View source for Inverse

{{also|inversé}}
==English==
{{wikipedia|Inverse}}

===Etymology===
Recorded since 1440, from {{uder|en|la|inversus}}, the past participle of {{m|la|invertere||to invert}}, itself from {{m|la|in-||in, on}} + {{m|la|vertere||to turn}}.

===Pronunciation===
* {{IPA|en|/ɪnˈvɝs/|/ˈɪnvɝs/|a=GA}}
* {{audio|en|en-ca-inverse.oga|a=CA}}
* {{rhymes|en|ɜ(ɹ)s|s=2}}

===Adjective===
[[File:Satisfaction sterility.jpg|thumb|right|200px|A chart depicting the strong '''inverse''' relationship between prosperity and fertility in existing societies]]
{{en-adj|-}}

# [[opposite|Opposite]] in effect, nature or order.
# [[reverse|Reverse]], opposite in order.
# {{lb|en|botany}} Inverted; having a position or mode of attachment the reverse of that which is usual.
# {{lb|en|mathematics}} Having the properties of an '''inverse'''; ''said with reference to any two operations, which, when both are performed in succession upon any quantity, reproduce that quantity''.
#: {{ux|en|Multiplication is the '''inverse''' operation to division.}}
# {{lb|en|geometry}} That has the property of being an '''inverse''' (the result of a [[circle inversion]] of a given point or geometrical figure); that is constructed by circle inversion. {{C|en|Curves}}
#: {{ux|en|A circle inversion maps a given [[generalized circle]] to its '''inverse''' generalized circle.}}
# {{lb|en|category theory|of a category}} Whose every element has an '''inverse''' ([[morphism]] which is both a [[left inverse]] and a [[right inverse]]).

====Derived terms====
{{col|en|direct-inverse marking|inverse number|inverse agonist<qq:pharmacology>|inverse attraction<qq:grammar>}}

====Related terms====
* {{l|en|inversion}}
* {{l|en|inversional}}
* {{l|en|inversive}}
* {{l|en|reverse}}
* {{l|en|reversion}}
* {{l|en|reversional}}
* {{l|en|reversive}}

====Translations====
{{trans-top|opposite in effect or nature}}
* Bulgarian: {{t+|bg|противоположен}}
* Catalan: {{t+|ca|invers}}
* Chinese:
*: Mandarin: {{t+|cmn|逆|tr=nì}}
* Czech: {{t|cs|inverzní}}
* Dutch: {{t+|nl|tegengesteld}}, {{t+|nl|tegenovergesteld}}
* Finnish: {{t+|fi|päinvastainen}}
* French: {{t+|fr|inversé}}
* German: {{t+|de|entgegengesetzt}}
* Spanish: {{t+|es|inverso}}
{{trans-bottom}}

{{trans-top|reverse, opposite in order}}
* Azerbaijani: {{t+|az|tərs}}
* Bulgarian: {{t+|bg|обратен}}
* Catalan: {{t+|ca|invers}}
* Chinese:
*: Mandarin: {{t+|cmn|逆|tr=nì}}
* Dutch: {{t+|nl|omgekeerd}}
* Esperanto: {{t|eo|inversa}}
* Finnish: {{t+|fi|käänteinen}}, {{t+|fi|päinvastainen}}
* French: {{t+|fr|inversé}}
* German: {{t+|de|umgekehrt}}
{{trans-bottom}}

{{trans-top|(''mathematics'') having the properties of an inverse}}
* Azerbaijani: {{t+|az|tərs}}
* Bulgarian: {{t+|bg|реципрочен}}
* Catalan: {{t+|ca|invers}}
* Chinese:
*: Mandarin: {{t+|cmn|逆|tr=nì}}
* Dutch: {{t+|nl|invers}}
* Finnish: {{t|fi|[[käänteinen|käänteis-]]}}, {{t+|fi|käänteinen}}
* Spanish: {{t+|es|inverso}}
* Swedish: {{t+|sv|invers|c}}
{{trans-bottom}}

{{checktrans-top}}
* Esperanto: {{t-check|eo|inversa}}
* French: {{t+check|fr|inverse}}
* Japanese: {{t+check|ja|あべこべ|tr=abekobe}}
{{trans-bottom}}

===Noun===
{{en-noun}}

# An [[invert]]ed [[state]]: a [[state]] in which [[something]] has been [[turn]]ed {{lb|en|properly}} [[upside down]] or {{lb|en|loosely}} [[inside out]] or [[backward]]s.
#: ''[[cowgirl|Cowgirl]] is the '''inverse''' of [[missionary]].''
#: ''321 is the '''inverse''' of 123.''
# The [[result]] of an [[inversion]], ''[[particularly]]'':
## The [[reverse]] of [[any]] [[procedure]] or [[process]].
##: ''Uninstalling is the '''inverse''' of installation.''
## {{lb|en|mathematics}} A [[ratio]] etc. in which the [[antecedent]]s and [[consequent]]s are [[switch]]ed.
##: ''The '''inverse''' of a:b is b:a.''
## {{lb|en|geometry}} The [[result]] of a [[circle inversion]]; the [[set]] of [[all]] [[such]] [[point]]s; the [[curve]] [[describe]]d by [[such]] a [[set]].
##: ''The '''inverse''' ''P‘'' of a point ''P'' is the point on a ray from the center ''O'' through ''P'' such that ''OP'' × ''OP‘'' = ''r²'' or the set of all such points.''
## {{lb|en|logic}} The [[non]]-[[truth]]-[[preserving]] [[proposition]] [[construct]]ed by [[negating]] [[both]] the [[premise]] and [[conclusion]] of an [[initially]] [[given]] [[proposition]].
##: ''"Anything that isn't a dog doesn't go to heaven" is the '''inverse''' of "All dogs go to heaven." More generally, <math>\lnot\mathsf{p}\to\lnot\mathsf{q}</math> is the '''inverse''' of <math>\mathsf{p}\to\mathsf{q}</math> and is equivalent to the [[converse]] proposition <math>\mathsf{q}\to\mathsf{p}</math>.''
##* {{quote-text|en|year=1896|author=James Welton|title=A Manual of Logic|edition=2nd|section=Bk iii, Ch. iii, §102
|passage=Inversion is the inferring, from a given proposition, another proposition whose subject is the contradictory of the subject of the original proposition. The given proposition is called the Invertend, that which is inferred from it is termed the '''Inverse'''... The rule for Inversion is: Convert either the Obverted Converse or the Obverted Contrapositive.}}
# {{lb|en|mathematics}} A [[second]] [[element]] which [[negate]]s a [[first]]; in a [[binary]] [[operation]], the [[element]] for which the [[binary]] [[operation]]—when [[applied]] to [[both]] it and an [[initial]]ly [[given]] [[element]]—[[yield]]s the [[operation]]'s [[identity element]], ''[[specifically]]:''
## {{lb|en|addition}} The [[negative]] of a [[given]] [[number]].
##: ''The [[additive]] '''inverse''' of <math>x</math> is <math>-x</math>, as <math>x-x=0</math>, as <math>0</math> is the additive identity element.''
## {{lb|en|multiplication}} [[one|One]] [[divide]]d by a [[given]] [[number]].
##: ''The [[multiplicative]] '''inverse''' of <math>x</math> is <math>x^{-1}</math>, as <math>x \times x^{-1}=1</math>, as <math>1</math> the multiplicative identity element.''
## {{lb|en|functions}} A [[second]] [[function]] which, when [[combine]]d with the [[initially]] [[given]] [[function]], [[yield]]s as its [[output]] [[any]] [[term]] [[input]]ted into the [[first]] [[function]].
##: ''The [[compositional]] '''inverse''' of a function <math>f</math> is <math>f^{-1}</math>, as <math>f\ f^{-1}=\mathit{I}</math>, as <math>\mathit{I}</math> is the identity function. That is, <math>\forall x,f(f^{-1}(x))=\mathit{I}(x)=x</math>.''
# {{lb|en|category theory}} A [[morphism]] which is both a [[left inverse]] and a [[right inverse]].
# {{lb|en|card games}} The [[win]]ning of the [[coup]] in a [[game]] of [[rouge et noir]] by a [[card]] of a [[color]] [[different]] from that [[first]] [[dealt]]; the [[area]] of the [[table]] [[reserve]]d for [[bet]]s upon [[such]] an [[outcome]].
#* {{quote-text|en|year=1850|author=Henry George Bohn|title=The Hand-book of Games|page=343
|passage=If the player... be determined to try his luck on the '''inverse''', he must place his money on a yellow circle, or rather a collection of circles, situated at the extremity of the table.}}
#* {{quote-text|en|year=1950|author=Lawrence Hawkins Dawson|title=Hoyle's Games Modernized|edition=20th|page=291
|passage=The ''[[tailleur]]'' never mentions the words ‘Black’ or ‘'''Inverse'''’, but always says that Red wins or Red loses, and that the colour wins or the colour loses.}}
# {{lb|en|linguistics|Kiowa-Tanoan}} A grammatical number marking that indicates the opposite grammatical number (or numbers) of the default number specification of noun class.

====Synonyms====
* {{sense|addition}} {{l|en|additive}} inverse
* {{sense|multiplication}} {{l|en|multiplicative}} inverse
* {{sense|composition}} {{l|en|compositional}} inverse
* {{sense|geometry}} inverse {{l|en|point}}, inverse {{l|en|curve}}

====Translations====
{{trans-top|opposite of a given due to contrary effects or nature}}
* Bulgarian: {{t+|bg|противоположност|f}}
* Dutch: {{t+|nl|tegendeel|n}}, {{t+|nl|tegengestelde|n}}, {{t+|nl|tegenovergestelde|n}}
* Finnish: {{t+|fi|vastakohta}}
{{trans-bottom}}

{{trans-top|reverse procedure}}
* Bulgarian: {{t+|bg|обръщане|n}}
* Dutch: {{t+|nl|omgekeerde|n}}, {{t+|nl|spiegelbeeld|n}}
* Finnish: {{t|fi|[[käänteinen]] [[toimenpide]]}}
* Trió: {{t|tri|da}}
{{trans-bottom}}

{{trans-top|math: element}}
* Bulgarian: {{t|bg|реципрочно число}}
* Chinese:
*: Mandarin: {{t+|cmn|逆元|tr=nìyuán}}
* Dutch: {{t+|nl|inverse|c}}
* Finnish: {{t+|fi|käänteisarvo}}
* Hindi: {{t+|hi|प्रतिलोम}}
{{trans-bottom}}

{{trans-top|logic: statement}}
* Chinese:
*: Mandarin: {{t|cmn|否命题|tr=fǒumìngtí}}
* Finnish: {{t|fi|[[vastakkainen]] [[lause]]}}, {{t|fi|[[vastakkainen]] [[väittämä]]}}
* Korean: {{t+|ko|이}}
{{trans-bottom}}

{{checktrans-top}}
* French: {{t+check|fr|inverse}}, {{t+check|fr|contraire}}
* Japanese: {{t+check|ja|あべこべ|tr=abekobe|sc=Jpan}}
* Swedish: {{t+check|sv|invers|c}}
{{trans-bottom}}

====See also====
* {{sense|logic}} {{l|en|obverse}}, {{l|en|converse}}, {{l|en|contraposition}}

===Verb===
{{en-verb}}

# {{lb|en|surveying}} To compute the [[bearing]] and [[distance]] between two [[point]]s.

====Antonyms====
* {{l|en|compute}} {{qualifier|a point}}.

====Derived terms====
{{col2|en|title=terms derived from all parts of speech|inverse function|additive inverse|conceptual inverse|inverse condemnation|inverse cosine|inverse Fourier transform|inverse hyperbolic function|inverse kinematics|inverse limit|inverse matrix|inverse proportion|inverse semigroup|inverse sine|inverse system|inverse tangent|inverse trigonometric function|inverse-square law|multiplicative inverse|inverse image|inverse spelling|inversely|inverse video}}

===Anagrams===
* {{anagrams|en|a=eeinrsv|Severin|enviers|inserve|veiners|venires|versine}}

==Danish==

===Adjective===
{{head|da|adjective form}}

# {{da-e-form of|invers}}

==Dutch==
{{wikipedia|lang=nl}}

===Pronunciation===
* {{audio|nl|Nl-inverse.ogg}}
* {{hyphenation|nl|in|ver|se}}

===Noun===
{{nl-noun-adj}}

# {{l|en|inverse}}

===Adjective===
{{head|nl|adjective form}}

# {{infl of|nl|invers||infl}}

===Anagrams===
* {{l|nl|viseren}}

==French==

===Pronunciation===
* {{fr-IPA}}
* {{audio|fr|Fr-inverse.ogg}}

===Etymology 1===
From {{der|fr|la|inversus}}.

====Adjective====
{{fr-adj}}

# {{l|en|inverse}}, the other way round

=====Derived terms=====
* {{l|fr|barre inverse}}
* {{l|fr|dérivation inverse}}
* {{l|fr|inversement}}

====Noun====
{{fr-noun|m}}

# the {{l|en|inverse}}, the [[contrary]]
#: {{syn|fr|contraire|envers}}

=====Derived terms=====
{{col3|fr
|à l'inverse
|à l'inverse de
}}

===Etymology 2===

====Verb====
{{head|fr|verb form}}

# {{inflection of|fr|inverser||1//3|s|pres|ind//sub|;|2|s|impr}}

===Further reading===
* {{R:fr:TLFi}}

===Anagrams===
* {{l|fr|enivres}}, {{l|fr|enivrés}}
* {{l|fr|reviens}}
* {{l|fr|Séverin}}
* {{l|fr|vernies}}

==German==

===Pronunciation===
* {{audio|de|De-inverse.ogg}}

===Adjective===
{{head|de|adjective form}}

# {{de-adj form of|invers}}

==Italian==

===Adjective===
{{head|it|adjective form}}

# {{feminine plural of|it|inverso}}

===Anagrams===
* {{anagrams|it|a=eeinrsv|svenire}}

==Latin==

===Participle===
{{head|la|participle form}}

# {{inflection of|la|inversus||voc|m|s}}