inverse
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in·verse
(ĭn-vûrs', ĭn'vûrs') 
adj.
adj.
- Reversed in order, nature, or effect.
- Mathematics. Of or relating to an inverse or an inverse function.
- Archaic. Turned upside down; inverted.
- Something that is opposite, as in sequence or character; the reverse.
- Mathematics. One of a pair of elements in a set whose result under the operation of the set is the identity element, especially:
- The reciprocal of a designated quantity. Also called multiplicative inverse.
- The negative of a designated quantity. Also called additive inverse.
[Middle English, from Latin inversus, past participle of invertere, to invert. See invert.]
inversely in·verse'ly adv.Random House Word Menu:
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categories related to 'inverse'
- Logic - inverse: statement reversed in order, directly opposite form of original
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Additive inverse
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In mathematics, the additive inverse, or opposite, of a number a is the number that, when added to a, yields zero. The additive inverse of a is denoted by unary minus: −a. This can be seen as a shorthand for a common subtraction notation:
- −a = 0 − a
with "0" omitted, although in a correct typography there should be no space after unary "−".
For example, the additive inverse of 7 is −7, because 7 + (−7) = 0, and the additive inverse of −0.3 is 0.3, because −0.3 + 0.3 = 0 .
In other words, the additive inverse of a number is the number's negative. For example, the additive inverse of 8 is −8, the additive inverse of 10002 is −10002 and the additive inverse of x² is −(x²).
The additive inverse of a number is defined as its inverse element under the binary operation of addition. It can be calculated using multiplication by −1; that is, −n = −1 × n .
Integers, rational numbers, real numbers, and complex number all have additive inverses, as they contain negative as well as positive numbers. Natural numbers, cardinal numbers, and ordinal numbers, on the other hand, do not have additive inverses within their respective sets. Thus, for example, we can say that natural numbers do have additive inverses, but because these additive inverses are not themselves natural numbers, the set of natural numbers is not closed under taking additive inverses.
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General definition
The notation + is reserved for commutative binary operations, i.e. such that x + y = y + x, for all x, y . If such an operation admits an identity element o (such that x + o ( = o + x ) = x for all x), then this element is unique ( o′ = o′ + o = o ). For a given x , if there exists x′ such that x + x′ ( = x′ + x ) = o , then x′ is called an additive inverse of x.
If + is associative (( x + y ) + z = x + ( y + z ) for all x, y, z), then an additive inverse is unique
- x″ = x″ + o = x″ + (x + x′) = (x″ + x) + x′ = o + x′ = x′
We often write x − y as x + (−y).[clarification needed]
For example, since addition of real numbers is associative, each real number has a unique additive inverse.
Other examples
All the following examples are in fact abelian groups:
- addition of real valued functions: here, the additive inverse of a function f is the function −f defined by (−f )(x) = − f (x) , for all x, such that f + (−f ) = o , the zero function ( o(x) = 0 for all x ).
- more generally, what precedes applies to all functions with values in an abelian group ('zero' meaning then the identity element of this group):
- complex valued functions,
- vector space valued functions (not necessarily linear),
- sequences, matrices and nets are also special kinds of functions.
- In a vector space the additive inverse −v is often called the opposite vector of v. Additive inversion corresponds to scalar multiplication by −1. For Euclidean space, it is inversion in the origin.
- In modular arithmetic, the modular additive inverse of x is also defined: it is the number a such that a + x ≡ 0 (mod n). This additive inverse does always exist. For example, the inverse of 3 modulo 11 is 8 because it is the solution to 3 + x ≡ 0 (mod 11).
See also
References
- Margherita Barile, "Additive Inverse" from MathWorld.
This entry is from Wikipedia, the leading user-contributed encyclopedia. It may not have been reviewed by professional editors (see full disclaimer)
Translations:
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Inverse
Dansk (Danish)
adj. - omvendt, invers, negativ
n. - invers
idioms:
- inverse proportion omvendt forhold
- inverse ratio omvendt forhold
- inverse square law omvendt lineær ensretning
- inverse video negativ billede
Nederlands (Dutch)
omgekeerde, inverse, omgekeerd
Français (French)
adj. - inverse, inversé, (Mus) renversé, renversé (une image), à l'envers (un objet)
n. - envers, (Math) inverse
idioms:
- inverse proportion inversement proportionnel
- inverse ratio inversement proportionnel
- inverse square law loi du carré inverse
- inverse video vidéo renversée
Deutsch (German)
n. - Gegenteil, Umkehrung
adj. - umgekehrt
idioms:
- inverse proportion umgekehrtes Verhältnis
- inverse ratio umgekehrtes Verhältnis
- inverse square law quadratisches Abstandsgesetz
- inverse video inverse Video
Ελληνική (Greek)
n. - το άκρως αντίθετο, το αντίστροφο
adj. - ανάστροφος, αντίστροφος, ανάποδος
idioms:
- inverse proportion αντίστροφος λόγος
- inverse ratio αντίστροφος λόγος
- inverse square law νόμος του αντίστροφου τετραγώνου
- inverse video (Η/Υ) ανάστροφη απεικόνιση
idioms:
- inverse proportion/ratio rapporto inverso
- inverse square law legge dell'inverso del quadrato
Português (Portuguese)
n. - inverso (m) (Mat.)
adj. - inverso
idioms:
- inverse proportion/ratio proporção/razão inversa
- inverse square law lei (f) da razão inversa do quadrado da distância
Русский (Russian)
противоположность, обратный, ставить вверх дном
idioms:
- inverse proportion/ratio обратная пропорциональность
- inverse square law закон обратных квадратов
Español (Spanish)
adj. - inverso, invertido
n. - lo inverso, lo contrario, (Matemáticas) propiedad o carácter recíproco, (Lógica) tipo de premisa
idioms:
- inverse proportion proporción inversa
- inverse ratio razón inversa
- inverse square law ley de la inversa de los cuadrados
- inverse video video invertido o inverso
Svenska (Swedish)
n. - omvänt proportionellt påstående
adj. - omkastad, motsatt
中文（简体）(Chinese (Simplified))
倒转的, 反向的, 反逆的, 反, 逆
idioms:
- inverse proportion 反比例商务专业反比例
- inverse ratio 反比
- inverse square law 平方反比率
- inverse video 反白影像
中文（繁體）(Chinese (Traditional))
adj. - 倒轉的, 反向的, 反逆的
n. - 反, 逆
idioms:
- inverse proportion 反比例商務專業反比例
- inverse ratio 反比
- inverse square law 平方反比率
- inverse video 反白影像
한국어 (Korean)
adj. - 반대의, 역수의
n. - 반대, 역수
日本語 (Japanese)
adj. - 逆の, 反対の, 倒錯の
n. - 逆, 反対
idioms:
- inverse proportion/ratio 反比例
- inverse square law 逆二乗則
العربيه (Arabic)
(الاسم) عكس, انقلاب (صفه) عكسي, مقلوبا على عقب
עברית (Hebrew)
adj. - הפוך, נגדי
n. - דבר הפוך
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