audio read-through Algebraic Definition of Absolute Value

Click here for a brief discussion of the algebraic definition of absolute value (printable version).

You may want to review some basic absolute value concepts:

Sometimes we don't care if a number is positive (i.e., lies to the right of zero) or negative (i.e., lies to the left of zero). We just want to know its size—its distance from zero.

In these situations, the concept of absolute value comes to the rescue. The notation $\,|x|\,$ denotes the absolute value of $\,x\,.$

The number $\,|x|\,$ gives the distance between $\,x\,$ and $\,0\,.$ This definition of absolute value as distance from zero is often the easiest one to use.

Whenever you see those two vertical bars that denote absolute value, I like to have my students do two ‘karate chops’ on them, saying: $$ \cssId{s13}{\overset{\text{karate chop #1:}}{\overset{\text{DISTANCE}}{\overbrace{\ \ \ \ |\ \ \ \ }}}} \cssId{s14}{\overset{\text{from}}{\ \ \ \ \ \strut\ \ \ \ \ }} \cssId{s15}{\overset{\text{karate chop #2:}}{\overset{\text{ZERO}}{\overbrace{\ \ \ \ |\ \ \ \ }}}} $$

However, there is also an algebraic definition of absolute value. In particular, the algebraic definition is needed in the derivation of the quadratic formula.

Piecewise-Defined Functions

The algebraic definition of absolute value uses a piecewise-defined function. A piecewise-defined function is used whenever different output formulas are needed for different inputs; that is, more than one piece is needed to give a full description of what the function does!

Whenever you need one rule for one set of inputs, and another rule for a different set of inputs, then a piecewise-defined function comes to the rescue.

As you'll observe in the definition below, the following notation is used for piecewise-defined functions:

I tend to prefer the simplest notation that clearly conveys meaning, so I opted for no commas in the definition below. However, some alternative presentations are offered following the definition.

Algebraic Definition of Absolute Value
Let $\,x\,$ be a real number. $$ \cssId{s38}{ {|x|} = \begin{cases} x &\text{if }\ x\ge 0 \\ -x &\text{if }\ x\lt 0 \end{cases}} $$

Here are some alternative presentations of this piecewise-defined function:

$$ \cssId{s40}{ {|x|} = \begin{cases} x &\text{for }\ x\ge 0 \\ -x &\text{for }\ x\lt 0 \end{cases}} $$
$$ \cssId{s41}{ {|x|} = \begin{cases} x\,, &\text{if }\ x\ge 0 \\ -x\,, &\text{if }\ x\lt 0 \end{cases}} $$
$$ \cssId{s42}{ {|x|} = \begin{cases} x\,, &\text{if }\ x\ge 0\,, \\ -x\,, &\text{if }\ x\lt 0 \end{cases}} $$
$$ \cssId{s43}{ {|x|} = \begin{cases} x\,, &\text{for }\ x\ge 0\,, \\ -x\,, &\text{for }\ x\lt 0 \end{cases}} $$

Comments on the Algebraic Definition of Absolute Value

How to use the formula

Here's how you'll use the formula to find $\,|x|\,.$

Start by looking inside the absolute value, at the number $\,x\,.$

The second line is most troublesome

The second line of the rule is most troublesome: $$\cssId{s56}{|x| = -x\,, \text{ if }\ x \lt 0}$$

One thing that leads to confusion is that people often read ‘$\,-x\,$’ as ‘negative $\,x\,$’, which makes it sound like the result is negative. Not so!! If $\,x\,$ is negative, then its opposite, $\,-x\,,$ is positive!

If you find yourself getting confused, just read ‘$\,-x\,$’ as ‘the opposite of $\,x\,$’, and this may clear up any difficulty.

The absolute value makes a useful appearance in the following formula:

For all real numbers $\,x\,$: $$ \cssId{s70}{\sqrt{x^2} = {|x|}} $$

So, what is $\,\sqrt{x^2}\,$?  There are two numbers which, when squared, give $\,x^2\,$:   $\,x\,$ and $\,-x\,.$

Which one do we want?  Answer:  Whichever one is nonnegative!

Unless we happen to know the sign of $\,x\,,$ then we won't know which one is nonnegative. Absolute value comes to the rescue! The number $\,|x|\,$ has the same size (distance from zero) as both $\,x\,$ and $\,-x\,,$ and is always nonnegative.

Don't ever fall into a common trap and think that $\,\sqrt{x^2}\,$ is just $\,x\,.$ Sure, this ‘formula’ works if we know, ahead of time, that $\,x\,$ is nonnegative. But, usually we're in situations where $\,x\,$ is allowed to be any real number.

For example, let $\,x = -5\,.$ Then: $$\cssId{s83}{x^2 = (-5)^2 = 25}$$

Is $\,\sqrt{x^2}\,$ equal to $\,x\,$ in this case? No! $$\cssId{s86}{\underbrace{\sqrt{25}}_{\sqrt{x^2}}\ne \underbrace{-5\strut}_{x}}$$ However, it is true that: $$ \cssId{s88}{\sqrt{x^2}} \cssId{s89}{= \sqrt{(-5)^2}} \cssId{s90}{= \overset{=\, |x|}{\overbrace{|-5|}}} \cssId{s91}{= 5} $$


Question: Let $\,x = 3\,.$  Is $\,|x|\,$ equal to $\,x\,$ or $\,-x\,$?
Solution: Since $\,x\,$ is positive, $\,|x| = x\,.$
Question: Let $\,x = -3\,.$  Is $\,|x|\,$ equal to $\,x\,$ or $\,-x\,$?
Solution: Since $\,x\,$ is negative, $\,|x| = -x\,.$

Concept Practice