# 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 $(1-x)$?

• It is the complement to 1. Sep 12, 2011 at 20:49
• Usually just $(1-x)$. Sep 12, 2011 at 20:51
• What is your $x$? And what is $1$? Are they allowed to be operators?
– user9464
Sep 12, 2011 at 20:55
• By the way, what makes $f(x)=1-x$ interesting is that $f(f(x))=x$, just like for $\frac{1}{x}$ and $-x$. Sep 12, 2011 at 20:56
• @Richard, it's an involution.
– user856
Sep 12, 2011 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.

• Congratulations on 10k reputation! Sep 12, 2011 at 20:59
• @Austin Mohr: Thank you! Sep 12, 2011 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, 2011 at 21:55
• @Ihf: here's a reference math.stackexchange.com/questions/63987/name-for-1-x/63992#63992  :) Sep 12, 2011 at 22:15
• @The Chaz: I [you] frequently relate to people with (what I [you] think is) humor :). Yes, it is! Sep 12, 2011 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 $1 - p$.

• I agree that I'd call it the complement; the further away from probability one gets, though, the less meaningful this name seems. Sep 12, 2011 at 23:21
• But how many people are far away from probability? z corresponding to $6\sigma$ ?!? Sep 13, 2011 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.