Editing Multiplicative inverse (section)

{{Short description|Number which when multiplied by x equals 1}}
{{Distinguish-redirect|Reciprocal (mathematics)|Pole and polar{{!}}Reciprocation (geometry)}}

[[Image:Hyperbola one over x.svg|thumbnail|right|300px|alt=Graph showing the diagrammatic representation of limits approaching infinity|The reciprocal function: {{nowrap|1=''y'' = 1/''x''}}. For every ''x'' except 0, ''y'' represents its multiplicative inverse. The graph forms a [[rectangular hyperbola]].]]

In [[mathematics]], a '''multiplicative inverse''' or '''reciprocal''' for a [[number]] ''x'', denoted by 1/''x'' or ''x''<sup>&minus;1</sup>, is a number which when [[Multiplication|multiplied]] by ''x'' yields the [[multiplicative identity]], 1. The multiplicative inverse of a [[rational number|fraction]] ''a''/''b'' is ''b''/''a''. For the multiplicative inverse of a real number, divide 1 by the number. For example, the reciprocal of 5 is one fifth (1/5 or 0.2), and the reciprocal of 0.25 is 1 divided by 0.25, or 4. The '''reciprocal function''', the [[Function (mathematics)|function]] ''f''(''x'') that maps ''x'' to 1/''x'', is one of the simplest examples of a function which is its own inverse (an [[Involution (mathematics)|involution]]).

Multiplying by a number is the same as [[Division (mathematics)|dividing]] by its reciprocal and vice versa. For example, multiplication by 4/5 (or 0.8) will give the same result as division by 5/4 (or 1.25). Therefore, multiplication by a number followed by multiplication by its reciprocal yields the original number (since the product of the number and its reciprocal is 1).

The term ''reciprocal'' was in common use at least as far back as the third edition of ''[[Encyclopædia Britannica]]'' (1797) to describe two numbers whose product is 1; geometrical quantities in inverse proportion are described as {{lang|enm|reciprocall}} in a 1570 translation of [[Euclid]]'s ''[[Euclid's Elements|Elements]]''.<ref>{{not a typo|"In equall Parallelipipedons the bases are reciprokall to their altitudes"}}. ''OED'' "Reciprocal" §3a. Sir [[Henry Billingsley]] translation of Elements XI, 34.</ref>

In the phrase ''multiplicative inverse'', the qualifier ''multiplicative'' is often omitted and then tacitly understood (in contrast to the [[additive inverse]]). Multiplicative inverses can be defined over many mathematical domains as well as numbers. In these cases it can happen that {{nowrap|''ab'' ≠ ''ba''}}; then "inverse" typically implies that an element is both a left and right [[inverse element|inverse]].

The notation ''f'' <sup>−1</sup> is sometimes also used for the [[inverse function]] of the function ''f'', which is for most functions not equal to the multiplicative inverse. For example, the multiplicative inverse {{nowrap|1=1/(sin ''x'') = (sin ''x'')<sup>−1</sup>}} is the [[cosecant]] of x, and not the [[Inverse trigonometric functions|inverse sine of ''x'']] denoted by {{nowrap|sin<sup>−1</sup> ''x''}} or {{nowrap|arcsin ''x''}}. The terminology difference ''reciprocal'' versus ''inverse'' is not sufficient to make this distinction, since many authors prefer the opposite naming convention, probably for historical reasons (for example in [[French language|French]], the inverse function is preferably called the {{langx|fr|bijection réciproque|link=no|label=none}}).