Opposite number

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

Definition

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

Opposite numbers are also known as additive inverses.

Properties

The opposite numbers satisfy the properties listed below.

1. The opposite of 0 is 0.
2. Two opposite numbers have the same absolute value. This follows from the fact that the [math]\displaystyle{ -a }[/math] is the opposite number of [math]\displaystyle{ a }[/math] and both have the same absolute value [math]\displaystyle{ |-a|=|a| }[/math].
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 [math]\displaystyle{ -a+a=0. }[/math]
6. The division of two non-zero opposite numbers is always [math]\displaystyle{ -1 }[/math], because [math]\displaystyle{ \frac{-a}{a}=\frac{a}{-a}=-1. }[/math]
7. Each number has a unique opposite number.

Opposite Number Media

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  These complex numbers, two of eight values of ⁸√1, are mutually opposite

Related pages

• Addition
• Reciprocal
• Absolute value
• Operation (mathematics)