compare and contrast the absolute value of a real number to that of a complex number.

Brief Explanations:

For a real number (x), the absolute - value (|x|) is its non - negative distance from 0 on the number line, so (|x|=\begin{cases}x, & x\geq0\ - x, & x < 0\end{cases}). For a complex number (z=a + bi) ((a,b\in R)), the absolute value (|z|) is the distance from the origin ((0,0)) to the point ((a,b)) in the complex plane, and (|z|=\sqrt{a^{2}+b^{2}}). When (b = 0) (i.e., (z) is a real number), (|z|=\sqrt{a^{2}}=|a|), so the absolute value of a real number is a special case of the absolute value of a complex number. The main difference is that the absolute value of a real number is defined based on the one - dimensional number line, while the absolute value of a complex number is defined based on the two - dimensional complex plane.

Answer:

The absolute value of a real number (x) is (|x|=\begin{cases}x, & x\geq0\ - x, & x < 0\end{cases}), representing its distance from 0 on the number line. The absolute value of a complex number (z=a + bi) is (|z|=\sqrt{a^{2}+b^{2}}), representing its distance from the origin in the complex plane. The absolute value of a real number is a special case of that of a complex number when the imaginary part is 0.