Multiplicative inverse

A multiplicative inverse of $x$ is an element $x^{-1}$ satisfyingFor a nonzero real number it is the reciprocal $1/x$, so the inverse of $2$ is $1/2$. The inverse must belong to the chosen structure: $2$ has no multiplicative inverse among the integers, though it does among the rational numbers. In a ring, an element with such an inverse is a unit.

The two equations matter when multiplication is noncommutative. If $y$ is a left inverse and $z$ a right inverse in a monoid, then $y=y(xz)=(yx)z=z$. Thus a two-sided multiplicative inverse is unique. A group requires every element to have one; a field requires one for every nonzero element.

The multiplicative inverse of a number \( x \) is another number, often denoted as \( \frac{1}{x} \) or \( x^{-1} \), such that when you multiply the two numbers together, the result is 1.