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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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    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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    Yes, that's fair.2011-11-22
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"Additive inversion."

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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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    "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