Opposite number

In mathematics, the opposite or additive inverse of a number ${\displaystyle k}$ is a number ${\displaystyle n}$ which, when added to ${\displaystyle k}$, results in 0. The opposite of ${\displaystyle a}$ is ${\displaystyle -a}$.[1][2] For example, −7 is the opposite of 7, because ${\displaystyle 7+-7=0}$.

Definition

A number ${\displaystyle x}$ is called an opposite number of another number ${\displaystyle y}$ if ${\displaystyle x+y=0}$. By definition, ${\displaystyle -x}$ is the opposite number of ${\displaystyle x}$. For example, ${\displaystyle -2}$ is the opposite number of ${\displaystyle 2}$ and vice-versa. This is because ${\displaystyle -2+2=0.}$

Opposite numbers are also known as additive inverses.

Properties

The opposite numbers satisfy the properties listed below.[3]

1. The opposite of 0 is 0.
2. Two opposite numbers have the same absolute value. This follows from the fact that the ${\displaystyle -a}$ is the opposite number of ${\displaystyle a}$ and both have the same absolute value ${\displaystyle |-a|=|a|}$.
3. The opposite of a positive number is the negative version of the number. The opposite of a negative number is the positive version of the number.
4. opposite numbers are located in the opposite direction on a number line having the same distance from the origin. That is, they are symmetric about the origin on a number line.
5. The sum of two opposite numbers is always zero, because ${\displaystyle -a+a=0.}$
6. The division of two non-zero opposite numbers is always ${\displaystyle -1}$, because ${\displaystyle {\frac {-a}{a}}={\frac {a}{-a}}=-1.}$
7. Each number has a unique opposite number.

References

1. Weisstein, Eric W. "Additive Inverse". mathworld.wolfram.com. Retrieved 2020-08-27.
2. "Additive Inverse". www.learnalberta.ca. Retrieved 2020-08-27.
3. Ghosh, N. "Opposite Numbers: Definition, Examples, and Properties". Mathstoon. Retrieved 6 February 2022.