How does a multiplicative inverse differ from an additive inverse?

In modern algebra, inverses are defined by their relationship to the identity element of a specific operation. The primary difference lies in the operation performed and the resulting identity element.

• Additive Inverse: An element that, when added to a given number, results in the additive identity, which is zero. For any number $a$, the additive inverse is $-a$, such that $a+(-a)=0$.1
• Multiplicative Inverse: An element that, when multiplied by a given element, yields the multiplicative identity, which is one. Also known as a reciprocal, the multiplicative inverse of a non-zero number $a$ is $\frac{1}{a}$, such that $a\times \frac{1}{a}=1$.2

While an additive inverse is found by changing the sign of a number, a multiplicative inverse is found by taking the reciprocal. Notably, while every number has an additive inverse, zero does not have a multiplicative inverse because division by zero is undefined.