# How do you find the additive and multiplicative inverse of -1 1/5?

Apr 9, 2017

Additive inverse: $\frac{11}{5}$
Multiplicative inverse: $- \frac{5}{11}$

#### Explanation:

The identity element $o$ for a set and a binary operation is the element such that $o \circ a = a \circ o = o$ for all $a$ in the set, where $\circ$ is the binary operator.

The inverse $b$ of a number $a$ under a binary operator $\circ$ is the number such that $a \circ b = b \circ a = o$, where $o$ is the identity element.

The additive inverse for a number $a$ is a number $b$ such that $a + b = b + a = 0$ (since $0$ is the identity element for addition under the reals).

The multiplicative inverse for a number $a$ is a number $b$ such that $a \cdot b = b \cdot a = 1$ (since $1$ is the identity element for addition under the reals).

The additive inverse of the number $- \frac{11}{5}$ is the number $b$ such that $b + \left(- \frac{11}{5}\right) = 0$. Add $\frac{11}{5}$ to both sides to get $b = \frac{11}{5}$.

The multiplicative inverse of the number $- \frac{11}{5}$ is the number $b$ such that $b \cdot \left(- \frac{11}{5}\right) = 1$. Multiply $- \frac{5}{11}$ to both sides to get $b = - \frac{5}{11}$.