Absolute Value

Absolute Value

Absolute value has many uses, but you probably won't see anything interesting for a few more classes yet. For now, you can view absolute value as the distance from zero. (There is a technical definition for absolute value, but you probably won't see this for quite a while yet, if ever.)

Look at the number line:

The absolute value of x, denoted "| x |" (and which is read as "the absolute value of x"), is regarded as the distance of x from zero. This is why absolute value is never negative; absolute value only asks "how far?", not "in which direction?". This means that | 3 | = 3, because 3 is three units to the right of zero, and also | –3 | = 3, because –3 is three units to the left of zero.

(Note: The absolute value notation is bars, not parentheses or brackets. Use the proper notation, as the other notations do not mean the same thing!)

It is important to note that the absolute value bars do NOT work in the same way as do parentheses. Whereas –(–3) = +3, this is NOT how it works for absolute value:

Simplify –| –3 |.

Given –| –3 |, first handle the absolute value part, taking the positive and converting the absolute value bars to parentheses:

–| –3 | = –(+3)

Now you can take the negative through the parentheses:

–| –3 | = –(3) = –3

So you see that if you take the negative of an absolute value, you will get a negative number for your answer!

(Note: When typing, such as in e-mail, the "pipe" is usually used to indicate absolute value. The "pipe" is probably a shift-key somewhere north of the "Enter" key on your keyboard. While the "pipe" denoted on the physical keyboard key may look broken, the typed character should display on your screen as a solid line. If you cannot locate a "pipe" character, you can use "abs()" instead, so that "the absolute value of negative 3" would be typed as "abs(–3)".)

Here are some more sample simplifications:

Simplify | –8 |.

Simplify | 0 – 6 |.

| 0 – 6 | = | –6 | = 6

Simplify | 5 – 2 |.

| 5 – 2 | = | 3 | = 3

Simplify | 2 – 5 |.

| 2 – 5 | = | –3 | = 3

Simplify | 0(–4) |.

| 0(–4) | = | 0 | = 0

Why is the absolute value of zero equal to "0"? Ask yourself: how far is zero from 0? Zero units, right? So | 0 | = 0.

Simplify | 2 + 3(–4) |.

| 2 + 3(–4) | = | 2 – 12 | = | –10 | = 10

Simplify –| –4 |.

–| –4| = –(4) = –4

Simplify –| (–2)² |.

–| (–2)² | = –| 4 | = –4

Simplify –| –2 |²

–| –2 |² = –(2)² = –(4) = –4

Simplify (–| –2 |)².

(–| –2 |)² = (–(2))² = (–2)² = 4

Sometimes you will be asked to insert an inequality sign between two absolute values, such as:

Insert the correct inequality: | –4 | _____ | –7 |

Whereas –4 > –7 (it is further to the right than is –7), we are dealing here with the absolute values. Since:

| –4 | = 4

| –7 | = 7,

...and 4 < 7, then the solution is:

| –4 | < | –7 |.

Note that, whenever we took the absolute value of a number, if it was positive to start with, we left the sign (an understood "+") alone, but if it was negative to start with, we changed the sign (the "minus" sign "–" was turned into an understood "+"). This leads to one fiddly point which may not come up now, but will probably come up later:

When you are dealing with variables, you cannot tell the sign of the number or value that is contained in the variable. For instance, given the variable x, you cannot tell by looking whether there is, say, a "2" or a "–4" contained inside. So if I ask you for the absolute value of x, what would you do? Since you cannot tell just by looking whether or not the variable contains a positive or negative value, you would have to consider these different cases.

If x > 0 (that is, if x is positive), then the value won't change when you take the absolute value. For instance, if x = 2, then you have | x | = | 2 | = 2 = x. In fact, for any positive (or zero) value of x, the sign would be unchanged, so:

For x > 0, | x | = x

On the other hand, if x < 0 (that is, if x is negative), then it will change its sign when you take the absolute value. For instance, if x = –4, then | x | = | –4 | = + 4 = –(–4) = –x. In fact, for any negative value of x, the sign would have to be changed, so:

For x < 0, | x | = –x

This is a case in which the "minus" sign does not indicate "a number to the left of zero", but "a change of the sign from whatever the sign originally was". This "–" does not mean "the number is negative" but instead means that "I've changed the sign on the original value".