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.
  • Formula vs. sentence Hierarchy
    In first-order logic, statements about objects are written as strings of symbols with variables and quantifiers. A formula is every such string that is built by the rules of the language, such as $x > 3$ or $\forall x \, (x^2 \geq 0)$, and the variables in a formula can be free or bound by quantifiers. A sentence is a formula in which a quantifier binds every variable, so a sentence is true or false once its symbols are interpreted, without assigning values to variables.
  • 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.
  • Operation vs. relation Siblings
    Symbols placed between two objects, such as $+$ and $<$, do different kinds of things. A binary operation takes two objects and produces a new object of the same type. A binary relation takes two objects and says whether they are related (true or false).
  • Proposition vs. predicate Siblings
    In logic, statements are classified by whether they include enough information to determine whether they are true or false. A proposition is a statement that is true or false, such as “$7 > 3$.” A predicate is a statement with variables, such as “$x > 3$,” that becomes a proposition once its variable $x$ is given a value, such as “$7 > 3$” (true) or bound by quantifiers, such as “$\forall x,\, x > 3$” (false).
  • Proposition vs. propositional formula Siblings
    In logic, statements are combined with logical connectives such as “and” ($\land$), “or” ($\lor$), and “not” ($\neg$). A proposition is a particular statement that is true or false, such as “$7$ is prime.” A propositional formula is a pattern, such as $p \land \neg q$, built from propositional variables that stand for propositions, and the formula has a truth value only once each variable is assigned a truth value. Substituting a proposition for each variable in a propositional formula gives a proposition.
  • Sentence vs. proposition Siblings
    In logic, a statement can be studied as a string of symbols or as a claim that is true or false. A sentence is a formula of first-order logic, a string of symbols in which a quantifier binds every variable. A proposition is a claim that is true or false. A sentence expresses a proposition only once its symbols are interpreted, for example by choosing the set that the variables range over, and the same sentence can express different propositions under different interpretations.
  • Sentence vs. propositional formula Siblings
    Formal logic has formulas at two levels: propositional logic (zeroth-order logic), whose variables stand for whole statements, and first-order logic, whose variables stand for objects such as numbers. A sentence is a formula of first-order logic in which a quantifier, such as “for all” ($\forall$) or “there exists” ($\exists$), binds every variable, and the sentence is true or false once its symbols are interpreted, for example as statements about real numbers. A propositional formula is a formula of propositional logic, built from propositional variables with logical connectives such as $\land$ and $\neg$, and the formula is true or false once each variable is assigned a truth value.
  • Term vs. formula Siblings
    In logic, the expressions of a formal language, which are the well-formed strings of symbols, come in two kinds. A term, such as $x^2 + 1$, names an object, so a term has a value, such as a number. A formula, such as $x^2 + 1 > 3$, makes a claim about objects, so a formula has a truth value. Applying a relation symbol, such as $>$ or $=$, to terms gives a formula.
  • Types of collections Siblings
    Mathematics has several kinds of “a bunch of things,” which differ in whether order and repetition matter. A set ignores order and repetition. A multiset keeps repetition. A sequence keeps order and repetition. “Collection” is informal, and “group” is a technical term for a set with an operation.
  • 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.
  • Types of sameness Family
    Mathematics has many ways to say that two objects are “the same.” Equal objects are the same object. The other words describe objects that are different but “the same in some respect,” and each one depends on what respect is meant.

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 an eigenvector: $Ax = \lambda x$. A singular value of a matrix $A$, square or not, is a factor $\sigma$ by which $A$ stretches vectors along some direction, allowing them to turn. Singular values are real and nonnegative, but eigenvalues can be negative or complex.
  • Linearly independent vs. orthogonal Hierarchy
    A list of vectors can point in “different directions” in two senses. A set of linearly independent vectors have no redundant vectors: no vector in the list is a linear combination of the others. Orthogonal vectors are pairwise at right angles: every two different vectors in the list have inner product $0$. Linear independence is defined in every vector space, but orthogonality requires an inner product.
  • Linearly independent vs. pairwise non-parallel Hierarchy
    A list of vectors can avoid redundancy as a whole or one pair at a time. A list of linearly independent vectors has no vector that is a linear combination of the others, which rules out redundancy among three or more vectors as well as among pairs. A list of pairwise linearly independent (pairwise non-parallel) vectors has no two vectors on the same line through the origin.
  • 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.
  • Right angle vs. perpendicular vs. orthogonal vs. normal Siblings
    Right angles, perpendicular lines, orthogonal vectors, and normal vectors all describe things that meet “squarely.” A right angle is an angle of $90^\circ$, so the term describes an angle. Perpendicular describes lines or planes that intersect and form right angles. Orthogonal describes vectors whose inner product is $0$, which extends the idea to vectors that do not meet, such as directions of skew lines, to the zero vector, and to vectors that are not arrows, such as functions. A normal vector is perpendicular to a line, curve, or surface at a point, so the term describes a vector relative to a shape.
  • 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.

Dynamical Systems

  • Periodic orbit vs. limit cycle Hierarchy
    In a dynamical system, a solution can return to its starting state and go around the same loop again and again. A periodic orbit is such a loop. A limit cycle is a periodic orbit that is isolated, with no other periodic orbits nearby, so that nearby solutions spiral toward the limit cycle or away from it.

Control Theory

  • Observable vs. reconstructible vs. detectable Hierarchy
    Observability, reconstructibility, and detectability describe how much the output of a system reveals about its state. An observable system’s initial state can be determined from its output, whereas a reconstructible system’s current state can be determined. A detectable system may have unobservable parts but those parts must that decay to zero so that an observer’s error still converges.
  • State feedback vs. output feedback Siblings
    A feedback controller computes its input from information about the system, which may be more or less complete. State feedback uses the full state, which is rarely measured directly. Output feedback uses only the measured output, and it may need an observer or other dynamics to do what state feedback can.
  • Types of control Siblings
    A controller can decide its input with or without measuring what the system is doing. Open-loop control computes the input without measuring the output, and closed-loop (feedback) control uses the measured output to correct errors. Feedforward acts on a measured disturbance or a known reference before it affects the output, and is usually combined with feedback.
  • Types of control objectives Siblings
    A feedback controller is designed to meet a goal, and the goals have different names. Stabilization makes an equilibrium asymptotically stable. Regulation holds an output at a constant set point despite disturbances. Tracking makes an output follow a reference that changes over time.
  • 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.
  • Types of exogenous inputs Family
    A control system is affected by signals that it does not generate itself, and these signals have different names depending on their role. Exogenous inputs are any inputs from outside the system. A control input is chosen; a disturbance acts on the dynamics; measurement noise only corrupts what is measured. A perturbation is a change to the system itself, which may or may not be modeled as an input.

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.
  • Right angle vs. perpendicular vs. orthogonal vs. normal Siblings
    Right angles, perpendicular lines, orthogonal vectors, and normal vectors all describe things that meet “squarely.” A right angle is an angle of $90^\circ$, so the term describes an angle. Perpendicular describes lines or planes that intersect and form right angles. Orthogonal describes vectors whose inner product is $0$, which extends the idea to vectors that do not meet, such as directions of skew lines, to the zero vector, and to vectors that are not arrows, such as functions. A normal vector is perpendicular to a line, curve, or surface at a point, so the term describes a vector relative to a shape.
  • 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