In this explainer, we will learn how to find the absolute value of an integer and represent it on the number line and compare and order the absolute values of different numbers.
Let us start with the definition.
Definition
The absolute value of a number is its distance from zero on a number line.
We write the absolute value of a number as .
For example, we write the absolute value of 1 as and the absolute value of as . Since both 1 and are a distance of 1 from zero (in opposite directions) both and are equal to 1.
Now, let us do some examples to see what this means and how we can use the absolute value.
Example 1: Comparing a Number and Its Absolute Value
Which of the following is true?
Answer
The absolute value of is the distance between 0 and on a number line.
Distances are always positive, so
We have been asked to compare and . Since is negative and is positive, the only true statement is that
Example 2: Comparing Numbers and Absolute Values Using Number Lines
Which of the following statments is true?
Answer
The absolute value of a number is its distance from zero on a number line. Remember that distances are always nonnegative numbers.
By looking at the number line, we can see that and the distance between 0 and is 6. Hence,
Similarly, and the distance between 0 and 5 is 5. Hence,
Let us now consider each of the statements to see if it is true or false.
Firstly, because is negative and the absolute value of is positive.
Since positive numbers are greater than negative numbers,
Since the distance between and 0 is greater than the distance between and 0 (as ), it is true that
Finally, again by considering the distances from zero, we see that and
So, the true statement is .
The above two examples should have helped you to see how to calculate the absolute value of a number by considering its distance from zero. What we find is that if a number is positive, then the absolute value of is equal to and if a number is negative, then the absolute value of is equal to the opposite (or additive inverse) of . For example,
Also, notice that when we take two numbers that are opposites (or additive inverses) like 6 and , then they have the same absolute value because they are both the same distance from zero on a number line.
Let us finish with a few more examples.
Example 3: Ordering Numbers According to Their Absolute Value
use the number line to order points from greatest absolute value to least absolute value.
Answer
To order the points from greatest absolute value to least absolute value means to order them from greatest distance from zero to smallest distance from zero. Let us “zoom in” on the number line.
The green line represents the absolute value of ; the distance is 3.
The orange line represents the absolute value of ; the distance is about 0.5.
The pink line represents the absolute value of ; the distance is 2.
From the picture, we can see that is closest to zero and is furthest from zero. Hence, when we order the points from greatest to least absolute value, we get because since
So, the order of the points from greatest absolute value to least absolute value is .
Example 4: Finding the Absolute Value of a Number
.
Answer
The absolute value of is the distance between 0 and on a number line, which is the same as the distance between 0 and 51 on a number line. This distance is 51. Hence,
We can summarize these observations formally.
Facts about the Absolute Value
- If , then .
- If , then .
- .
- If then .
