# Definition:Additive Inverse

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## Definition

### Additive Inverse in Ring

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 in Field

Let $\struct {F, +, \times}$ be a field whose addition operation is $+$.

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

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

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

where $0_F$ 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 number systems: $\N$, $\Z$, $\Q$, $\R$, $\C$.

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

- 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**additive inverse**