# Name for $(1-x)$?

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. – Américo Tavares Sep 12 '11 at 20:49
Usually just $(1-x)$. – Asaf Karagila Sep 12 '11 at 20:51
@Americo: Ah yes, thank you. (Happy to accept if you make it an answer...) – Richard Inglis Sep 12 '11 at 20:52
By the way, what makes $f(x)=1-x$ interesting is that $f(f(x))=x$, just like for $\frac{1}{x}$ and $-x$. – Rasmus Sep 12 '11 at 20:56
@Richard, it's an involution. – Rahul Sep 12 '11 at 21:27

$1-x$ 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! – Austin Mohr Sep 12 '11 at 20:59
@Austin Mohr: Thank you! – Américo Tavares Sep 12 '11 at 21:01
Funny that that equation is called Euler's reflection formula in English... Also funny is that fórmula de reflexão de Euler is mentioned in the PT as pt.wikipedia.org/wiki/Função_gama. But that probably is just a translation of the EN page. – lhf Sep 12 '11 at 21:55
@Ihf: here's a reference math.stackexchange.com/questions/63987/name-for-1-x/63992#63992  :) – The Chaz 2.0 Sep 12 '11 at 22:15
@The Chaz: I [you] frequently relate to people with (what I [you] think is) humor :). Yes, it is! – Américo Tavares Sep 12 '11 at 22:20

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. – Michael Lugo Sep 12 '11 at 23:21
But how many people are far away from probability? z corresponding to $6\sigma$ ?!? – The Chaz 2.0 Sep 13 '11 at 1:05

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.

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