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What is the function called, when the function effectively multiplies its input by $-1$?

i.e. $f(x) = -x$.

Similar terminology being the inverse of a number, i.e. $f(x) = 1/x$.

There may not be one, I'm just convinced there is, and no one I ask can give me a straight answer.

Thanks,

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    You could say $f(x) = -x$ is 'inversion', if the domain of $f$ has the structure of an additive group.2011-11-21
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    I think $-x$ is called the opposite of $x$.2011-11-21
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    $1/x$ is the "reciprocal" of $x$. $-x$ is the "additive inverse" of $x$. The simplest way to call the function $f(x)=-x$ is "multiplication by $-1$", but you can call it the function that "gives the additive inverse"; the simplest way to call $f(x)=1/x$ is "reciprocal", but you can call it the function that "gives the multiplicative inverse (if $x\neq 0$)".2011-11-21
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    Call it **negation** if you must.2011-11-21
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    A geometric name for $-x$ might be "reflection in 0".2011-11-21

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This is quite simply the negation function. Alternative names include just "negation", or either "negative $x$" or "minus $x$" (in analogy to the terminology "$x$ squared" for the function $x \mapsto x^2$).

I would apply this terminology in any context where a mapping to an additive inverse makes sense.

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    “Negation” can be a little ambiguous. While I agree it’s used for this, it’s probably more commonly used for the negation of truth values in logic; so to anyone who thinks of the standard truth values as being 0 and 1 (eg many computer scientists), “the negation of 1” is 0, not –1.2011-11-21
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    @PeterLeFanuLumsdaine: Yes, it is true that the word 'negation' is used for the concept of logical negation, for instance the function $x \mapsto 1-x$ over $\mathbb F_2$ where we wish to apply the semantics $1 \sim $"true", $0 \sim $"false". But this is no more vicious an ambiguity than that surrounding the term "normal" (which has more than one *standard* meaning). Given enough context, I'm confident that the usage would be unambiguous.2011-11-22
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    Yes, that's fair.2011-11-22
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"Additive inversion."

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    My original comment, "+1 satire" was blocked as 6 characters too short.2011-11-21
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    Maybe I should put it this way: The _function_ is called "additive inversion"; any _value of_ the function is called an "additive inverse".2011-11-21
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    I've edited accordingly.....2011-11-21
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Multiplication by –1.

A little less snappy than the other suggestions, but (a) completely standard; (b) quite unambiguous; and (c) understandable by anyone mathematically literate, not just mathematicians.

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    This is less general than "additive inverse", which is defined in any ring (or commutative group), while $(-1)$ may not exist in a ring without unit.2011-11-21
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    It’s true that –1 may not exist in a ring without unit, but “multiplication by –1” still makes sense, interpreting –1 as living in ℤ, and multiplication in the sense of the natural ℤ-module structure. This structure is defined on any Abelian group, so in exactly the same generality as “additive inverse”, as far as I can see.2011-11-21
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    The possibility of a ring without unit is slightly pedantic, but accomodating it through the reference to Z-modules actually makes the phrase "mult by -1" less accurate, since the thing multiplied is no longer the element $x$, but the function $f(x)=x$ *denoted* (by abuse of notation) as $x$. The question was how to describe a function on elements, not an operator on functions. Academic details but maybe worth mentioning none the less.2011-11-21
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    @zyx: I don't follow what you're saying about this requiring seeing it as an operator on functions. Say *A* is an Abelian group. Then the the function *A*→*A* sending *x* to its additive inverse can be described as sending *x* to *(−1)x*, using the multiplication of the natural Z-action. In other words, the "additive inverse" function *A*→*A* is the same as the function "multiplication by −1". Right?2011-11-22
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    The element $x$ is not multiplied in any sense (e.g. in A, which has no multiplication operation) to get the additive inverse element $(-x)$. The things that are multiplied are maps $x \to nx$ for different $n$, through composition of functions, or multiplication in the endomorphism ring (into which that Z is mapped homomorphically) of $A$. The endomorphism corresponding to -1, multiplied by the one corresponding to 1 (sometimes also called $x$ through abuse of notation), gives the one corresponding to -1.2011-11-22
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    "The element x is not multiplied in any sense" - I'm sorry, that's just not correct! I'm not claiming this is multiplication *within* any ring; rather, it's multiplication of an element of the module by an element of the ring, aka scalar multiplication. // "Also, this ℤ-module structure is constructed from additive inverse, not vice versa." Yes, this I agree with (at least, this is how it's standardly presented, although a real monoidophile might reasonably see it the other way round). I like "additive inverse", and for talking to fellow mathematicians agree it's probably the best name.2011-11-22
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Antipode also-rans:

flip, reverse, opposite, anti-, evil twin, 180,

reversal, switcheroo, NOT, change of direction, Nemesis,

turnabout, Bizarro, topsy-turve, the world turned upside down.

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    At least one side of one sheep could use some laughter. And *antipode* is arguably the most general mathematical term that applies, as it includes groups through its use for Hopf algebras, and n-dimensional space through its use for spheres. It may also have the longest history, coming from ancient Greek and later used in the cartographic sense.2011-11-22