Math glossary
Every recurring term across algebra, trigonometry, calculus, precalculus, statistics and linear algebra, each with its formula, a worked example and the solvers that use it. Jump to any term from the contents below.
Algebra
Discriminant
The expression Δ = b² − 4ac that determines the nature of the roots of a quadratic equation.
The discriminant tells you, before you compute anything else, what kind of roots a quadratic equation ax² + bx + c = 0 has. A positive discriminant means two distinct real roots; zero means a single repeated real root; a negative discriminant means two complex conjugate roots.
Formula Δ = b² − 4ac
Example For x² − 5x + 6 = 0, Δ = (−5)² − 4·1·6 = 25 − 24 = 1. Positive, so two distinct real roots: 2 and 3.
See also Coefficient
Coefficient
A constant multiplier in front of a variable or a term of a polynomial.
In an expression such as 3x² − 5x + 7, the numbers 3 and −5 are coefficients of x² and x; the constant 7 is sometimes called the constant term. Coefficients carry the magnitude information of a term while the variable carries the structure.
Example In −4x³ + x, the coefficient of x³ is −4 and the coefficient of x is 1 — not 0, because the x is present.
See also Discriminant
Slope
The steepness of a line — its rise over its run.
The slope m of a line measures how quickly y changes as x changes. For two points (x₁, y₁) and (x₂, y₂) on the line, m = (y₂ − y₁) / (x₂ − x₁). A vertical line has undefined slope; a horizontal line has slope 0.
Formula m = (y₂ − y₁) / (x₂ − x₁)
Example Through (1, 2) and (4, 11): m = (11 − 2) / (4 − 1) = 9 / 3 = 3.
See also Y-intercept · Derivative
Y-intercept
The y-coordinate where a curve crosses the y-axis.
For a line y = mx + b, the y-intercept is b — the value of y when x = 0. More generally, any curve's y-intercept is the function's value at x = 0; a curve may have one, none or several depending on whether the y-axis is crossed.
Example For y = 3x − 7, setting x = 0 gives y = −7, so the line crosses the y-axis at (0, −7).
See also Slope
Logarithm
The inverse of exponentiation: logₐ(x) is the exponent to which a must be raised to get x.
Logarithms turn multiplication into addition: logₐ(x·y) = logₐ(x) + logₐ(y). Common bases are 10 (the common logarithm log) and e (the natural logarithm ln). The change-of-base formula logₐ(x) = ln(x)/ln(a) lets any logarithm be computed from any other.
Formula logₐ(x) = y ⇔ a^y = x
Example log₂(32) = 5, because 2⁵ = 32. Changing base: log₂(32) = ln(32) / ln(2) gives the same 5.
Trigonometry
Sine
In a right triangle, the ratio of the side opposite an angle to the hypotenuse.
Sine is one of the three primary trig functions: sin θ = opposite / hypotenuse. Extended to the unit circle, sin θ is the y-coordinate of the point reached after rotating θ radians from the positive x-axis. It is periodic with period 2π and takes values in [−1, 1].
Formula sin θ = opposite / hypotenuse
Example In a 3–4–5 right triangle, the angle opposite the side of length 3 has sin θ = 3 / 5 = 0.6.
See also Cosine · Tangent (trig function) · Radian
Cosine
In a right triangle, the ratio of the side adjacent to an angle to the hypotenuse.
Cosine is the second primary trig function: cos θ = adjacent / hypotenuse. On the unit circle it is the x-coordinate after rotating θ radians from the positive x-axis. Sine and cosine are connected by sin²θ + cos²θ = 1.
Formula cos θ = adjacent / hypotenuse
Example In that same 3–4–5 triangle the angle has cos θ = 4 / 5 = 0.8, and 0.6² + 0.8² = 1.
See also Sine · Tangent (trig function)
Tangent (trig function)
The ratio sin θ / cos θ — undefined when cos θ = 0.
The tangent of an angle in a right triangle equals the opposite side divided by the adjacent side. It blows up at angles where the cosine is zero (90°, 270°, …) and repeats every π radians, half the period of sine and cosine.
Formula tan θ = sin θ / cos θ = opposite / adjacent
Example tan 45° = sin 45° / cos 45° = (√2/2) / (√2/2) = 1. At 90° the cosine is 0, so the tangent is undefined.
Radian
The natural unit of angle: a full turn is 2π radians.
One radian is the angle subtended at the centre of a circle by an arc equal in length to the radius. The conversion 1° = π/180 rad makes radians the unit of choice in calculus, where the derivative formulas for sin and cos assume the input is in radians.
Formula θ (rad) = θ (deg) × π / 180
Example 180° × π/180 = π radians. A right angle is π/2 ≈ 1.5708 radians, and a full turn is 2π ≈ 6.2832.
Pythagorean theorem
In a right triangle, the square of the hypotenuse equals the sum of the squares of the legs.
The Pythagorean theorem states a² + b² = c², where a and b are the legs and c is the hypotenuse of a right triangle. It also gives the Euclidean distance between two points in the plane and underpins the formulas for vector magnitudes.
Formula a² + b² = c²
Example With legs 5 and 12: 5² + 12² = 25 + 144 = 169, and √169 = 13, so the hypotenuse is 13.
Calculus
Derivative
The instantaneous rate of change of a function at a point.
The derivative f'(a) measures how fast f(x) is changing at x = a, defined as the limit of the average rate of change over an interval shrinking to zero. Geometrically it is the slope of the tangent line to the graph at that point.
Formula f'(a) = lim h→0 (f(a + h) − f(a)) / h
Example For f(x) = x² the difference quotient is ((a + h)² − a²) / h = 2a + h, which approaches 2a as h → 0.
See also Limit · Integral · Tangent line
Integral
The accumulation of a function — the signed area under its graph between two points.
The definite integral ∫ₐᵇ f(x) dx is the signed area between the graph of f and the x-axis from a to b. The fundamental theorem of calculus links it to the antiderivative: ∫ₐᵇ f(x) dx = F(b) − F(a), where F'(x) = f(x).
Formula ∫ₐᵇ f(x) dx = F(b) − F(a)
Example ∫₀¹ x² dx = [x³/3]₀¹ = 1/3 − 0 = 1/3.
See also Antiderivative · Derivative
Limit
The value a function approaches as the input gets arbitrarily close to a point.
The two-sided limit lim x→a f(x) = L means f(x) can be made as close as desired to L by taking x close enough to a. A limit may exist even when f(a) is undefined; it fails to exist when the left and right limits disagree.
Example limₓ→₂ (x² − 4)/(x − 2) = limₓ→₂ (x + 2) = 4, even though the original expression is undefined at x = 2.
See also Derivative
Antiderivative
A function whose derivative is the given function.
If F'(x) = f(x) then F is an antiderivative of f. Any two antiderivatives of the same function differ only by a constant, so the indefinite integral is written ∫ f(x) dx = F(x) + C.
Formula ∫ f(x) dx = F(x) + C
Example d/dx (x³/3 + C) = x² for every constant C, so x³/3 + C is the general antiderivative of x².
See also Integral · Derivative
Tangent line
The line that touches a curve at a point and has the same slope as the curve there.
At a point (a, f(a)) on a differentiable curve, the tangent line has equation y = f(a) + f'(a)(x − a). It is the best linear approximation to the curve near that point — close to (a, f(a)) the curve and the line are nearly indistinguishable.
Formula y = f(a) + f'(a)(x − a)
Example For f(x) = x² at a = 3: f(3) = 9 and f′(3) = 6, so the tangent is y = 9 + 6(x − 3) = 6x − 9.
See also Derivative · Slope
Precalculus
Binomial theorem
A formula that expands (a + b)ⁿ as a sum of terms with binomial coefficients.
The binomial theorem states (a + b)ⁿ = Σ C(n, k)·a^(n−k)·b^k for k from 0 to n. The coefficients C(n, k) — the binomial coefficients — are the entries of Pascal's triangle.
Formula (a + b)ⁿ = Σ C(n, k)·a^(n−k)·b^k
Example (a + b)³ = a³ + 3a²b + 3ab² + b³. The coefficients 1, 3, 3, 1 are C(3, 0) through C(3, 3).
See also Combination · Factorial
Complex number
A number of the form a + bi, where i is the imaginary unit with i² = −1.
A complex number has a real part a and an imaginary part b. Complex numbers extend the reals so that every polynomial equation has a root and so that operations like square roots of negative numbers make sense. The modulus |a + bi| = √(a² + b²) is its distance from the origin in the complex plane.
Formula z = a + bi, |z| = √(a² + b²)
Example For z = 3 + 4i, |z| = √(3² + 4²) = √25 = 5. Its conjugate is 3 − 4i, and z·z̄ = 25.
Factorial
The product of all positive integers from 1 up to n, written n!.
By convention 0! = 1, and n! = n × (n − 1)!. Factorials count the number of orderings of n distinct objects and appear in the formulas for permutations and combinations, the Taylor series and many probability distributions.
Formula n! = 1 · 2 · 3 · ⋯ · n
Example 5! = 1·2·3·4·5 = 120. By convention 0! = 1, which is what makes the combination formula work at r = 0.
See also Permutation · Combination
Permutation
An ordered arrangement of r objects chosen from n.
The number of permutations nPr = n!/(n − r)!. Permutations count outcomes where the order matters — for example, the number of different ways to award first, second and third place from a group of contestants.
Formula nPr = n! / (n − r)!
Example 5P3 = 5! / (5 − 3)! = 120 / 2 = 60 ordered arrangements of three objects chosen from five.
See also Combination · Factorial
Combination
An unordered selection of r objects chosen from n.
The number of combinations nCr = n! / (r!(n − r)!). Combinations count outcomes where the order does not matter — for example, the number of possible five-card poker hands from a deck of 52.
Formula nCr = n! / (r!(n − r)!)
Example 5C3 = 5! / (3!·2!) = 120 / 12 = 10 — one sixth of the 60 permutations, since 3! = 6 orderings collapse into one.
See also Permutation · Factorial · Binomial theorem
Statistics
Mean
The arithmetic average of a set of numbers — their sum divided by the count.
The mean x̄ of a data set is the simplest measure of its centre. It is sensitive to outliers: a single extreme value can pull the mean far from where most of the data lies, which is why the median is often reported alongside it.
Formula x̄ = (Σ xᵢ) / n
Example For 4, 8, 9 and 11: x̄ = (4 + 8 + 9 + 11) / 4 = 32 / 4 = 8.
See also Standard deviation
Standard deviation
A measure of how spread out the values in a data set are around the mean.
The standard deviation is the square root of the variance. Population standard deviation divides the sum of squared deviations by n; sample standard deviation divides by n − 1 to correct bias when estimating from a sample.
Formula σ = √( Σ(xᵢ − x̄)² / n )
Example For 2, 4, 4, 4, 5, 5, 7, 9 the mean is 5, the squared deviations sum to 32, and σ = √(32/8) = √4 = 2.
Z-score
The number of standard deviations a value lies above or below the mean.
A z-score standardizes a value so that data sets with different units and scales can be compared. A z of 0 is at the mean, positive z is above, negative below. For a normal distribution, about 68% of values lie within |z| < 1 and about 95% within |z| < 2.
Formula z = (x − μ) / σ
Example With μ = 100 and σ = 15, a value of 130 has z = (130 − 100) / 15 = 2 — two standard deviations above the mean.
See also Mean · Standard deviation · Normal distribution
Normal distribution
The bell-shaped probability distribution defined by a mean and a standard deviation.
The normal distribution N(μ, σ) has a symmetric bell-shaped density centred at μ with spread controlled by σ. It is fundamental in statistics because, under mild assumptions, the sample mean of many real-world processes follows a normal distribution (the central limit theorem).
Example A normal curve is symmetric about its mean, so mean, median and mode coincide. Standardising with z = (x − μ)/σ maps any normal curve onto μ = 0, σ = 1.
See also Mean · Standard deviation · Z-score
Correlation coefficient
A number between −1 and 1 measuring the strength and direction of a linear relationship.
Pearson's correlation coefficient r is +1 for a perfect upward straight line, −1 for a perfect downward straight line and 0 for no linear relationship. Its square r² is the fraction of the variation in y explained by a linear fit in x.
Formula r = Σ(x − x̄)(y − ȳ) / √(Σ(x − x̄)² · Σ(y − ȳ)²)
Example The points (1, 2), (2, 4), (3, 6) lie exactly on a rising line, so r = 1. Reversing to (1, 6), (2, 4), (3, 2) gives r = −1.
See also Mean
Linear Algebra
Matrix
A rectangular array of numbers arranged in rows and columns.
An m × n matrix has m rows and n columns. Matrices encode systems of linear equations, linear transformations, and data tables. Matrices of compatible dimensions can be added, subtracted and multiplied; square matrices may have an inverse.
Example A 2×3 matrix has two rows and three columns, so it holds six entries. The entry in row 2, column 1 is written a₂₁.
See also Determinant · Vector
Determinant
A single number computed from a square matrix that signals invertibility.
A square matrix is invertible if and only if its determinant is non-zero. Geometrically, the absolute value of the determinant is the volume-scaling factor of the linear transformation the matrix represents.
Formula det(A) ≠ 0 ⇔ A is invertible
Example det [[1, 2], [3, 4]] = 1·4 − 2·3 = 4 − 6 = −2. Non-zero, so the matrix is invertible.
See also Matrix
Vector
An ordered list of numbers representing a point or direction in space.
A vector in n dimensions is written as a tuple of n components. Vectors can be added entrywise and scaled by a number. Two key operations between vectors are the dot product (a scalar) and, in three dimensions, the cross product (another vector).
Example The vector (3, 4) has magnitude √(3² + 4²) = √25 = 5 and points from the origin into the first quadrant.
See also Dot product · Cross product
Dot product
A single number obtained by multiplying matching components of two vectors and adding the results.
For vectors a and b, the dot product a · b = Σ aᵢ bᵢ equals |a||b|cos θ, where θ is the angle between them. It is zero exactly when the vectors are perpendicular and is the cornerstone of orthogonality in linear algebra.
Formula a · b = Σ aᵢ bᵢ = |a||b| cos θ
Example (1, 2, 3) · (4, −5, 6) = 1·4 + 2·(−5) + 3·6 = 4 − 10 + 18 = 12.
See also Vector · Cross product
Cross product
An operation on two 3D vectors that produces a third vector perpendicular to both.
The cross product a × b is a vector with magnitude |a||b| sin θ, perpendicular to both a and b, with direction given by the right-hand rule. It is used to find normal vectors to planes and to compute areas and volumes.
Formula |a × b| = |a||b| sin θ
Example (1, 0, 0) × (0, 1, 0) = (0, 0, 1). The result is perpendicular to both factors, which the dot product confirms: it is 0 with each.
See also Vector · Dot product