Lesson 1: Relate Integers and Their Opposites

Property

Two integers are opposites if they are each the same distance away from zero, but on opposite sides of the number line.
The absolute value $|a|$ of an integer $a$ is the distance from the point on the line to 0.
A number and its opposite have the same absolute value.
Since the positive integers are the opposites of the negative integers, we conclude that $-(-5) = 5$, and in general $-(-a) = a$ for any integer $a$.

Examples

• The opposite of 9 is $-9$. The absolute value $|9|$ is 9, and the absolute value $|-9|$ is also 9.
• The expression $-(-25)$ means the opposite of negative 25, which simplifies to 25.
• If two numbers on a number line are opposites and one is 14, the other must be $-14$, as they are both 14 units from 0.

Explanation

Opposites are mirror images across zero on the number line. The absolute value is just the distance to zero, which is why it is always a positive value. Taking the opposite of a negative flips you back to positive.

Property

The sum of an integer and its opposite is always zero. For any integer $a$:

Examples