Absolute Value

The absolute value of a Real Number $x$ is denoted $\vert x\vert$ and given by

$$\vert x\vert= x\mathop{\rm sgn}\nolimits (x) = \cases{ -x & for $x\leq 0$\cr x & for $x\geq 0$,\cr}$$

where Sgn is the sign function.

The same notation is used to denote the Modulus of a Complex Number $z=x+iy$, $\vert z\vert\equiv\sqrt{x^2+y^2}$, a p-adic Number absolute value, or a general Valuation. The Norm of a Vector ${\bf x}$ is also denoted $\vert{\bf x}\vert$, although $\vert\vert{\bf x}\vert\vert$ is more commonly used.

Other Notations similar to the absolute value are the Floor Function $\left\lfloor{x}\right\rfloor$, Nint function $[x]$, and Ceiling Function $\left\lceil{x}\right\rceil$.

See also Absolute Square, Ceiling Function, Floor Function, Modulus (Complex Number), Nint, Sgn, Triangle Function, Valuation