < 0 a sink. Foundational for fluid dynamics and electromagnetism.

The domain of a function is the set of all valid inputs; the range is the set of all possible outputs. Together they fully describe what the function maps.

An exponent indicates how many times a base is multiplied by itself. In aⁿ, n is the exponent and a is the base. Example: 2³ = 2·2·2 = 8.

Factoring an expression means rewriting it as a product of simpler expressions, e.g. x²+5x+6 = (x+2)(x+3). It is the reverse of expansion.

A function is a rule that assigns exactly one output to each input. Notation: f(x) = ... means "the output of f when x is the input."

The gradient of a multivariable function f(x,y,...) is the vector of partial derivatives. It points in the direction of steepest ascent and is the foundation of gradient descent.

Hypothesis testing decides between two competing claims about a population using sample data. We compute a test statistic and reject the null hypothesis if the p-value is small.

Implicit differentiation finds dy/dx when y is defined implicitly by an equation (like x²+y²=25), without first solving for y.

An improper integral has either an infinite limit or an integrand that is unbounded somewhere on the interval. Evaluated as a limit of proper integrals.

An inequality compares two expressions using <, ≤, >, or ≥. Solutions form intervals or unions of intervals on the number line.

An integral is the continuous analogue of summation — most commonly the area under a curve. Definite integrals produce numbers; indefinite integrals produce antiderivative functions.

Inverse trig functions (arcsin, arccos, arctan) recover the angle from a trig ratio. arcsin(y) = x means sin(x) = y, with restricted output range.

L'Hôpital's rule resolves indeterminate limits of the form 0/0 or ∞/∞ by replacing the limit with the limit of the derivatives' ratio.

The law of cosines generalises the Pythagorean theorem to any triangle: c² = a² + b² − 2ab cos(C). Use for SSS or SAS triangle problems.

The law of sines relates the sides of any triangle to the sines of opposite angles: a/sin(A) = b/sin(B) = c/sin(C).

A limit describes the value a function approaches as its input gets arbitrarily close to a target — without necessarily reaching it. Limits underpin both derivatives and integrals.

A linear equation is an equation whose graph is a straight line. In one variable: ax + b = 0. In two variables: ax + by = c.

Linear regression fits a straight line to data: y = mx + b. The line minimises the sum of squared vertical distances to the points (least squares).

A logarithm is the inverse of exponentiation: log_a(b) = c means a^c = b. It answers "what power of a gives b?"

The mean — also called the arithmetic average — is the sum of a set of values divided by the count of values. It is the most common single-number summary of a data set.

The Mean Value Theorem states that for a smooth function on [a,b], there is a point c where f'(c) equals the average rate of change (f(b)−f(a))/(b−a).

The median is the middle value of a sorted dataset. For even-sized data, it is the average of the two middle values. Robust to outliers.

The mode is the value that appears most frequently in a dataset. A dataset can have one mode, multiple modes, or no mode. Useful for categorical data.

The normal (Gaussian) distribution is a bell-shaped probability curve fully described by its mean μ and standard deviation σ. Foundation of much of statistics.

Optimization in calculus means finding maximum or minimum values of a function. Set f'(x) = 0 to find critical points, then test for max/min.

A p-value is the probability of observing data at least as extreme as your sample, assuming the null hypothesis is true. Small p means evidence against H₀.

A parallelogram is a quadrilateral with both pairs of opposite sides parallel. Includes rectangles, rhombi, and squares as special cases.

A partial derivative measures how a multivariable function changes when only one variable changes, holding others constant. Notation: ∂f/∂x.

The k-th percentile is the value below which k% of observations fall. The 50th percentile is the median; the 25th and 75th are the quartiles.

Perimeter is the total length around a 2D shape. For a circle the perimeter is called the circumference: C = 2πr.

The period is the horizontal length over which a trig function completes one full cycle. sin and cos have period 2π; tan has period π.

Phase shift is a horizontal translation of a periodic function. For y = sin(Bx + C), the phase shift is -C/B (positive = right, negative = left).

A polygon is a closed 2D shape with straight sides. Common types: triangle (3), quadrilateral (4), pentagon (5), hexagon (6), and so on.

A polynomial is a sum of terms, each consisting of a constant times a variable raised to a non-negative integer power. Examples: 3x²+2x-7, x³-4x+1.

The degree of a polynomial is the highest exponent on its variable. Degree 1 = linear, 2 = quadratic, 3 = cubic, 4 = quartic.

The Pythagorean theorem states that in any right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides: a² + b² = c².

A quadratic equation is a second-degree polynomial equation in one variable, written as ax² + bx + c = 0 with a ≠ 0. Its graph is a parabola.

Quartiles split a dataset into four equal parts. Q1 (25th percentile), Q2 (median, 50th), Q3 (75th). The interquartile range Q3-Q1 is a robust spread measure.

A radian is the angle subtended by an arc whose length equals the radius. A full circle is 2π radians (≈ 6.28). Required unit for calculus.

A radical denotes a root: √a is the square root, ∛a the cube root, and ⁿ√a the n-th root. Radicals are inverses of exponentiation.

A rational expression is a fraction whose numerator and denominator are polynomials, e.g. (x²-1)/(x+2). Simplify by factoring and cancelling common factors.

Related rates problems link the rates of change of two or more variables connected by an equation. Use implicit differentiation with respect to time.

A Riemann sum approximates the area under a curve by dividing the region into rectangles. As the rectangles get thinner, the sum converges to the definite integral.

Secant is the reciprocal of cosine: sec(θ) = 1/cos(θ). Domain excludes angles where cos = 0 (π/2 + kπ).

A series is the sum of a sequence — finite or infinite. Whether an infinite series adds up to a finite number is determined by convergence tests.

Two figures are similar if one is a scaled copy of the other — same shape, possibly different size. All corresponding angles are equal; all corresponding sides are proportional.

Sine, cosine, and tangent are the three basic trigonometric functions, defined as ratios of sides in a right triangle and extended to all real numbers via the unit circle.

Standard deviation measures how spread out a data set is around its mean. A small standard deviation means values cluster tightly; a large one means they are scattered.

The t-distribution is bell-shaped like the normal but with heavier tails. Used for inference about means when sample size is small or σ is unknown.

Surface area is the total area of all faces of a 3D solid. Different from volume: surface area is in squared units (cm²), volume in cubed.

A tangent line touches a curve at exactly one point and matches the curve's direction there. For circles, a tangent is perpendicular to the radius at the point of tangency.

A Taylor series approximates a smooth function as an infinite polynomial built from its derivatives at a single point. Truncating gives polynomial approximations.

A trapezoid is a quadrilateral with at least one pair of parallel sides (called bases). Area = (1/2)(b₁+b₂)h.

A triangle is a three-sided polygon whose interior angles always sum to 180°. Classified by sides (equilateral, isosceles, scalene) or angles (acute, right, obtuse).

Trig identities are equations relating trigonometric functions that hold for all valid angles, e.g. sin²θ + cos²θ = 1. Used to simplify expressions and solve equations.

A trinomial is a polynomial with exactly three terms, e.g. x² + 5x + 6. The most common type encountered in factoring practice.

The unit circle is the circle of radius 1 centered at the origin. It defines the trigonometric functions for all real angles, not just acute ones.

Variance measures the spread of a dataset around its mean. It is the average of squared deviations. Standard deviation is the square root of variance.

A vector is a quantity with both magnitude and direction. Notation: ⟨x, y⟩ or ⟨x, y, z⟩. Vectors add componentwise and underpin physics, graphics, and ML.

Volume measures the 3D space occupied by a solid. Units are cubed (cm³, m³). Each shape has its own formula; calculus generalises via integration.

A z-score measures how many standard deviations a value lies above or below the mean. z = (x − μ) / σ. Used for comparing values across distributions and probability lookups.