In mathematics, the additive inverse of an element x, denoted -x, [1] is the element that when added to x, yields the additive identity, 0. [2] In the most familiar cases, this is the number 0, 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. [3] [4] The concept is closely related to subtraction [5] and is important in solving algebraic equations. [6] Not all sets where addition is defined have an additive inverse, such as the natural numbers. [7]
When working with integers, rational numbers, real numbers, and complex numbers, the additive inverse of any number can be found by multiplying it by −1. [6]
The concept can also be extended to algebraic expressions, which is often used when balancing equations.
The additive inverse is closely related to subtraction, which can be viewed as an addition using the inverse:
Conversely, the additive inverse can be thought of as subtraction from zero:
This connection lead to the minus sign being used for both opposite magnitudes and subtraction as far back as the 17th century. While this notation is standard today, it was met with opposition at the time, as some mathematicians felt it could be unclear and lead to errors. [8]
Given an algebraic structure defined under addition with an additive identity , an element has an additive inverse if and only if , , and . [7]
Addition is typically only used to refer to a commutative operation, but it is not necessarily associative. When it is associative, so , the left and right inverses, if they exist, will agree, and the additive inverse will be unique. In non-associative cases, the left and right inverses may disagree, and in these cases, the inverse is not considered to exist.
The definition requires closure, that the additive element be found in . This is why despite addition being defined over the natural numbers, it does not an additive inverse for its members. The associated inverses would be the negative numbers, which is why the integers do have an additive inverse.
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In mathematics, particularly in algebra, a field extension is a pair of fields , such that the operations of K are those of L restricted to K. In this case, L is an extension field of K and K is a subfield of L. For example, under the usual notions of addition and multiplication, the complex numbers are an extension field of the real numbers; the real numbers are a subfield of the complex numbers.
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In mathematics, the concept of an inverse element generalises the concepts of opposite and reciprocal of numbers.
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GF(2) is the finite field with two elements.