Simple Proof Methods

Definition. [Absolute values] The absolute value of a real number $x$, denoted $|x|$, is defined as: $$|x| = \begin{cases}x & \text{ if } x \ge 0, \\-x & \text{ if } x < 0.\end{cases}$$(

$$|x| = \begin{cases}x & \text{ if } x \ge 0, \\-x & \text{ if } x < 0.\end{cases}$$

)

Geometrically, the absolute value represents the distance of a number from zero on the number line.

For example, $|3| = 3$ and $|-8| = 8$.

Since the absolute value is a piecewise defined function (specifically, it has 2 pieces,) this means that proofs concerning absolute values will sometimes need to be done in at least two pieces. A proof of this kind is called a proof by cases.