Absolute value Notes

Absolute Value Measures Distance, Not Direction

1. Absolute value is also called the modulus or numerical value.
2. Because it represents a distance, it is never negative.

Two Equivalent Definitions Give You Two Ways to Work

1. One standard definition is piecewise (it depends on whether $x$ is negative or not): $$|x|=\begin{cases} -x, & x<0 \\ \phantom{-}x, & x\ge 0 \end{cases}$$
2. A second equivalent definition is: $$|x|=\sqrt{x^2}$$
3. These two definitions agree because squaring makes any real number non-negative, and then the square root returns the non-negative value.

What Numbers Can Be Absolute Values?

1. Since $|x|\ge 0$, any non-negative real number could be an absolute value.
   1. $\sqrt{25}=5$ can be an absolute value.
   2. $\sqrt{17}$ can be an absolute value (it is positive).
   3. $-\sqrt{2}$ cannot be an absolute value (it is negative).
2. This leads to an important property: $$|x|\ge 0\text{ for all real }x,\quad \text{and }|x|=0\text{ only when }x=0.$$

Absolute Value Works Predictably with Multiplication and Division

Absolute value interacts nicely with multiplication and division.

Multiplication Property

1. For any real numbers $a$ and $b$, $$|ab|=|a|\,|b|$$
2. This is true whether $a$ and $b$ are positive, negative, or zero.

Division Property

For any real numbers $a$ and $b$ with $b\ne 0$, $$\left|\frac{a}{b}\right|=\frac{|a|}{|b|}$$

Powers: Even Exponents Remove Signs, Odd Exponents Keep Them

Because absolute value removes sign, it has a clear relationship with powers.

Squaring (and Any Even Power)

1. For any real $a$, $$|a^2|=|a|^2=a^2$$
2. The key idea is that $a^2\ge 0$ already, so the absolute value does nothing.
3. More generally, for any integer $n$,
   1. If $n$ is even, then $a^n\ge 0$ and $|a^n|=|a|^n=a^n.$
   2. If $n$ is odd, then $|a^n|=|a|^n,$ but $a^n$ may be negative (it has the same sign as $a$).

Absolute Value Measures Error and "How Far Apart" Two Values Are

Absolute value is especially useful in approximation and measurement, because it measures size of an error without caring whether you were above or below the true value.

The expression $|A-T|$ is the distance between $A$ and $T$ on the number line.

Square Roots and Absolute Value: What Is Always Allowed?

Square roots and absolute values are related, but they have different domain rules.

1. You can take $|a|$ for any real number $a$.
2. You cannot take $\sqrt{a}$ for negative real $a$ (in real-number work).
3. But you can always take $\sqrt{|a|}$ for real $a$, because $|a|\ge 0$.