Explainer: Absolute Values of Integers

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 π‘₯ as |π‘₯|.

For example, we write the absolute value of 1 as |1| and the absolute value of βˆ’1 as |βˆ’1|. Since both 1 and βˆ’1 are a distance of 1 from zero (in opposite directions) both |1| and |βˆ’1| 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?

  1. βˆ’96<|βˆ’96|
  2. βˆ’96=|βˆ’96|
  3. βˆ’96>|βˆ’96|

Answer

The absolute value of βˆ’96 is the distance between 0 and βˆ’96 on a number line.

Distances are always positive, so |βˆ’96|=96.

We have been asked to compare βˆ’96 and |βˆ’96|. Since βˆ’96 is negative and |βˆ’96| is positive, the only true statement is that βˆ’96<|βˆ’96|.

Example 2: Comparing Numbers and Absolute Values Using Number Lines

Which of the following statments is true?

  1. 𝐴=|𝐡|
  2. 𝐴>|𝐡|
  3. |𝐴|>|𝐡|
  4. |𝐴|=|𝐡|
  5. |𝐴|<|𝐡|

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 𝐴=βˆ’6 and the distance between 0 and βˆ’6 is 6. Hence, |𝐴|=6.

Similarly, 𝐡=5 and the distance between 0 and 5 is 5. Hence, |𝐡|=5.

Let us now consider each of the statements to see if it is true or false.

Firstly, 𝐴|𝐡|isequaltonot because 𝐴 is negative and the absolute value of 𝐡 is positive.

Since positive numbers are greater than negative numbers, 𝐴|𝐡|.isgreaterthannot

Since the distance between 𝐴 and 0 is greater than the distance between 𝐡 and 0 (as 6>5), it is true that |𝐴|>|𝐡|.

Finally, again by considering the distances from zero, we see that |𝐴||𝐡|isequaltonot and |𝐴||𝐡|.islessthannot

So, the true statement is |𝐴|>|𝐡|.

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 π‘₯ is positive, then the absolute value of π‘₯ is equal to π‘₯ and if a number 𝑦 is negative, then the absolute value of 𝑦 is equal to the opposite (or additive inverse) of 𝑦. For example, |3|=3,|17|=17,|261|=261,|βˆ’5|=5,|βˆ’82|=82,|βˆ’349|=349.

Also, notice that when we take two numbers that are opposites (or additive inverses) like 6 and βˆ’6, 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.

  1. 𝐢,𝐴,𝐡
  2. 𝐴,𝐡,𝐢
  3. 𝐴,𝐢,𝐡
  4. 𝐢,𝐡,𝐴
  5. 𝐡,𝐢,𝐴

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 β€œzoom in” on the number line.

The green line represents the absolute value of 𝐴; the distance is 3.

The orange line represents the absolute value of 𝐡; the distance is about 0.5.

The pink line represents the absolute value of 𝐢; the distance is 2.

From the picture, we can see that 𝐡 is closest to zero and 𝐴 is furthest from zero. Hence, when we order the points from greatest to least absolute value, we get 𝐴,𝐢,𝐡 because |𝐴|>|𝐢|>|𝐡| since 3>2>0.5.

So, the order of the points from greatest absolute value to least absolute value is 𝐴,𝐢,𝐡.

Example 4: Finding the Absolute Value of a Number

|βˆ’51|=.

Answer

The absolute value of βˆ’51 is the distance between 0 and βˆ’51 on a number line, which is the same as the distance between 0 and 51 on a number line. This distance is 51. Hence, |βˆ’51|=51.

We can summarize these observations formally.

Facts about the Absolute Value

  1. If π‘₯>0, then |π‘₯|=π‘₯.
  2. If βˆ’π‘₯<0, then |βˆ’π‘₯|=π‘₯.
  3. |0|=0.
  4. If π‘₯β‰₯0 then |π‘₯|=π‘₯=|βˆ’π‘₯|.

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