For questions about or involving the absolute value function also known as modulus function.
The absolute value function, usually denoted by $|x|$, is a function $\mathbb{R} \to [0, \infty)$ which can be defined in three equivalent ways:
$|x| = \begin{cases}x &\ \text{if $x \ge 0$, and} \\ -x &\ \text{if $x < 0$.} \end{cases}$
$|x| = \sqrt{x^2}$, and
$|x| = \max \{x, -x\}$.
This definition extends to complex numbers as the square root of the norm: $|x+iy|=\sqrt{x^2+y^2}$. In both cases, the function may be interpreted geometrically as the distance of the input number from the origin.
More generally, an absolute value may be defined on a field (or an integral domain) $k$ as a function $|\cdot | : k \to \mathbb{R}$ which satisfies the axioms
(nonnegativity) $|x| \ge 0$ for all $x \in k$,
(definiteness) $|x| = 0 \iff x = 0 \in k$,
(multiplicativity) $|x y| = |x||y| $ for all $x,y\in k$ ), and
(triangle inequality) $|x+y| \le |x| + |y|$ for all $x,y\in k$.
For example, if $p$ is a fixed prime number and $x \in \mathbb{Q}$, then there exists a unique $n \in \mathbb{Z}$ such that $x$ may be written as $$ x = p^n \frac{a}{b}, $$ where $\gcd(p, a) = \gcd(p, b) = 1$. The function which maps $x$ to $p^{-n}$ is an absolute value on $\mathbb{Q}$, called the $p$-adic absolute value.
