inverse

Etymology

Recorded since 1440, from Latin inversus, the past participle of invertere (“to invert”), itself from in- (“in, on”) + vertere (“to turn”).

adj

1. Opposite in effect, nature or order.
2. Reverse, opposite in order.
3. (botany) Inverted; having a position or mode of attachment the reverse of that which is usual.
4. (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.
   Multiplication is the inverse operation to division.
5. (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.
   A circle inversion maps a given generalized circle to its inverse generalized circle.
6. (category theory, of a category) Whose every element has an inverse (morphism which is both a left inverse and a right inverse).

noun

1. An inverted state: a state in which something has been turned (properly) upside down or (loosely) inside out or backwards.
   Cowgirl is the inverse of missionary.
   321 is the inverse of 123.
2. The result of an inversion
   1. The reverse of any procedure or process.
      Uninstalling is the inverse of installation.
   2. (mathematics) A ratio etc. in which the antecedents and consequents are switched.
      The inverse of a:b is b:a.
   3. (geometry) The result of a circle inversion; the set of all such points; the curve described 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.
   4. (logic) The non-truth-preserving proposition constructed 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, ¬ mathsf p→¬ mathsf q is the inverse of mathsf p→ mathsf q and is equivalent to the converse proposition mathsf q→ mathsf p.
      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. 1896, James Welton, A Manual of Logic, 2nd edition, Bk iii, Ch. iii, §102
3. (mathematics) A second element which negates a first; in a binary operation, the element for which the binary operation—when applied to both it and an initially given element—yields the operation's identity element, specifically:
   1. (addition) The negative of a given number.
      The additive inverse of x is -x, as x-x=0, as 0 is the additive identity element.
   2. (multiplication) One divided by a given number.
      The multiplicative inverse of x is x⁻¹, as x×x⁻¹=1, as 1 the multiplicative identity element.
   3. (functions) A second function which, when combined with the initially given function, yields as its output any term inputted into the first function.
      The compositional inverse of a function f is f⁻¹, as ff⁻¹= mathit I, as mathit I is the identity function. That is, ∀x,f(f⁻¹(x))= mathit I(x)=x.
4. (category theory) A morphism which is both a left inverse and a right inverse.
5. (card games) The winning 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 reserved for bets upon such an outcome.
   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. 1850, Henry George Bohn, The Hand-book of Games, page 343
   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. 1950, Lawrence Hawkins Dawson, Hoyle's Games Modernized, 20th edition, page 291
6. (linguistics, Kiowa-Tanoan) A grammatical number marking that indicates the opposite grammatical number (or numbers) of the default number specification of noun class.

verb

1. (surveying) To compute the bearing and distance between two points.