4-2 Opposites and Absolute Value

Property

Negative numbers are numbers less than 0.
The opposite of a number is the number that is the same distance from zero on the number line but on the opposite side of zero.
The notation $-a$ is read as “the opposite of $a$.”
The whole numbers and their opposites are called the integers.
The integers are the numbers $\ldots, -3, -2, -1, 0, 1, 2, 3, \ldots$.

Examples

• The opposite of $15$ is $-15$, as both are 15 units from zero.
• The opposite of $-9$ is $9$. This can be written as $-(-9) = 9$.
• If $y = -25$, then $-y$ means the opposite of $-25$, which is $-(-25) = 25$.

Explanation

Integers expand our number system to include negative values, which are like mirror images of positive numbers across zero. The 'opposite' of a number is simply its reflection on the other side of the number line.

Property

Two numbers are opposites when they are the same distance away from zero, but in opposite directions.
For example, “3” represents the point that is 3 units to the right of 0, and “$-3$” is its opposite, three units to the left of 0.
Zero is its own opposite.

Examples

• The opposite of 9 is $-9$, as both numbers are 9 units away from 0.
• The opposite of the opposite of $-4$ is $-(-(-4)) = -4$. The first opposite is 4, and the opposite of 4 is $-4$.
• If earning 5 points in a game is represented by $+5$, its opposite is losing 5 points, represented by $-5$.

Explanation

Opposites are mirror images of each other across zero. They have the same distance from zero, just in different directions. Taking the opposite of an opposite brings you right back to where you started!