Mathematical terminology

rigorous demonstration that a mathematical statement follows from its premises

thumb|The Pythagorean theorem has at least 370 known proofs.

thumb|A definition states the meaning of a word using other words. This is sometimes challenging. Common dictionaries contain lexical descriptive definitions, but there are various types of definition – all with different purposes and focuses.

mathematical relation comparing two different values

In mathematics, a coefficient is a multiplicative factor involved in some term of a polynomial, a series, or any other type of expression. It may be a number without units, in which case it is known as a numerical factor. It may also be a constant with units of measurement, in which it is known as a constant multiplier. In general, coefficients may be any expression (including variables such as , and ). When the combination of variables and constants is not necessarily involved in a product, it may be called a parameter. For example, the polynomial 2x^2-x+3 has coefficients 2, −1, and 3, and

in mathematics, two variables are proportional if a change in one is always accompanied by a change in the other, and if the changes are always related by use of a constant

description of a system using mathematical concepts and language

thumb|350px|The real part (red) and imaginary part (blue) of the Riemann zeta function along the critical line Re(s) = 1/2. The first non-trivial zeros can be seen at Im(s) = ±14.135, ±21.022 and ±25.011. The Riemann hypothesis, a famous conjecture, says that all non-trivial zeros of the zeta function lie along the critical line.

logical connective

theorem for proving more complex theorems

A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when identifying the system, or when evaluating its performance, status, condition, etc.

Q.E.D. or QED is an initialism of the Latin phrase , meaning "that which was to be demonstrated". Literally, it states "what was to be shown". Traditionally, the abbreviation is placed at the end of mathematical proofs and philosophical arguments in print publications, to indicate that the proof or the argument is complete.

unit for the arithmetic difference of two percentages

In mathematics and logic, a corollary (, ; , ) is a proposition which can be readily deduced from a previous, already proven proposition. A corollary could be a proposition that is incidentally proved while proving another proposition; it might also be used more casually to refer to something which naturally or incidentally accompanies something else.

property of mathematical objects that remains unchanged under certain transformations applied to the objects

concept in mathematical modeling, statistical modeling and experimental sciences

relative standard deviation: standard deviation divided by the mean

conditional or implicational relationship between two statements: a necessary condition is one which must be present in order for another condition to occur, while a sufficient condition is one which produces the said condition

A counterexample is a specific example that contradicts a claim, hypothesis, or generalization. In logic a counterexample disproves a universally stated claim, and does so rigorously in the fields of mathematics and philosophy. For example, the statement that "student John Smith is not lazy" is a counterexample to the generalization "students are lazy", and both a counterexample to, and disproof of, the universal quantification "all students are lazy."

thumb|400px|right|The lemniscate of Bernoulli and its two foci

collection of maps in which all map compositions starting from the same set and ending with the same set give the same result

notion that some mathematicians may derive aesthetic pleasure from mathematics

Stochastic (; ) is the property of being well-described by a random probability distribution. Stochasticity and randomness are technically distinct concepts. Stochasticity refers to a modeling approach, while randomness describes phenomena. These terms are often used interchangeably. In probability theory, the formal concept of a stochastic process is also referred to as a random process.

set of pseudo units to describe small values of miscellaneous dimensionless quantities

frequently used expression in mathematics

In mathematics, the term linear is used in two distinct senses for two different properties: linearity of a function (or mapping); linearity of a polynomial. An example of a linear function is the function defined by f(x)=(ax,bx) that maps the real line to a line in the Euclidean plane R2 that passes through the origin. An example of a linear polynomial in the variables X, Y and Z is aX+bY+cZ+d.

A proof by contradiction used to show that a statement cannot be true for all numbers. If a statement is true for one number, it will also be true for a smaller number, up to infinity, leading to a contradiction and disproving the original statement.

description to what extent a mathematical statement or complication can be disregarded due to simplicity

indicator for how well data points fit a line or curve

In mathematics, an inequation is a statement that either an inequality (relations "greater than" and "less than", ) or a relation "not equal to" (≠) holds between two values. It is usually written in the form of a pair of expressions denoting the values in question, with a relational sign between the two sides, indicating the specific inequality relation. Some examples of inequations are:

logical quantification stating that a statement holds for exactly one object

mathematical term in the context of differential equations

every element of a partially ordered set A which is greater (resp. lower) than every element of a subset B included in A

idempotent mapping of a set into its subset

standard (often unique) way of presenting an object as a mathematical expression

The significand (also coefficient, sometimes argument, or more ambiguously mantissa, fraction, or characteristic) is the first (left) part of a number in scientific notation or related concepts in floating-point representation, consisting of its significant digits. For negative numbers, it does not include the initial minus sign.

event that happens with probability one

process of extracting the underlying essence of a mathematical concept

In physics and mathematics, an ansatz (; , meaning: "initial placement of a tool at a work piece", plural ansatzes or, from German, ansätze ; ) is an educated guess or an additional assumption made to help solve a problem, and which may later be verified to be part of the solution by its results.

Independent of choice of representatives.

central object of study in category theory

matrix satisfying some of the criteria of an inverse

mathematical statement of uniqueness, except for an equivalent structure (equivalence relation)

term used to describe a property on a set that is false only on a measurable set with zero measure

mathematical symbol "∂", used for partial derivatives and other concepts

mathematical expression

numerical measure of some type of correlation

In mathematics, connectedness is used to refer to various properties meaning, in some sense, "all one piece". When a mathematical object has such a property, we say it is connected; otherwise it is disconnected. When a disconnected object can be split naturally into connected pieces, each piece is usually called a component (or connected component).

deliberately simplistic model with many details removed so that it can be used to explain a mechanism concisely, also useful in a description of the fuller model

informal mathematical concept of an object that behaves in an exceptional way

mathematical term around presentation of concepts

rewriting of an expression into a simpler form

In mathematics, a univariate object is an expression, equation, function or polynomial involving only one variable. Objects involving more than one variable are multivariate. In some cases the distinction between the univariate and multivariate cases is fundamental; for example, the fundamental theorem of algebra and Euclid's algorithm for polynomials are fundamental properties of univariate polynomials that cannot be generalized to multivariate polynomials.

type of abstraction in science, mathematics, and philosophy

Wikimedia disambiguation page about mathematical concepts

a coincidence in mathematics

mathematical notion of infinitesimal difference

space of possible parameter values that define a particular mathematical model

In logic, a metatheorem is a statement about a formal system proven in a metalanguage. Unlike theorems proved within a given formal system, a metatheorem is proved within a metatheory, and may reference concepts that are present in the metatheory but not the object theory.

tongue-in-cheek description of category theory and abstract mathematics