The absolute value of a number is its distance from zero on the number line. To illustrate this, let's take a look at a number on the number line, and identify its distance from zero:
As we can see in the number line above, the number 4 is 4 units away from zero. This means that the absolute value of 4 is 4. This seems easy, and with positive numbers, it is very straightforward. But what about -4? The magnitude of the number is the same (the actual number we see, ignoring if it's positive or negative), but it is on the opposite side of 0 on the number line, as shown below:
We see that -4 is also 4 units away from 0 on the number line. This means that the absolute value of -4 is 4. Notice that with both 4 and -4, the absolute value is non-negative.
When taking the absolute value of a number, it is always non-negative. A simple way to think about evaluating the absolute value of a number is to rewrite the number without regard to its sign (do not write positive or negative, just the number).
Adding and Subtracting with Absolute Value
Now that we have an understanding of the absolute value of a number, let's use absolute value in addition and subtraction problems. The important thing here is to evaluate the absolute value first, and then perform the addition or subtraction. This is shown in the following examples:
If there are multiple number and operations within absolute value bars, we must evaluate the expression inside before taking any absolute value. This is because absolute value bars also act as grouping symbols, and must be evaluated first according to the order of operations.
Multiplying and Dividing with Absolute Value
There are a couple of special properties with absolute value when multiplying and dividing two numbers. Let's take a look at multiplication first:
Let's take a look at another way to approach the same problem:
With products, we can say that | a•b | = | a | • | b |
Next, let's see if the same property holds true with division:
Do you think we will arrive at the same solution if we first rewrite the expression using two sets of absolute value bars?
With quotients, we can say that | a / b | = | a | / | b |
The distance between a number and zero on the number line; it is always non-negative.