I should frame this question in the context of dealing with probabilities: I've read the wikipedia entry on the multiplicative inverse: Where it clearly says that $x^{-1}$ is the inverse. But I feel like x-1 or 1-x has a name and I keep wanting to call it an inverse, when it seems like I shouldn't. Is there another name for this x-1 or 1-x phenomena?
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$\begingroup$If $x$ is a probability, then $1-x$ is the complementary probability.
$\endgroup$ 1 $\begingroup$To the best of my knowledge, there is no commonly used term for the relationship between $x$ and $1-x$.
$\endgroup$ $\begingroup$If you add $x-1$ and $1-x$, you get $$(1-x)+(x-1) = 0$$ Indeed, $(x-1)= -(1-x)$.
Each is the additive inverse of the other.
$\endgroup$ $\begingroup$Maybe what you mean is that $1-x$ is a sort of additive inverse of $x$ with respect to $1$. Moreover, the function is an involution as long as $x\in[0, 1]$, so applying this "inverse" twice gets you $x$ back. This operation is useful in probability, for instance, and I don't think it has a name, although if you called it the complement people wouldn't think you were crazy. If you're working with the reals or rationals modulo $1$, it's literally an additive inverse.
$\endgroup$ $\begingroup$If you're speaking of $x^{-1}$, then that's been answered in the comments. If you're talking about $x - 1$ and $1 - x$, then the correct answer is they are each other's additive inverses. The additive inverse of $x$ is the number $-x$ such that $x + (-x) = 0$. If we add $(x - 1) + (1 - x)$, we have $$(x - 1) + (1 - x) = x - 1 + 1 - x = x - x + 1 - 1 = 0$$Also notice that $(1-x) = -(x -1)$.
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