# Definition:Additive Inverse

## Definition

Let $\struct {R, +, \circ}$ be a ring whose ring addition operation is $+$.

Let $a \in R$ be any arbitrary element of $R$.

The **additive inverse** of $a$ is its inverse under ring addition, denoted $-a$:

- $a + \paren {-a} = 0_R$

where $0_R$ is the zero of $R$.

### Additive Inverse of Number

The concept is often encountered in the context of numbers:

Let $\Bbb F$ be one of the standard sets of numbers

Let $a \in \Bbb F$ be any arbitrary number.

The **additive inverse** of $a$ is its inverse under addition, denoted $-a$:

- $a + \paren {-a} = 0$

## Sources

- 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next): Entry:**additive inverse** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**additive inverse**