Absolute Value

Why

We want a notion of distance between elements of the real line.

Definition

The absolute value of a real number is the greater of itself and its additive inverse. In other words, if $x$ is positive, then the absolute value of $x$ is $x$. If $x$ is negative, then the absolute value of $x$ is $-x$ (a positive real number).

Notation

We denote the absolute value of a real number $x \in \R $ by $\abs{x}$.

Distance

The absolute value can be interpreted as the distance between the point corresponding to the real number and the point corresponding to 0. We can generalize this idea. Consider $x, y \in \R $. If $x > y$, then $x - y > 0$ and so the distance between the corresponding points is $x - y$. If $x < y$ then $y - x > 0$, and so the distance is $y - x$.

The observation is that $\abs{-x} = \abs{x}$. So

\[ \abs{y - x} = \abs{-(x - y)} = \abs{x-y}. \]