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The multiplicative inverse of $x$ is $\frac{1}{x}$,

and the additive inverse of $x$ is $-x$,

is there a similar term for $(1-x)$?

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    It is the complement to 1.2011-09-12
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    Usually just $(1-x)$.2011-09-12
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    What is your $x$? And what is $1$? Are they allowed to be operators?2011-09-12
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    By the way, what makes $f(x)=1-x$ interesting is that $f(f(x))=x$, just like for $\frac{1}{x}$ and $-x$.2011-09-12
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    @Rasmus: ...and what's the word to describe that property?2011-09-12
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    @Richard, it's an [involution](http://en.wikipedia.org/wiki/Involution_%28mathematics%29).2011-09-12

3 Answers 3

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$1-x$ is could be known called as the "complement to $1$ of $x$".

Added: In English this designation is likely not generally used.

But "one's complement" and "complementary angles" are, according to English Wikipedia. In French the "Euler's reflection formula" is known as "Formule des compléments".

Added 2: This designation would be more natural for $0\le x\le 1$, similarly to complementary angles: An acute angle is "filled up" by its complement to form a right angle.

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    Congratulations on 10k reputation!2011-09-12
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    @Austin Mohr: Thank you!2011-09-12
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    I can accept *complement* but not *is known as*. Any references?2011-09-12
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    @lhf: It is a direct translation from the Portuguese/French. I have thought of the "[Formule des compléments"](http://fr.wikipedia.org/wiki/Formule_des_compl%C3%A9ments) $$\forall z\ 0<\text{Re}(z)<1,\Gamma (1-z)\Gamma (z)=\frac{\pi }{\sin \pi z}$$2011-09-12
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    Funny that that equation is called [Euler's *reflection* formula](http://en.wikipedia.org/wiki/Reflection_formula) in English... Also funny is that *fórmula de reflexão de Euler* is mentioned in the PT as http://pt.wikipedia.org/wiki/Função_gama. But that probably is just a translation of the EN page.2011-09-12
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    @lhf: That's right!2011-09-12
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    @lhf: ... and there's the [one's complement](http://en.wikipedia.org/wiki/One%27s_complement)2011-09-12
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    @lhf: ... as well as [complementary angles](http://en.wikipedia.org/wiki/Complementary_angles).2011-09-12
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    @lhf *Fórmula de reflexão de Euler* is a more recent Portuguese translation than *fórmula (de Euler) dos complementos*.2011-09-12
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    @Ihf: here's a reference http://math.stackexchange.com/questions/63987/name-for-1-x/63992#63992 $$$$ :)2011-09-12
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    @The Chaz: *I [you] frequently relate to people with (what I [you] think is) humor :)*. Yes, it is!2011-09-12
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    Well thanks! And seriously, nomenclature has to start *somewhere*. I'm all in favor of this expression. Or we could call it the "American complement". Ok that last part was not as serious...2011-09-12
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I would call it the complement. One motivation is that if some event occurs with probability $p$, the complementary event occurs with probability $1 - p$.

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    I agree that I'd call it the complement; the further away from probability one gets, though, the less meaningful this name seems.2011-09-12
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    But how many people are far away from probability? z corresponding to $6\sigma$ ?!?2011-09-13
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I'd call it a complement or a negation. You don't need to stand all that close to probability for these names to seem meaningful, in my opinion. If we have 0 as indicating falsity, and 1 as indicating truth, then the negation of a proposition x has truth value of (1-x). The same holds if we have 1 as indicating falsity, and 0 as indicating truth. In fuzzy logic, which has truth values of the unit interval [0, 1], (1-x) also comes as the fuzzy complement most commonly considered.

Also, consider classical or crisp sets under their characteristic function representation. The characteristic function assigns 1 to each element of the universal set which belongs to the subset under consideration, and 0 to each element of the universal set which does not belong to the subset under consideration. For example, if we have {a, b, c, d} as our universal set, and {a, b} as the subset under consideration, the characteristic function assigns 1 to a, 1 to b, 0 to c, and 0 to d, or equivalently {(a, 1), (b, 1), (c, 0), (d, 0)}. Now, the complement of {a, b} for this universal set equals {c, d}. Well, (1-x) on the values defined by the characteristic function for {a, b}={(a, 1), (b, 1), (c, 0), (d, 0)} gives us {(a, 0), (b, 0), (c, 1), (d, 1)}={c, d} the complement of {a, b}. This does generalize also such that (1-x) here always gives us the complement of the subset under consideration.