1.11 Opposites of integers and rational numbers
Introduction
In this lesson, we'll be using number lines to explore opposite integers and rational numbers, along with zero pairs.
Opposite integers and rational numbers
When looking for the opposite of a meaning we usually try to reverse it. For example, the opposite of left is right because we can reverse moving to the left by moving to the right. When trying to find opposites on a number line, we can use the same approach.
Consider the integer 3. On this number line, the integer 3 represents "the location 3 units to the right of 0", shown in green. The opposite of this would involve reversing the direction. In other words, the opposite would be "the location 3 units to the left of 0, shown in blue.
This example shows that the opposite of the integer 3 is the integer -3.
We can use the same method to find the opposite of a negative integer, as well as any rational number.
Consider the number -3. This number represents "the location 3 units to the left of 0", shown in blue, so its opposite will be "the location 3 units to the right of 0".
Two numbers are opposite if their locations on the number line are the same distance from 0, but on different sides of 0.
What about 0 itself? We can think about the opposite of 0 as being the number -0. But since -0 is the same as 0, the opposite of 0 is again 0. That is, the integer 0 is its own opposite.
Exploration
This applet lets you visualize the idea of opposites. Slide point A and see its opposite (point B) move.
- What patterns do you see with the numbers on the number line?
- What is the relationship between any number and its opposite when plotted on a number line?
Examples
Example 1
State the opposite of the following number: -\dfrac{2}{3}
Example 2
Think about the following statement:
"Arriving 14 minutes late."
Pick the statement that describes the opposite of "Arriving 14 minutes late".
Suppose "Arriving 14 minutes late" is represented by the number 14. What integer should represent "Arriving 14 minutes early"?
Two integers are opposite if their locations on the number line are the same distance from 0, but on different sides of 0.