Definition → notation → example → calculator
A mathematics dictionary you can calculate with.
Look up a term, see what its symbols mean, follow a compact example, then open a calculator that puts the concept to work.
Browse A–ZBrowse symbolsMathematics calculatorsLearn at the right depth
Browse by the stage where a topic is usually introduced
“Introduced” does not mean finished. Set theory can begin with sorting before high school and return as axiomatic foundations at university.
Cornerstone guide
Set theory: the map beneath mathematics
Build live Venn diagrams, calculate set operations, read every major symbol and explore the laws, proofs, infinities and paradoxes that shaped modern mathematical foundations.
Explore the interactive set theory guideBig ideas worth exploring
Interactive cornerstone guides
These are the dictionary’s deeper maps: three explanation levels, a local interactive experiment, school-stage guidance, worked reasoning and connections toward quantum mathematics.
Function
A rule that assigns exactly one output to every permitted input.
y = f(x)Explore the full guide Pre-high school → college depthInfinity
The mathematical study of unbounded processes and infinite collections, including limits and distinct cardinalities that show some infinities are strictly larger than others.
∞; ℵ₀; |ℕ|<|ℝ|Explore the full guide High school → college depthLogic and proof
The formal study of valid reasoning and the methods used to establish that a mathematical statement follows from definitions, axioms and previously proved results.
P⇒Q; P⇔Q; ∀x; ∃x; ¬PExplore the full guide College → college depthMathematical structures
Sets equipped with chosen relations, operations or notions of distance, size or continuity that determine how their elements may interact.
(S, operations, relations)Explore the full guide Pre-high school → college depthNumber systems
Nested families of numbers—natural, whole, integer, rational, irrational, real and complex—created to make progressively more operations and equations possible.
ℕ ⊂ ℤ ⊂ ℚ ⊂ ℝ ⊂ ℂExplore the full guide Pre-high school → college depthSymmetry
A property that remains unchanged under an allowed transformation such as reflection, rotation, translation or a more abstract operation.
T(x) preserves a stated propertyExplore the full guide High school → college depthUnsolved problems
Mathematical questions for which no generally accepted proof or disproof is currently known, even when enormous computational evidence supports a conjectured answer.
evidence ≠ proofExplore the full guide Search by term, symbol or meaning
78 mathematical terms
78 terms
A
Absolute value
The non-negative distance of a real number from zero on the number line.
|x|Usually introduced: Pre-high schoolRead and calculate Geometry and trigonometryAngle
The amount of rotation between two rays that share an endpoint.
θUsually introduced: Pre-high schoolRead and calculate Symbols and notationApproximately equal symbol
The symbol ≈ states that two quantities are close enough for the stated purpose but are not necessarily exactly equal.
a ≈ bUsually introduced: High schoolRead and calculate Geometry and trigonometryArea
The two-dimensional measure of the region enclosed by a shape.
square unitsUsually introduced: Pre-high schoolRead and calculate Numbers and algebraArithmetic sequence
A sequence in which consecutive terms differ by the same constant amount.
aₙ = a₁ + (n−1)dUsually introduced: Pre-high schoolRead and calculate C
Circle
The set of all points in a plane that are the same distance from one centre point.
(x−h)²+(y−k)²=r²Usually introduced: Pre-high schoolRead and calculate Statistics and probabilityCombination
A selection in which order does not matter.
C(n,r)=n!/[r!(n−r)!]Usually introduced: High schoolRead and calculate Waves and mathematical physicsComplex number
A number with real and imaginary components, where the imaginary unit satisfies i²=−1.
z=a+biUsually introduced: CollegeRead and calculate Vectors and matricesCross product
A three-dimensional vector perpendicular to two input vectors, with magnitude equal to the parallelogram area they span.
a×bUsually introduced: High schoolRead and calculate D
Delta symbol
Uppercase delta usually represents a finite change or difference, while lowercase delta often represents a small change, variation or parameter whose precise meaning depends on context.
Δx=x₂−x₁; δUsually introduced: High schoolRead and calculate Calculus and changeDerivative
The instantaneous rate of change of a function, defined as a limit of average rates where that limit exists.
f′(x) = lim[h→0] (f(x+h)−f(x))/hUsually introduced: High schoolRead and calculate Vectors and matricesDeterminant
A scalar associated with a square matrix that describes signed scale change and indicates invertibility when non-zero.
det(A) or |A|Usually introduced: High schoolRead and calculate Calculus and changeDifferential equation
An equation that relates an unknown function to one or more of its derivatives.
F(x,y,y′,…)=0Usually introduced: High schoolRead and calculate Geometry and trigonometryDistance formula
The Pythagorean relationship used to find straight-line distance between two coordinate points.
d=√((x₂−x₁)²+(y₂−y₁)²)Usually introduced: Pre-high schoolRead and calculate Vectors and matricesDot product
A scalar product that combines corresponding vector components and also equals the product of magnitudes times the cosine of their angle.
a·b=ΣaᵢbᵢUsually introduced: High schoolRead and calculate E
Eigenvalue
A scalar by which a linear transformation stretches an associated non-zero eigenvector without changing its direction line.
Av=λvUsually introduced: High schoolRead and calculate Symbols and notationEpsilon symbol
Epsilon commonly denotes a small positive quantity, an error tolerance or an arbitrarily small bound, although individual formulas may define it differently.
ε>0Usually introduced: High schoolRead and calculate Symbols and notationEquals and not-equal symbols
The equals sign asserts that two expressions have the same value, while the not-equal sign asserts that their values differ.
a=b; a≠bUsually introduced: High schoolRead and calculate Numbers and algebraEquation
A statement that two mathematical expressions have equal value.
left side = right sideUsually introduced: Pre-high schoolRead and calculate Calculus and changeEuler's method
A first-order numerical method that advances an approximate differential-equation solution using the current slope.
yₙ₊₁=yₙ+h f(tₙ,yₙ)Usually introduced: High schoolRead and calculate Numbers and algebraExponent
A number that states how many times a base is multiplied by itself, with broader definitions extending to zero, negative and non-integer powers.
bⁿUsually introduced: Pre-high schoolRead and calculate Numbers and algebraExponential function
A function in which the variable appears in the exponent and changes by a constant multiplicative factor over equal intervals.
f(x) = abˣUsually introduced: Pre-high schoolRead and calculate F
Factorial
The product of every positive integer from 1 through a non-negative integer n, with 0! defined as 1.
n!Usually introduced: Pre-high schoolRead and calculate Waves and mathematical physicsFourier transform
A transformation that represents a signal or function through frequency components.
Xₖ=Σxₙe⁻ⁱ²πkn/NUsually introduced: CollegeRead and calculate Numbers and algebraFraction
A quotient that represents one number divided by another non-zero number.
a/b, b ≠ 0Usually introduced: Pre-high schoolRead and calculate Waves and mathematical physicsFrequency
The number of cycles or repeated events per unit time.
f=1/TUsually introduced: CollegeRead and calculate Numbers and algebraFunction
A rule that assigns exactly one output to every permitted input.
y = f(x)Usually introduced: Pre-high schoolRead and calculate G
Geometric sequence
A sequence in which each non-zero term is obtained by multiplying the previous term by a constant ratio.
aₙ = a₁rⁿ⁻¹Usually introduced: Pre-high schoolRead and calculate Numbers and algebraGreatest common factor
The largest positive integer that divides each supplied integer without a remainder.
gcf(a,b)Usually introduced: Pre-high schoolRead and calculate I
Infinity
The mathematical study of unbounded processes and infinite collections, including limits and distinct cardinalities that show some infinities are strictly larger than others.
∞; ℵ₀; |ℕ|<|ℝ|Usually introduced: Pre-high schoolRead and calculate Symbols and notationInfinity symbol
The infinity symbol describes unbounded growth, extent or continuation; it is not an ordinary real number on which standard arithmetic rules always apply.
∞Usually introduced: High schoolRead and calculate Calculus and changeIntegral
A mathematical construction representing accumulation; a definite integral is the signed limit of sums over an interval.
∫ₐᵇf(x)dxUsually introduced: High schoolRead and calculate Symbols and notationIntegral sign
The elongated integral sign indicates continuous accumulation, with limits marking the interval for a definite integral and a differential identifying the variable.
∫f(x)dx; ∫ₐᵇf(x)dxUsually introduced: High schoolRead and calculate Vectors and matricesInverse matrix
A matrix that reverses an invertible square matrix under multiplication.
AA⁻¹=IUsually introduced: High schoolRead and calculate L
Lambda symbol
Lambda is a Greek letter commonly used for wavelength, an eigenvalue, a rate parameter or another quantity explicitly defined by its formula.
λUsually introduced: High schoolRead and calculate Numbers and algebraLogarithm
The exponent to which a positive base must be raised to produce a positive value.
log_b(x) = y ⇔ bʸ = xUsually introduced: Pre-high schoolRead and calculate Foundations, logic and setsLogic and proof
The formal study of valid reasoning and the methods used to establish that a mathematical statement follows from definitions, axioms and previously proved results.
P⇒Q; P⇔Q; ∀x; ∃x; ¬PUsually introduced: High schoolRead and calculate M
Mathematical structures
Sets equipped with chosen relations, operations or notions of distance, size or continuity that determine how their elements may interact.
(S, operations, relations)Usually introduced: CollegeRead and calculate Vectors and matricesMatrix
A rectangular arrangement of values used to represent linear maps, systems, data and other structured relationships.
A=[aᵢⱼ]Usually introduced: High schoolRead and calculate Statistics and probabilityMean
The arithmetic average found by adding values and dividing by the number of values.
x̄=Σxᵢ/nUsually introduced: High schoolRead and calculate Statistics and probabilityMedian
The central value in an ordered dataset, or the mean of the two central values when the count is even.
middle ordered valueUsually introduced: High schoolRead and calculate Waves and mathematical physicsMomentum
A vector quantity equal to mass times velocity in classical mechanics.
p=mvUsually introduced: CollegeRead and calculate Symbols and notationMu symbol
Mu is a Greek letter commonly used for a population mean, a coefficient or a physical parameter whose meaning must be read from context.
μUsually introduced: High schoolRead and calculate N
Nabla symbol
Nabla is the vector differential operator used to write gradients, divergence and curl in multivariable calculus and mathematical physics.
∇f; ∇·F; ∇×FUsually introduced: High schoolRead and calculate Waves and mathematical physicsNormalisation
Scaling a mathematical state so that its total norm or probability satisfies a required convention, commonly one.
Σ|aᵢ|²=1Usually introduced: CollegeRead and calculate Foundations, logic and setsNumber systems
Nested families of numbers—natural, whole, integer, rational, irrational, real and complex—created to make progressively more operations and equations possible.
ℕ ⊂ ℤ ⊂ ℚ ⊂ ℝ ⊂ ℂUsually introduced: Pre-high schoolRead and calculate P
Partial derivative symbol
The rounded symbol ∂ marks differentiation with respect to one variable while other independent variables are held constant.
∂f/∂xUsually introduced: High schoolRead and calculate Numbers and algebraPercentage
A dimensionless ratio expressed as a number of parts per hundred, allowing proportions with different totals to be compared.
percentage = part/whole × 100%Usually introduced: Pre-high schoolRead and calculate Statistics and probabilityPermutation
An ordered arrangement of selected objects, so changing the order creates a different outcome.
P(n,r)=n!/(n−r)!Usually introduced: High schoolRead and calculate Symbols and notationPi symbol
Lowercase pi usually denotes the circle constant, while uppercase pi can denote multiplication across an indexed collection of factors.
π≈3.14159; ∏ᵢaᵢUsually introduced: High schoolRead and calculate Symbols and notationPlus-minus symbol
The plus-minus sign compactly represents two values or cases, one formed with addition and the other with subtraction.
a±bUsually introduced: High schoolRead and calculate Statistics and probabilityProbability
A number from zero to one that measures how likely an event is under a stated model.
P(A)Usually introduced: High schoolRead and calculate Numbers and algebraProportion
An equation stating that two ratios are equal.
a/b = c/dUsually introduced: Pre-high schoolRead and calculate Geometry and trigonometryPythagorean theorem
In a right triangle, the square of the hypotenuse equals the sum of the squares of the two shorter sides.
a²+b²=c²Usually introduced: Pre-high schoolRead and calculate R
Radian
An angle unit defined by the ratio of arc length to circle radius.
θ=s/rUsually introduced: Pre-high schoolRead and calculate Numbers and algebraRatio
A comparison of two quantities by division.
a:b or a/bUsually introduced: Pre-high schoolRead and calculate Numbers and algebraRoot
A value that reverses a power, or a value that makes an equation or function equal zero, depending on context.
ⁿ√x or f(x)=0Usually introduced: Pre-high schoolRead and calculate S
Scalar
A single quantity that can scale another mathematical object and has magnitude but no vector direction.
kUsually introduced: High schoolRead and calculate Foundations, logic and setsSet theory
The branch of mathematics that studies collections of distinct objects, membership, relationships between collections and the operations that combine or compare them.
x∈A; A⊆B; A∪B; A∩B; Aᶜ; ℘(A); |A|Usually introduced: Pre-high schoolRead and calculate Symbols and notationSigma symbols
Uppercase sigma commonly denotes a sum, while lowercase sigma commonly denotes standard deviation; context determines the intended quantity.
Σᵢxᵢ; σUsually introduced: High schoolRead and calculate Geometry and trigonometrySlope
The rate at which a line's vertical coordinate changes relative to its horizontal coordinate.
m=(y₂−y₁)/(x₂−x₁)Usually introduced: Pre-high schoolRead and calculate Symbols and notationSquare-root symbol
The radical sign requests a root; without a written index it means the principal square root, and with an index it specifies another root degree.
√x; ⁿ√xUsually introduced: High schoolRead and calculate Statistics and probabilityStandard deviation
The square root of variance, measuring the typical spread of observations around their mean in the original units.
σ=√(Σ(xᵢ−μ)²/N)Usually introduced: High schoolRead and calculate Foundations, logic and setsSymmetry
A property that remains unchanged under an allowed transformation such as reflection, rotation, translation or a more abstract operation.
T(x) preserves a stated propertyUsually introduced: Pre-high schoolRead and calculate T
Tangent
In right-triangle trigonometry, tangent is opposite side divided by adjacent side; in geometry it can also mean a line touching a curve locally.
tan(θ)=opposite/adjacentUsually introduced: Pre-high schoolRead and calculate Symbols and notationTheta symbol
Theta is a Greek letter frequently used for an unknown angle, phase or parameter whose role is defined by the surrounding formula.
θUsually introduced: High schoolRead and calculate Geometry and trigonometryTrigonometry
The study of relationships among angles, side lengths and periodic functions.
sin, cos, tanUsually introduced: Pre-high schoolRead and calculate U
Uncertainty principle
A quantum-mechanical bound on the product of standard deviations of conjugate observables such as position and momentum.
ΔxΔp≥ℏ/2Usually introduced: CollegeRead and calculate Symbols and notationUnion and intersection symbols
Union combines everything belonging to either set, while intersection keeps only elements belonging to both sets.
A∪B; A∩BUsually introduced: High schoolRead and calculate Foundations, logic and setsUnsolved problems
Mathematical questions for which no generally accepted proof or disproof is currently known, even when enormous computational evidence supports a conjectured answer.
evidence ≠ proofUsually introduced: High schoolRead and calculate V
Variance
The average squared deviation from the mean under the selected population or sample convention.
σ²=Σ(xᵢ−μ)²/NUsually introduced: High schoolRead and calculate Vectors and matricesVector
An ordered mathematical object with components, commonly used to represent magnitude and direction.
v=(v₁,v₂,…)Usually introduced: High schoolRead and calculate Symbols and notationVertical bars
Vertical bars can denote absolute value, a determinant, a vector magnitude, divisibility or set size, so their meaning is determined by the enclosed object and context.
|x|; |A|; ‖v‖Usually introduced: High schoolRead and calculate W
Wavelength
The spatial distance over which a periodic wave repeats one complete cycle.
λ=v/fUsually introduced: CollegeRead and calculate Statistics and probabilityWeighted mean
A mean in which values contribute in proportion to assigned weights.
x̄_w=Σwᵢxᵢ/ΣwᵢUsually introduced: High schoolRead and calculate Clear answers
About the mathematics dictionary
What is the MW SysArc Mathematics Dictionary?
It is a concise A–Z reference for mathematical terms, notation and examples, with each concept linked to calculators that let you use it.
Can I search by a symbol or another name?
Yes. Search includes terms, definitions, notation, examples and aliases such as gradient for slope or GCD for greatest common factor.
How does the dictionary connect to the calculators?
Every term page links to one or more relevant calculators, so you can move directly from a definition to a worked numerical model.