Absolute value on ℝ

The absolute value of a real number $x\in\mathbb{R}$ is

Equivalently, $|x|=\sqrt{x^2}$ and $|x|$ is the distance from $x$ to $0$ in $\mathbb{R}$ with the usual metric $d(x,y)=|x-y|$. Absolute value is fundamental for defining convergence, continuity, and error bounds in one-variable analysis.

Examples:

• $|3|=3$ and $|-3|=3$.
• $|0|=0$.
• $|x-y|$ measures the distance between $x$ and $y$ in $\mathbb{R}$.