In mathematics, many terms are similar but not interchangeable. This dictionary groups easily confused terms, defines each one, and explains the distinction with examples to illustrate the difference. This work was inspired by the Dictionary of Fine Distinctions: Nuances, Niceties, and Subtle Shades of Meaning by Eli Burnstein.

See also the lists of all definitions and all examples.

Distinctions by subject

Numbers

  • Natural numbers: starting at 0 or at 1 Conflicting definitions
    Whether the natural numbers include $0$ depends on the author. Set theory, logic, and computer science usually include it, while number theory and analysis often start at $1$.
  • Whole numbers vs. natural numbers Synonyms
    “Whole numbers” and “natural numbers” are used synonymously by some others to indicate all nonnegative integers, $0, 1, 2, 3, \dots$. “Whole numbers” is the usual name in primary schools, and “natural numbers” (or “nonnegative integers”) is the usual name in mathematics. “Natural numbers” is, however, used by some authors to indicate the non-negative integers, $1, 2, 3, \dots$ and “whole numbers” is used to indicate all integers, “$\dots, -2, -1, 0, 1, 2, \dots$.

Sets and Logic

  • Disjoint union: union of disjoint sets vs. tagged union Conflicting definitions
    “Disjoint union” can mean an ordinary union of sets that happen to be disjoint, or a construction that tags each element with the set it came from, so that copies of overlapping sets are kept apart.
  • Function vs. map Synonyms
    “Function” and “map” (or “mapping”) mean exactly the same thing. “Map” is more common when the function preserves some structure or acts on geometric objects, but every statement about functions holds for maps, and vice versa.
  • Types of infinity Family
    Infinity appears in several different senses: as the size of a set, as unboundedness, and as the limit of a divergent sequence. An infinite set has infinitely many elements, but it can be bounded. Infinite sets come in different sizes: countable and uncountable. A divergent sequence does not converge, but it doesn’t necessarily go to infinity.

Analysis

Topology

Linear Algebra

  • Array vs. matrix vs. linear map Siblings
    Grids of numbers appear throughout linear algebra and computing, but a grid can be just data, a matrix with its own arithmetic, or a way of writing a linear map. An array is a grid of data. A matrix is a grid of numbers with matrix arithmetic. A linear map is a function between vector spaces. A matrix represents a linear map as an array of scalars with respect to a given vector basis.
  • Dot product vs. inner product vs. bilinear form Hierarchy
    Bilinear forms, inner products, and the dot product are all ways of multiplying two vectors to get a number, from the most general to the most specific. The dot product is one particular inner product on $\reals^n$. An inner product is a bilinear form that is also symmetric and positive definite.
  • Eigenvalue vs. singular value Siblings
    Eigenvalues and singular values both describe how a matrix stretches vectors, but they agree only for special matrices. An eigenvalue of a square matrix $A$ is a factor $\lambda$ by which $A$ scales a particular nonzero vector $x$, called a eigenvalue: $Ax = \lambda x$. A singular value of a matrix $A$, square or not, is a factor $\sigma$ by which $A$ stretches vectors along some direction, allowing it them turn. Singular values are real and nonnegative, but eigenvalues can be negative or complex.
  • Norm vs. metric vs. inner product Siblings
    Norms, metrics, and inner products measure size, distance, and angle, and each one gives the next in a natural way. A norm measures the lengths of vectors, a metric measures distances between points, and an inner product measures both lengths and angles. An inner product gives a norm, and a norm gives a metric, but not the reverse.
  • Positive definite vs. positive semidefinite Hierarchy
    Positive definiteness and semidefiniteness describe the sign of the quadratic form $x^{\top} A x$ of a symmetric matrix, and they appear in optimization, stability, and statistics. A symmetric matrix $A$ is positive semidefinite if $x^{\top} A x \geq 0$ for every vector $x$, and positive definite if $x^{\top} A x > 0$ for every nonzero $x$. Equivalently, the eigenvalues of $A$ are nonnegative or positive.
  • Types of matrix decompositions Siblings
    A matrix decomposition writes a matrix in terms of simpler matrices. Most are products: LU, PLU, and Cholesky decompositions use triangular factors, QR and singular value decompositions use orthogonal factors, and eigendecompositions, Schur decompositions, and Jordan normal forms are similarities $A = P M P^{-1}$ to a simpler matrix $M$ (diagonal, triangular, or made of Jordan blocks). In contrast, the Jordan–Chevalley decomposition is a sum of commuting diagonalizable and nilpotent parts, and the Toeplitz decomposition is a sum $H + K$ of a symmetric matrix $H$ and a skew-symmetric matrix $K$.
  • Types of norms Siblings
    A norm measures the size of a vector, and $\reals^n$ has several common norms that measure size in different ways. The Euclidean norm is the straight-line length, the 1-norm adds the sizes of the entries, and the infinity norm takes the largest entry. Each is a norm, so each satisfies the triangle inequality, and for every vector, $\norm{x}_{\infty} \leq \norm{x}_2 \leq \norm{x}_1$.
  • Vector vs. coordinate vector Siblings
    Vectors in a vector space are often described by lists of numbers, which depend on a choice of basis. A vector is an element of a vector space. Its coordinate vector is a list of numbers (coordinates) that describes it relative to a chosen basis. Changing the basis changes the coordinates, not the vector.
  • Vector vs. covector Siblings
    Vectors and covectors are both described by lists of numbers in coordinates, so they are easy to confuse. A covector is a linear function that takes a vector and returns a number. In coordinates, both look like lists of numbers, but when the basis changes, their coordinates change in opposite ways.
  • Vector vs. point Siblings
    Points and vectors are both written as lists of coordinates in $\reals^n$, but they behave differently. A point is a location. A vector is a displacement, defined by the operations of addition and scaling. The difference of two points is a vector. Two vectors can be added, and a vector can be added to a point, but the sum of two points is not meaningful.

Control Theory

  • Types of estimators Family
    When the state of a system is not measured directly, it can be estimated in several ways. An estimator is any rule for estimating an unknown quantity. An observer is a dynamical system whose state estimate converges to the true state. A filter estimates the current state from noisy measurements, and a predictor estimates a future state.

Statistics

  • Measures of center Siblings
    The mean, median, and mode are different ways to summarize the middle of data, and they can be far apart. “Average” usually means the mean. The geometric mean averages growth factors and ratios, and the expected value is the mean of a probability distribution rather than of data.
  • Poisson vs. exponential distribution Siblings
    Consider events that happen independently of each other at a constant average rate $\lambda$, such as customers arriving at a shop (a Poisson process). The Poisson distribution describes the number of events in a fixed length of time, which is a whole number. The exponential distribution describes the time until the next event, which is a positive real number.
  • Topology vs. σ-algebra Siblings
    Topologies and σ-algebras are both collections of subsets that are closed under some operations. A topology is closed under arbitrary unions and finite intersections, and describes nearness. A σ-algebra is closed under complements and countable unions, and describes what can be measured.
  • Variance vs. sample variance Siblings
    The spread of data is measured by the average squared distance of the data from their mean. The variance of the data divides the sum of the squared distances by the number $n$ of data points, which describes the data themselves. The sample variance divides by $n - 1$ instead, which gives an unbiased estimate of the variance of the population that the data were sampled from.

Mathematical Language

  • Conjecture vs. hypothesis vs. law vs. theory Siblings
    Mathematics and science both have words for statements that are proposed, tested, or established, and the words are often confused. A conjecture is a statement believed true but not yet proved, and it belongs to mathematics, where statements are settled by proof. In science, where statements are tested by evidence, a hypothesis is a testable proposal, and a theory is a well-supported explanation of many observations. A law describes a pattern that holds in every tested case, often as an equation such as $F = G m_1 m_2 / r^2$, without explaining why the pattern holds; explaining it is the job of a theory. In everyday speech, “theory” means a guess, but in science a theory is an a principle or body of principles that explains observed data and phenomenon.
  • Types of difference Family
    Many quantities compare two things by “how far apart” they are. A difference is the result of subtraction and can be negative. No distance is negative. An error compares an approximation with the truth, a change compares the end with the start, and a set difference removes elements.

Geometry

  • Convex set vs. convex function Siblings
    “Convex” describes both sets and functions, and the two meanings are linked. A convex set contains the segment between every two of its points, and a convex function lies on or below the segment between every two points on its graph. A function is convex exactly when the region on and above its graph is a convex set.
  • Types of distances Siblings
    A metric measures how far apart two points are, and the same two points can be at different distances under different metrics. The Euclidean distance is the straight-line distance, the taxicab distance travels along the axes, and the Chebyshev distance takes the largest difference in one coordinate. The geodesic distance is the length of the shortest path that stays on a surface, such as the great-circle distance on the Earth. The discrete metric only says whether two points are equal.
  • Vector vs. point Siblings
    Points and vectors are both written as lists of coordinates in $\reals^n$, but they behave differently. A point is a location. A vector is a displacement, defined by the operations of addition and scaling. The difference of two points is a vector. Two vectors can be added, and a vector can be added to a point, but the sum of two points is not meaningful.

Computing