Negation¶

Definition¶

The negation of an integer $x\text{,}$ written $-x\text{,}$ is the unique integer satisfying

The number $-x$ is also called the additive inverse or opposite of $x\text{.}$ The same definition is used for rational and real numbers.

Interpretation¶

Negation reverses a number's direction on the number line while preserving its distance from zero: $x$ and $-x$ lie on opposite sides of $0$, and $|-x|=|x|$$|x|$. Adding $-x$ undoes adding $x$, which is why subtraction is defined by $x-y=x+(-y)$$(-y)$.

The minus sign has two roles that should be kept apart: in $-x$ it negates one number, in $x-y$ it subtracts one number from another. A common slip is to read $-x$ as "a negative number". If $x=-3$$-3$, then $-x=3$ is positive; this is exactly why $|x|=-x$ when $x<0$$0$. In Math II, $-x$ returns as an example of an additive inverse element.

Key Results¶

• The negation of a number is unique.
• $-0=0$ and $-(-x)=x\text{.}$
• If $x<y\text{,}$ then $-x>-y\text{;}$ negation reverses order.
• $-(x+y)=(-x)+(-y)$ and $(-x)y=-(xy)\text{.}$

Examples¶

• The negation of $5$ is $-5$ because $5+(-5)=0\text{.}$
• $-(-3)=3\text{.}$
• $-(-a)=a\text{.}$

Prerequisites¶

• Integer
• Addition and Sum

Related Concepts¶

• Absolute Value
• Complex Conjugate

Sources¶

• Math I: Calculus · Lecture 2

Aliases: opposite number, additive inverse

Last updated: 20 October 2026 03:02