Inverse element
For addition of numbers, for every number $a$, there is another number $-a$ such that adding $a + (-a)$ is zero. We call $a$ the additive inverse $a$. For example, the additive inverse of $5$ is $-5$.
The multiplicative inverse (or reciprocal) of a nonzero number $a$ is the number $\frac{1}{a}$ that you multiply by $a$ to get $1$:
\[a \times \frac{1}{a} = 1.\]For example, the multiplicative inverse of $5$ is $\frac{1}{5}$. The number $0$ has no multiplicative inverse, because $0$ times every number is $0$.
Consider a way of combining two things, such as addition of numbers, that has an element that changes nothing, such as $0$ for addition. An inverse of an element $a$ is an element that undoes $a$: combining $a$ with the inverse gives the element that changes nothing. For example, the additive inverse of $5$ is $-5$, because $5 + (-5) = 0$.
Consider a set $S$ with a binary operation $\ast$ and an identity element $e$. An inverse of an element $a \in S$ is an element $b \in S$ such that
\[a \ast b = b \ast a = e.\]Examples
Worked examples
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The number $5$ under multiplication of real numbers
: $\frac{1}{5}$, because $5 \times \frac{1}{5} = 1$
Elementary
For multiplication of real numbers, the identity element is $1$, so the inverse of $5$ is the number $\frac{1}{5}$ whose product with $5$ is $1$; the number $0$ has no inverse for multiplication, because $0$ times every number is $0$.
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The function $f(x) = 2x + 1$ on $\reals$ under composition of functions
: $f^{-1}(x) = \frac{x - 1}{2}$
High school
For composition of functions $\reals \to \reals$, the identity element is the identity function $x \mapsto x$, so the inverse element of $f(x) = 2x + 1$ is the inverse function $f^{-1}(x) = \frac{x - 1}{2}$, because $f(f^{-1}(x)) = f^{-1}(f(x)) = x$ for all $x$.
Properties
- If $\ast$ is associative, then each element has at most one inverse, which is written $a^{-1}$ (or $-a$ when the operation is addition).