Math Glossary
7 Things Nobody Tells You About GRE Math Prep
How to Prepare for the CLEP Math ExamsMath Glossary — Terms & Definitions
Clear, plain-English definitions of the math terms students meet from elementary school through college — each with a worked example. Searchable, sorted A–Z, and grouped by topic.
A
Absolute value |x|
Numbers
The distance of a number from zero on the number line; always zero or positive. Example: |−5| = 5.
Addition +
Arithmetic
Combining two or more numbers into a total called the sum. Example: 3 + 4 = 7.
Angle ∠
Geometry
The opening between two rays that share an endpoint (the vertex), measured in degrees or radians. Example: ∠ABC = 30°.
Area
Geometry
The amount of surface a flat shape covers, measured in square units. Example: a 3×4 rectangle has area = 12.
B
Base
Algebra
The number being raised to a power (in geometry, the bottom side of a shape). Example: in 2³, the base is 2.
Biconditional ⇔
Logic
“If and only if” — true when both statements have the same truth value. Example: p ⇔ q.
C
Cardinality |A|
Sets
The number of elements in a set. Infinite sets use “aleph” numbers, e.g. ℵ₀ for the counting numbers. Example: |{a,b,c}| = 3.
Cartesian product A × B
Sets
The set of all ordered pairs (a, b) with a in A and b in B. Example: {1,2}×{x} = {(1,x),(2,x)}.
Circumference
Geometry
The distance around a circle. Example: C = 2πr.
Combination ₙCₖ
Combinatorics
A selection of items where order does not matter. Example: ₅C₃ = 10.
Complement Aᶜ
Sets
Everything in the universal set that is not in A. Example: if U = {1,2,3} and A = {1}, then Aᶜ = {2,3}.
Complex number ℂ
Numbers
A number of the form a + bi, combining a real part and an imaginary part. Example: 3 + 2i.
Congruent ≅
Geometry
Having exactly the same shape and size. Example: △ABC ≅ △XYZ.
Convolution ∗
Analysis
An operation that blends two functions to produce a third, widely used in signal processing. Example: (f ∗ g)(t).
Correlation ρ
Statistics
A measure of how strongly two variables move together, from −1 to +1. Example: ρ = 0.6 (moderate positive).
Cross product a × b
Linear Algebra
A product of two 3-D vectors that gives a new vector perpendicular to both. Example: a × b.
Cube root ∛
Numbers
The number that, used as a factor three times, gives the original value. Example: ∛8 = 2.
D
Decimal .
Numbers
A number written with a decimal point separating whole units from fractional parts. Example: 2.5 = 2½.
Delta Δ
Algebra
The Greek letter used to mean “change in” a quantity. Example: Δx = x₂ − x₁.
Denominator
Numbers
The bottom number of a fraction; it tells how many equal parts make a whole. Example: in 3/4, the denominator is 4.
Derivative dy/dx
Calculus
The instantaneous rate at which a function changes — the slope of its graph at a point. Example: (x³)′ = 3x².
Determinant det(A)
Linear Algebra
A single number computed from a square matrix that indicates scaling and whether the matrix is invertible. Example: det = ad − bc.
Diameter
Geometry
A straight line through the center of a circle, joining two points on it; twice the radius. Example: d = 2r.
Dirac delta δ
Analysis
An idealized “spike” that is zero everywhere except at one point yet integrates to 1. Example: ∫ δ(x) dx = 1.
Direct sum ⊕
Abstract Algebra
A way of combining structures (e.g. vector spaces) so each piece keeps its identity. Also means XOR in logic. Example: U ⊕ V.
Distribution ∼
Statistics
A description of how likely each value of a random variable is. Example: X ∼ N(0,1).
Division ÷
Arithmetic
Splitting a quantity into equal parts; the inverse of multiplication. Example: 12 ÷ 4 = 3.
Dot product a · b
Linear Algebra
A product of two vectors giving a single number (scalar), related to the angle between them. Example: a·b = |a||b|cosθ.
E
e (Euler’s number) e
Numbers
An irrational constant ≈ 2.71828, the base of natural growth and the natural logarithm. Example: eˣ grows continuously.
Empty set ∅
Sets
The unique set containing no elements. Example: ∅ = { }.
Epsilon ε
Calculus
A symbol for an arbitrarily small positive quantity, central to the definition of limits. Example: ε → 0.
Equation =
Algebra
A statement that two expressions are equal, joined by an equals sign. Example: 2x + 1 = 7.
Euler–Mascheroni constant γ
Numbers
A constant ≈ 0.57722 that links the harmonic series with the natural logarithm. Example: γ ≈ 0.5772.
Even & odd numbers
Numbers
Even numbers are divisible by 2; odd numbers are not. Example: 6 is even, 7 is odd.
Expected value E(X)
Statistics
The long-run average outcome of a random variable, weighted by probability. Example: a fair die: E(X) = 3.5.
Exponent aⁿ
Algebra
A small raised number showing how many times the base is multiplied by itself. Example: 2³ = 8.
F
Factorial n!
Algebra
The product of all positive whole numbers up to n. Example: 5! = 120.
Floor & ceiling ⌊x⌋ ⌈x⌉
Algebra
Floor rounds a number down to the nearest integer; ceiling rounds up. Example: ⌊4.7⌋ = 4, ⌈4.2⌉ = 5.
Fourier transform ℱ
Analysis
A tool that decomposes a signal into the frequencies that make it up. Example: X(ω) = ℱ{f(t)}.
Fraction a/b
Numbers
A number written as one integer over another, showing part of a whole. Example: 3/4 = 0.75.
Function f(x)
Functions
A rule that assigns exactly one output to each input. Example: f(x) = 3x + 5.
Function composition f∘g
Functions
Applying one function to the result of another. Example: (f∘g)(x) = f(g(x)).
G
Golden ratio φ
Numbers
The constant φ = (1+√5)/2 ≈ 1.618, common in art, nature, and geometry. Example: φ ≈ 1.618.
Gradient (del) ∇
Calculus
A vector of partial derivatives pointing in the direction of steepest increase. Example: ∇f = (∂f/∂x, ∂f/∂y).
Greatest common factor (GCF)
Numbers
The largest whole number that divides two or more numbers evenly. Example: GCF(12,18) = 6.
H
Hermitian conjugate A†
Linear Algebra
The conjugate transpose of a matrix — transpose it, then conjugate each entry. Example: (A†)ᵢⱼ = conj(Aⱼᵢ).
Hypotenuse
Geometry
The longest side of a right triangle, opposite the right angle. Example: c in a²+b²=c².
I
Identity ≡
Algebra
An equation true for every value of its variables. Example: (a+b)² ≡ a²+2ab+b².
Imaginary unit i
Numbers
The number defined by i² = −1, the basis of imaginary and complex numbers. Example: √−9 = 3i.
Implication ⇒
Logic
“If… then…”; false only when a true premise leads to a false conclusion. Example: p ⇒ q.
Inequality < > ≤ ≥
Algebra
A statement that one value is greater or less than another. Example: x ≥ 5.
Infinity ∞
Numbers
A concept describing something without bound — larger than any number. Example: x → ∞.
Inner product ⟨x,y⟩
Linear Algebra
A generalized dot product that defines length and angle in a vector space. Example: ⟨x, y⟩.
Integer ℤ
Numbers
A whole number, positive, negative, or zero — no fractions. Example: …−2,−1,0,1,2….
Integral ∫
Calculus
The accumulated area under a curve; the reverse of differentiation. Example: ∫ 2x dx = x² + C.
Intersection ∩
Sets
The set of elements common to both sets. Example: {1,2,3}∩{2,3,4} = {2,3}.
Interval [a,b]
Sets
The set of all numbers between two endpoints; brackets include them, parentheses exclude them. Example: x ∈ [2, 6].
Inverse matrix A⁻¹
Linear Algebra
The matrix that “undoes” A; multiplying them gives the identity matrix. Example: A·A⁻¹ = I.
L
Laplace transform ℒ
Analysis
A transform that turns differential equations into algebra, key in engineering. Example: F(s) = ℒ{f(t)}.
Least common multiple (LCM)
Numbers
The smallest number that two or more numbers all divide into. Example: LCM(4,6) = 12.
Limit lim
Calculus
The value a function approaches as its input nears a point. Example: lim(x→0) sin x / x = 1.
Logical AND ∧
Logic
A connective that is true only when both statements are true. Example: p ∧ q.
Logical OR ∨
Logic
A connective that is true when at least one statement is true. Example: p ∨ q.
M
Mean (average) x̄ , μ
Statistics
The sum of values divided by how many there are. Example: (80+90+70)/3 = 80.
Median
Statistics
The middle value when data is ordered from least to greatest. Example: median of 3,5,9 = 5.
Mode
Statistics
The value that appears most often in a data set. Example: mode of 2,4,4,5 = 4.
Modulo mod
Numbers
The remainder left after dividing one number by another. Example: 7 mod 2 = 1.
Multiplication ×
Arithmetic
Repeated addition of the same number; the result is the product. Example: 6 × 3 = 18.
N
Natural number ℕ
Numbers
A counting number: 1, 2, 3, … (sometimes including 0). Example: 6 ∈ ℕ.
Negation ¬
Logic
The “not” operator, which reverses a statement’s truth value. Example: ¬(true) = false.
Norm ‖x‖
Linear Algebra
The length or magnitude of a vector. Example: ‖(3,4)‖ = 5.
Normal distribution N(μ,σ²)
Statistics
The symmetric “bell curve” describing many natural data sets. Example: X ∼ N(0, 1).
Numerator
Numbers
The top number of a fraction; how many parts are taken. Example: in 3/4, the numerator is 3.
O
Order of operations ( )
Arithmetic
The agreed order for evaluating an expression: PEMDAS — Parentheses, Exponents, Multiply/Divide, Add/Subtract. Example: 2 + 3 × 4 = 14.
P
Parallel ∥
Geometry
Lines in a plane that never meet and stay the same distance apart. Example: AB ∥ CD.
Partial derivative ∂
Calculus
The derivative of a multivariable function with respect to one variable, others held fixed. Example: ∂(x²+y²)/∂x = 2x.
Percent %
Numbers
A ratio expressed out of 100. Example: 25% = 25/100 = 0.25.
Perimeter
Geometry
The total distance around the outside of a 2-D shape. Example: a 3×4 rectangle has perimeter = 14.
Permutation ₙPₖ
Combinatorics
An arrangement of items where order does matter. Example: ₅P₃ = 60.
Perpendicular ⊥
Geometry
Two lines that meet at a right angle (90°). Example: AB ⊥ CD.
Pi π
Geometry
The ratio of a circle’s circumference to its diameter ≈ 3.14159. Example: C = πd.
Polygon
Geometry
A closed 2-D shape with straight sides. Example: triangle, square, pentagon.
Power set 𝒫(A)
Sets
The set of all subsets of A, including ∅ and A itself. Example: a 3-element set has 2³ = 8 subsets.
Prime number
Numbers
A whole number greater than 1 with exactly two factors: 1 and itself. Example: 2, 3, 5, 7, 11….
Probability P(A)
Statistics
A measure of how likely an event is, from 0 (impossible) to 1 (certain). Example: P(heads) = 0.5.
Product (series) ∏
Algebra
The result of multiplying terms together; capital pi (∏) means “multiply all”. Example: ∏ᵢ₌₁³ i = 6.
Proof (Q.E.D.) ∎
Logic
A logical argument establishing that a statement is true; ∎ marks its end. Example: … ∴ true. ∎.
Entailment (models) ⊨
Logic
A structure satisfies, or a set of premises semantically guarantees, a statement. Example: M ⊨ φ.
Provability (turnstile) ⊢
Logic
A statement can be derived from given axioms using rules of inference. Example: Γ ⊢ φ.
Proportion ∝
Algebra
A statement that two ratios are equal; “∝” means “varies directly with”. Example: y ∝ x.
Q
Quadratic
Algebra
A polynomial of degree 2, graphing as a parabola. Example: ax² + bx + c = 0.
Quantifier ∀ ∃
Logic
A symbol stating how many: ∀ (“for all”) or ∃ (“there exists”). Example: ∀x, x² ≥ 0.
R
Radius
Geometry
The distance from the center of a circle to any point on it; half the diameter. Example: r = d/2.
Rank rank(A)
Linear Algebra
The number of linearly independent rows (or columns) of a matrix. Example: rank(A) = 2.
Ratio :
Numbers
A comparison of two quantities by division. Example: 3 : 2.
Rational number ℚ
Numbers
Any number that can be written as a fraction of two integers. Example: ½, −7, 0.25.
Real number ℝ
Numbers
Any number on the continuous number line, rational or irrational. Example: √2, π, −3.
Right angle ∟
Geometry
An angle of exactly 90°. Example: the corner of a square is 90°.
Root (radical) ⁿ√
Numbers
A value that, raised to the nth power, gives the original number. Example: ⁴√16 = 2.
Rounding ≈
Arithmetic
Replacing a number with a nearby simpler one to a chosen place value. Example: 3.14159 ≈ 3.14.
S
Set { }
Sets
A well-defined collection of distinct objects called elements. Example: A = {1, 2, 3}.
Set difference A \ B
Sets
Elements in A that are not in B. Example: {1,2,3}\{2} = {1,3}.
Similar ~
Geometry
Same shape but possibly different size; corresponding angles equal, sides proportional. Example: △ABC ~ △XYZ.
Slope
Algebra
The steepness of a line: rise over run. Example: m = (y₂−y₁)/(x₂−x₁).
Square root √
Numbers
A value that multiplied by itself gives the original number. Example: √9 = 3.
Standard deviation σ
Statistics
A measure of how spread out data is around the mean. Example: σ = 2.
Subset ⊂ ⊆
Sets
A set whose every element is also in another set. Example: {1,2} ⊂ {1,2,3}.
Subtraction −
Arithmetic
Taking one quantity away from another; the result is the difference. Example: 9 − 4 = 5.
Summation ∑
Algebra
Adding a sequence of terms; capital sigma (∑) means “sum of”. Example: ∑ᵢ₌₁³ i = 6.
Superset ⊃ ⊇
Sets
A set that contains another set entirely. Example: {1,2,3} ⊃ {1,2}.
Symmetric difference A △ B
Sets
Elements in either set but not in both. Example: {1,2}△{2,3} = {1,3}.
T
Tensor product ⊗
Linear Algebra
An operation combining vector spaces or matrices into a larger one. Example: A ⊗ B.
Theta θ
Trigonometry
The Greek letter conventionally used for an unknown angle. Example: sin θ = 0.5.
Transpose Aᵀ
Linear Algebra
Flipping a matrix over its diagonal, turning rows into columns. Example: (Aᵀ)ᵢⱼ = Aⱼᵢ.
Triangle △
Geometry
A polygon with three sides and three angles that always sum to 180°. Example: △ABC.
U
Union ∪
Sets
All elements that are in either set (or both). Example: {1,2}∪{2,3} = {1,2,3}.
V
Variable x
Algebra
A letter that represents an unknown or changing value. Example: 2x = 4 → x = 2.
Variance σ²
Statistics
The average of the squared distances from the mean; the square of the standard deviation. Example: σ² = 4.
Vector →a
Linear Algebra
A quantity with both magnitude and direction. Example: →a = (3, 4).
Volume
Geometry
The amount of space a 3-D object occupies, in cubic units. Example: a cube of side 2 has volume = 8.
W
Whole number
Numbers
The counting numbers together with zero: 0, 1, 2, 3, … — with no negatives or fractions. Example: 0, 1, 2, 3, ….
Weighted mean
Statistics
An average in which some values count more than others according to assigned weights. Example: a grade that is 70% exams + 30% homework.
X
x-axis
Geometry
The horizontal axis of a coordinate plane. Example: the point (3, 0) lies on the x-axis.
x-intercept
Algebra
Where a graph crosses the x-axis (where y = 0). Example: y = x − 2 has x-intercept 2.
Y
y-axis
Geometry
The vertical axis of a coordinate plane. Example: the point (0, 4) lies on the y-axis.
y-intercept
Algebra
Where a graph crosses the y-axis (where x = 0); the “b” in y = mx + b. Example: y = 2x + 5 has y-intercept 5.
Z
Zero
Numbers
The number representing none; the additive identity, since adding 0 changes nothing. Example: 7 + 0 = 7.
Z-score (standard score)
Statistics
How many standard deviations a value lies from the mean. Example: z = (x − μ) / σ.
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