Mathematics Dictionary
175 terms, every letter from A to Z. A plain-language answer first, the formal statement second — short enough to read in one breath, precise enough to trust.
Absolute Value
The distance of a number from zero on the number line, always non-negative. Written |x|; e.g. |−5| = 5.
Algorithm
A finite, precise sequence of steps that solves a class of problems, such as long division or the Euclidean algorithm.
Amplitude
Half the distance between the maximum and minimum values of a periodic function; for y = A sin x, the amplitude is |A|.
Argand Plane
The plane used to represent complex numbers, with the real part on the horizontal axis and the imaginary part on the vertical axis.
Arithmetic Sequence
A sequence in which consecutive terms differ by a fixed constant d; the nth term is a₁ + (n−1)d.
Asymptote
A line that a curve approaches arbitrarily closely without touching it as the input grows or nears a forbidden value.
Axiom
A statement accepted without proof as a starting point of a mathematical theory, such as Euclid's postulates.
Base (of a Logarithm)
The number b in log_b(x); the exponent to which b must be raised to obtain x.
Bernoulli Equation
An energy conservation statement for steady, inviscid, incompressible flow along a streamline: p + ½ρv² + ρgz stays constant.
Binomial Theorem
The expansion (a+b)ⁿ = Σ C(n,k)aⁿ⁻ᵏbᵏ, giving every term of an integer power of a binomial via combinations.
Bisector
A line, ray, or segment that divides an angle or another segment into two equal parts.
Boolean Algebra
The algebra of the values true and false with operations AND, OR, and NOT; the mathematical basis of digital circuits.
Boundary Layer
The thin region next to a solid surface where viscosity brings the flow from zero at the wall to the free-stream speed.
Bounded Set
A set of numbers contained within some finite interval; it has both an upper and a lower bound.
Cardinality
The size of a set — the number of elements it contains, extended by Cantor to compare infinite sets.
Coefficient
The numerical factor multiplying a variable in a term; in 7x², the coefficient is 7.
Combination
A selection of items where order does not matter; the count of k-item choices from n is C(n,k) = n!/(k!(n−k)!).
Continuity Equation
The mass-conservation law of fluid motion; for incompressible flow it reads ∇·u = 0.
Convergence
The property of a sequence or series approaching a specific finite limit as its index grows without bound.
Cosine
For an acute angle in a right triangle, the ratio of the adjacent side to the hypotenuse; extended to all angles by the unit circle.
Critical Point
A point where a function's derivative is zero or undefined; candidates for maxima, minima, and saddle points.
Cross Product
A product of two 3-D vectors yielding a vector perpendicular to both, with magnitude equal to the parallelogram area they span.
Derivative
The instantaneous rate of change of a function — the limit of the difference quotient, and the slope of the tangent line.
Determinant
A scalar computed from a square matrix that measures volume scaling and invertibility; zero determinant means no inverse.
Dimensionless Number
A pure-number ratio of physical effects, such as the Reynolds or Prandtl number, that lets different systems be compared.
Discriminant
The quantity b² − 4ac of a quadratic; its sign tells whether the roots are two real, one repeated, or a complex pair.
Divergence
A scalar field ∇·F measuring the net outflow of a vector field per unit volume at each point.
Eigenvalue
A scalar λ for which a matrix A has a nonzero vector v with Av = λv; the factor by which that direction is stretched.
Ellipse
The set of points whose distances to two fixed foci sum to a constant; the closed conic section.
Euler's Number
The constant e ≈ 2.71828, the base of natural logarithms, defined by the limit of (1 + 1/n)ⁿ.
Even Function
A function with f(−x) = f(x), symmetric about the vertical axis, such as x² or cos x.
Expected Value
The long-run average of a random quantity — each outcome weighted by its probability.
Exponent
The power to which a base is raised; in xⁿ, n counts repeated multiplication and extends to all real values.
Extrapolation
Estimating values beyond the range of observed data — useful but riskier than interpolation.
Factorial
The product n! = n×(n−1)×…×1, counting the arrangements of n distinct objects; 0! is defined as 1.
Fibonacci Sequence
The sequence 1, 1, 2, 3, 5, 8, … where each term is the sum of the previous two; its term ratios approach the golden ratio.
Fluid
A substance — liquid or gas — that deforms continuously under any applied shear stress, however small.
Fourier Series
The representation of a periodic function as an infinite sum of sines and cosines.
Frequency
The number of times a value occurs in a data set; relative frequency divides by the total count.
Frustum
The solid left when a cone or pyramid is cut by a plane parallel to its base and the top removed.
Gaussian Distribution
The normal bell-curve distribution, fixed by its mean and standard deviation; the limit shape of many averaged effects.
Geometric Sequence
A sequence in which each term is the previous one times a fixed ratio r; the nth term is a₁rⁿ⁻¹.
Golden Ratio
The constant φ = (1+√5)/2 ≈ 1.618, satisfying φ² = φ + 1; the limit of Fibonacci term ratios.
Gradient
The vector ∇f of partial derivatives, pointing in the direction of steepest increase of a scalar field.
Graph
The set of all points (x, f(x)) picturing a function, or, in discrete mathematics, a network of vertices and edges.
Greatest Common Divisor
The largest integer dividing two given integers exactly; computed efficiently by the Euclidean algorithm.
Harmonic Mean
The reciprocal of the average of reciprocals; the correct average for rates, such as speeds over equal distances.
Hartmann Number
The dimensionless group Ha = BL√(σ/μ) comparing magnetic (Lorentz) force with viscous force in MHD flow.
Heron's Formula
The triangle-area formula √(s(s−a)(s−b)(s−c)) using only the three side lengths, with s the semi-perimeter.
Histogram
A bar chart of frequencies over numeric intervals, showing the shape of a data distribution.
Hydrostatics
The study of fluids at rest, where pressure increases linearly with depth as p = p₀ + ρgh.
Hyperbola
The conic section of points whose distances to two foci differ by a constant; it has two branches and two asymptotes.
Identity Matrix
The square matrix with ones on the diagonal and zeros elsewhere; multiplying by it changes nothing.
Inflection Point
A point where a curve changes concavity — the second derivative changes sign.
Integral
The accumulation of a quantity — geometrically the signed area under a curve; the inverse operation of differentiation.
Interpolation
Estimating a value between known data points, e.g. by fitting a line or polynomial through them.
Inverse Function
The function f⁻¹ that undoes f, so f⁻¹(f(x)) = x; it exists exactly when f is one-to-one.
Irrational Number
A real number that cannot be written as a ratio of integers, such as √2 or π; its decimal never repeats.
Jacobian
The matrix of all first partial derivatives of a vector-valued map; its determinant scales areas and volumes under change of variables.
Jerk
The rate of change of acceleration — the third derivative of position with respect to time.
Joint Probability
The probability that two events occur together, written P(A ∩ B); for independent events it equals P(A)P(B).
Julia Set
The boundary set of points where iterating a complex map like z² + c switches between stable and chaotic behaviour; a classic fractal.
Jump Discontinuity
A discontinuity where the left and right limits both exist but differ, so the graph leaps between two values.
Kepler's Laws
Three laws of planetary motion: elliptical orbits, equal areas in equal times, and period squared proportional to semi-axis cubed.
Kinematic Viscosity
Dynamic viscosity divided by density, ν = μ/ρ; the diffusivity of momentum, measured in m²/s.
Kite
A quadrilateral with two pairs of adjacent equal sides; its diagonals meet at right angles.
Knudsen Number
The ratio of molecular mean free path to a flow's length scale; small values justify treating a fluid as a continuum.
Kurtosis
A measure of how heavy a distribution's tails are compared with the normal distribution.
Laminar Flow
Smooth, layered fluid motion without mixing between layers; occurs at low Reynolds numbers.
Least Squares
The fitting principle that chooses parameters minimising the sum of squared differences between model and data.
Limit
The value a function approaches as its input approaches a point; the foundational concept beneath derivatives and integrals.
Linear Equation
An equation whose variables appear only to the first power, such as 3x + 2y = 6; its graph is a straight line.
Locus
The set of all points satisfying a stated geometric condition, such as constant distance from a centre (a circle).
Logarithm
The inverse of exponentiation: log_b(x) is the exponent giving x when b is raised to it.
Lorentz Force
The force qE + qv×B on moving charge; per unit volume in MHD it is J×B and brakes or pumps conducting fluids.
Magnetohydrodynamics
The study of electrically conducting fluids interacting with magnetic fields — MHD for short.
Material Derivative
The rate of change following a moving fluid particle: D/Dt = ∂/∂t + (u·∇).
Matrix
A rectangular array of numbers that represents a linear transformation or a system of equations.
Modulus
Either the absolute value of a number, the remainder operation in modular arithmetic, or the magnitude of a complex number.
Nanofluid
A base liquid seeded with nanometre-scale particles to raise its thermal conductivity for cooling applications.
Newton's Method
An iterative root-finder x_{n+1} = x_n − f(x_n)/f′(x_n) that typically doubles correct digits each step.
Normal Distribution
The symmetric bell-shaped distribution described by its mean and standard deviation; ubiquitous by the central limit theorem.
Numerator
The top part of a fraction, counting how many of the denominator's parts are taken.
Nusselt Number
The dimensionless convective heat-transfer rate Nu = hL/k, comparing convection with pure conduction.
ODE
An ordinary differential equation — one relating a function of a single variable to its derivatives.
Odd Function
A function with f(−x) = −f(x), symmetric about the origin, such as x³ or sin x.
Order of Operations
The agreed sequence for evaluating expressions: brackets, exponents, multiplication/division, addition/subtraction.
Ordinate
The y-coordinate of a point; its partner, the x-coordinate, is the abscissa.
Outlier
A data value lying unusually far from the rest; it can distort the mean while barely moving the median.
Parabola
The graph of a quadratic function; the set of points equidistant from a focus and a directrix.
Permutation
An ordered arrangement of items; the count of k-item orderings from n is P(n,k) = n!/(n−k)!.
Polynomial
A sum of terms, each a constant times a whole-number power of the variable, such as 4x³ − x + 7.
Prandtl Number
The fluid property Pr = ν/α comparing momentum and heat diffusion; it sets the relative thickness of velocity and thermal boundary layers.
Prime Number
An integer greater than 1 divisible only by 1 and itself; the multiplicative atoms of the integers.
Probability
A number between 0 and 1 measuring how likely an event is; the ratio of favourable to total equally likely outcomes.
Pythagorean Theorem
In a right triangle, a² + b² = c²: the squares on the legs sum to the square on the hypotenuse.
QED
The traditional close of a proof, from the Latin quod erat demonstrandum — 'which was to be demonstrated'.
Quadrant
One of the four regions into which the coordinate axes divide the plane, numbered anticlockwise from the upper right.
Quadratic Equation
A degree-two equation ax² + bx + c = 0, solved by factoring, completing the square, or the quadratic formula.
Quartile
One of the three values splitting ordered data into four equal parts; Q2 is the median, and Q3 − Q1 is the interquartile range.
Quaternion
A four-part extension of complex numbers (a + bi + cj + dk) used to represent 3-D rotations without gimbal lock.
Radian
The angle subtending an arc equal in length to the radius; a full circle is 2π radians.
Range
In statistics, the difference between the largest and smallest data values; for a function, the set of all outputs.
Rational Number
A number expressible as a fraction of integers; its decimal expansion terminates or repeats.
Reynolds Number
The dimensionless ratio Re = ρvL/μ of inertial to viscous effects; it predicts whether a flow is laminar or turbulent.
Root
A value of the variable making an equation true; the roots of f(x) = 0 are where its graph crosses the x-axis.
Rounding
Replacing a number by a nearby simpler one to a chosen precision, following the half-up or half-even convention.
Scalar
A quantity with magnitude only, such as temperature or mass — contrasted with a vector.
Set
A well-defined collection of distinct objects, the basic building block of modern mathematics.
Sine
For an acute angle in a right triangle, the ratio of the opposite side to the hypotenuse; the fundamental periodic function.
Slope
The steepness of a line: rise over run, or the change in y per unit change in x.
Standard Deviation
The square root of the variance; the typical distance of data values from their mean.
Stream Function
A function ψ whose derivatives give the velocity components of a 2-D incompressible flow; its level curves are streamlines.
Symmetry
Invariance of a figure under a transformation such as reflection, rotation, or translation.
Tangent
The ratio sin θ/cos θ; geometrically, a line touching a curve at exactly one local point, whose slope is the derivative.
Triangle Inequality
Any side of a triangle is shorter than the sum of the other two; also |a + b| ≤ |a| + |b| for numbers and vectors.
Truth Table
A table listing the truth value of a compound statement for every combination of its components' truth values.
Turbulence
Chaotic, swirling fluid motion with strong mixing, arising at high Reynolds numbers.
Uniform Distribution
A distribution in which every outcome in the range is equally likely, like a fair die.
Unit Circle
The circle of radius 1 centred at the origin; the stage on which sine and cosine are defined for all angles.
Unit Vector
A vector of length 1 used to specify direction; any nonzero vector is normalised by dividing by its magnitude.
Upper Bound
A number at least as large as every element of a set; the least such number is the supremum.
Variance
The average of squared deviations from the mean; the square of the standard deviation.
Vector
A quantity with both magnitude and direction, such as velocity or force, drawn as an arrow.
Venn Diagram
Overlapping circles picturing the relationships — unions, intersections, complements — among sets.
Viscosity
A fluid's internal resistance to shearing motion; the coefficient μ in Newton's law of viscosity τ = μ du/dy.
Wavelength
The distance over which a periodic wave repeats; the spatial period of the wave.
Weighted Average
An average in which each value counts in proportion to an assigned weight, such as credit hours in a GPA.
Womersley Number
The dimensionless group α = r√(ω/ν) comparing pulsation frequency with viscous diffusion in oscillatory pipe flow, central to blood-flow modelling.
Work
Energy transferred by a force acting through a distance: W = F·d for a constant aligned force.
Wronskian
A determinant built from functions and their derivatives that tests whether solutions of a linear ODE are independent.
Xi (ξ)
A Greek letter commonly used for random variables, similarity variables, and dummy variables of integration.
x-axis
The horizontal reference line of the Cartesian plane, along which the y-coordinate is zero.
x-intercept
A point where a graph crosses the x-axis; found by setting y = 0 and solving.
xy-plane
The flat two-dimensional coordinate plane spanned by the x- and y-axes; in 3-D, the plane z = 0.
Y-hat (ŷ)
The symbol for a predicted value from a regression model, distinguishing it from the observed y.
Yaw
Rotation of a body about its vertical axis — one of the three orientation angles alongside pitch and roll.
Yield (Percentage)
In financial mathematics, the annualised return of an investment expressed as a percentage of its price.
y-axis
The vertical reference line of the Cartesian plane, along which the x-coordinate is zero.
y-intercept
The point where a graph crosses the y-axis; for y = mx + c it is (0, c).
Z-axis
The third coordinate axis, perpendicular to the xy-plane, giving depth in three-dimensional space.
Z-score
The number of standard deviations a value lies from the mean: z = (x − μ)/σ; it standardises different scales.
Zeta Function
Riemann's function ζ(s) = Σ 1/nˢ, whose zeros encode the distribution of prime numbers; subject of the most famous open problem in mathematics.
Zone (of a Sphere)
The band of a sphere's surface between two parallel planes; its area is 2πRh, depending only on the band's height.