The absolute value \(|x|\) of a real number \(x\) provides its magnitude, i.e. how large it is without regard to being positive or negative. Geometrically, it represents the distance the number is from zero along the real number line.
We can express this precisely with the conditional expression
Note that \(|0| = 0\text{.}\) Formulas like Equation (1.7) which involving multiple cases are common in engineering. Take some time to carefully understand the notation used above.
Distance between numbers.
We can measure the distance between two real numbers \(a\) and \(b\) on the real line by subtracting them and then computing the absolute value to obtain a distance. This ensures that we do not need to worry about the order the subtraction was performed.
\begin{equation*}
\left(\text{distance between $a$ and $b$}\right) = \left| a-b\right| = \left|b-a\right|.
\end{equation*}
Example1.52.
If we divide the portion of the real line between \(-6\) and \(8\) into ten equal width parts, what is the width of each part?
Be careful simplifying the expression \(\sqrt{x^2}\text{.}\) First, note that \(x\) is squared (making the result non-negative) and then we want the principal square root of \(x^2\text{.}\) Consider two examples, first with \(x=4\)
which does not result in \(-4\text{.}\) In both cases, we finished with the positive value \(4\text{,}\) i.e. the absolute value of \(x\text{.}\) In general, for any real number \(x\text{,}\)
