Identity elementIn mathematics, an identity element or neutral element of a binary operation is an element that leaves unchanged every element when the operation is applied. For example, 0 is an identity element of the addition of real numbers. This concept is used in algebraic structures such as groups and rings.
Inverse functionIn mathematics, the inverse function of a functionf (also called the inverse of f) is a function that undoes the operation of f. The inverse of f exists if and only iff is bijective, and if it exists, is denoted by f−1.{\displaystyle f^{-1}.}For a function f:X→Y{\displaystyle f\colon X\to Y}, its inverse f−1:Y→X{\displaystyle f^{-1}\colon Y\to X} admits an explicit description: it sends each element y∈Y{\displaystyle y\in Y} to the unique element x∈X{\displaystyle x\in X} such that f(x) = y. As an example, consider the real-valued function of a real variable given by f(x) = 5x − 7.
Multiplicative inverseIn mathematics, a multiplicative inverse or reciprocal for a numberx, denoted by 1x{\displaystyle {\tfrac {1}{x}}} or x, is a number which when multiplied byx yields the multiplicative identity, 1. The multiplicative inverse of a fractionab{\displaystyle {\tfrac {a}{b}}} is ba.{\displaystyle {\tfrac {b}{a}}.} Dividing 1 by a real number yields its multiplicative inverse.
Ring (mathematics)Algebraic structure with addition and multiplicationIn mathematics, a ring is an algebraic structure consisting of a set with two binary operations typically called addition and multiplication and denoted like addition and multiplication of integers. They work similarly to integer addition and multiplication, except that multiplication in a ring does not need to be commutative.
Group (mathematics)Set with associative invertible operationThe manipulations of the Rubik's Cube form the Rubik's Cube group.In mathematics, a group is a set with an operation that combines any two elements of the set to produce a third element within the same set. The following conditions must hold: the operation is associative, it has an identity element, and every element of the set has an inverse element. For example, the integers with the addition operation form a group.Invertible matrixIn linear algebra, an invertible matrix (non-singular, non-degenerate or regular) is a square matrix that has an inverse. In other words, if a matrix is invertible, it can be multiplied by its inverse matrix to yield the identity matrix. Invertible matrices are the same size as their inverse.
Operation (mathematics)In mathematics, an operation is a function that takes as input a fixed number of elements of a set and returns an element of the same set. For example, addition on real numbers is an operation that accepts two real numbers and returns a real number. In general, the input values may be called "operands" or "arguments". The number of operands is the arity of the operation. The arity is usually one of 0,1,2,…{\displaystyle 0,1,2,\ldots }.
Additive inverseIn mathematics, the additive inverse of an elementx, denoted −x, is the element that when added to x, yields the additive identity. This additive identity is often the number 0 (zero), but it can also refer to a more generalized zero element. In elementary mathematics, the additive inverse is often referred to as the opposite number, or the negative of a number.
Unit (ring theory)In algebra, a unit or invertible element of a ring is an invertible element for the multiplication of the ring.
MathematicsMathematics is a field of knowledge concerned with abstract concepts such as numbers, geometric shapes, sets, functions, and probabilities. It uses logical reasoning and proof to study and establish their properties, often expressed as theorems, formulas, and equations. Mathematics is used to model and solve problems in science, engineering, technology, economics, and everyday life.Associative propertyIn mathematics, the associative property is a property of some binary operations that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a validrule of replacement for expressions in logical proofs. Within an expression containing two or more occurrences in a row of the same associative operator, the order in which the operations are performed does not matter as long as the sequence of the operands is not changed.
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