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Additive inverse

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From Wikipedia, the free encyclopedia

Number that, when added to the original number, yields the additive identity

"Opposite number" redirects here. For other uses, seeanalog andcounterpart.

In mathematics, theadditive inverse of anelementx, denoted−x,^[1] is the element that whenadded tox, yields theadditive identity.^[2] This additive identity is often the number0 (zero), but it can also refer to a more generalizedzero element.

Inelementary mathematics, the additive inverse is often referred to as theopposite number,^[3]^[4] or thenegative of a number.^[5] Theunary operation ofarithmetic negation^[6] is closely related tosubtraction^[7] and is important insolving algebraic equations.^[8] Not allsets where addition is defined have an additive inverse, such as thenatural numbers.^[9]

Common examples

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When working withintegers,rational numbers,real numbers, andcomplex numbers, the additive inverse of any number can be found by multiplying it by−1.^[8]

These complex numbers, two of eight values of⁸√1, are mutually opposite

Simple cases of additive inverses
$n {\displaystyle n}$	$-n$
$7 {\displaystyle 7}$	$-7$
$0.35 {\displaystyle 0.35}$	$-0.35$
${\frac {1}{4}}$	$-{\frac {1}{4}}$
$\pi$	$-\pi$
$1+2i$	$-1-2i$

The concept can also be extended to algebraic expressions, which is often used when balancingequations.

Additive inverses of algebraic expressions
$n {\displaystyle n}$	$-n$
$a-b$	$-(a-b)=-a+b$
$2x^{2}+5$	$-(2x^{2}+5)=-2x^{2}-5$
${\frac {1}{x+2}}$	$-{\frac {1}{x+2}}$
${\sqrt {2}}\sin {\theta }-{\sqrt {3}}\cos {2\theta }$	$-({\sqrt {2}}\sin {\theta }-{\sqrt {3}}\cos {2\theta })=-{\sqrt {2}}\sin {\theta }+{\sqrt {3}}\cos {2\theta }$

Relation to subtraction

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The additive inverse is closely related tosubtraction, which can be viewed as an addition using the inverse:

a −b = a + (−b).

Conversely, the additive inverse can be thought of as subtraction from zero:

−a = 0 −a.

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.^[10]

Formal definition

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Given an algebraic structure defined under addition $(S,+)$ with an additive identity $e\in S$ , an element $x\in S$ has an additive inverse $y {\displaystyle y}$ if and only if $y\in S$ , $x+y=e$ , and $y+x=e$ .^[9]

Addition is typically only used to refer to acommutative operation, but for some systems of numbers, such asfloating point, it might not beassociative.^[11] When it is associative, so $(a+b)+c=a+(b+c)$ , 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 requiresclosure, that the additive element $y {\displaystyle y}$ be found in $S {\displaystyle S}$ . However, despite being able to add the natural numbers together, the set of natural numbers does not include the additive inverse values. This is because the additive inverse of a natural number (e.g., $-3$ for $3 {\displaystyle 3}$ ) is not a natural number; it is aninteger. Therefore, the natural numbers in set $S {\displaystyle S}$ do have additive inverses and their associated inverses arenegative numbers.