The abs function returns the absolute value of the expression x.

You can enter the command abs using either the 1-D or 2-D calling sequence. For example, abs(-11) is equivalent to $\left|\mathrm{-11}\right|$.

If x is an rtable (Array, Matrix, or Vector), the abs function applies the abs function to each entry in the table, and returns the resulting rtable.

If x includes a function f, then abs will attempt to execute the procedure abs/f to determine the absolute value of the corresponding part of x.  The user can thus easily extend the functionality of abs.

The derivative of abs is denoted by abs(1, x). This is signum(x) for all non-0 real numbers, and is undefined otherwise. Neither first order nor higher order derivatives of abs can be determined if x is an rtable.

Higher order derivatives of abs are denoted by abs(n, x), where n is a positive integer.  When n is known, the expression is automatically simplified to the appropriate expression in a derivative of either signum or abs.

$\mathrm{abs}\left(-11\right)$

$11$

$\mathrm{abs}\left(\cos \left(3\right)\right)$

$-\cos \left(3\right)$

$\mathrm{evalf}\left(\right)$

$0.9899924966$

$\mathrm{abs}\left(-2\cdot 10\right)$

$20$

$a≔\mathrm{abs}\left(2x-3\right)$

$a≔\left|2x-3\right|$

$x≔1$

$x≔1$

$a$

$1$

$\mathrm{abs}\left(3-4\mathrm{I}\right)$

$5$

$\mathrm{expr}≔\mathrm{abs}\left(\mathrm{sqrt}\left(2\right)\mathrm{I}{u}^{2}v\right)$

$\mathrm{expr}≔\sqrt{2}{\left|u\right|}^2\left|v\right|$

$\mathrm{expand}\left(\mathrm{expr}\right)$

$\sqrt{2}{\left|u\right|}^2\left|v\right|$

$\mathrm{combine}\left(\mathrm{expr}\,\mathrm{abs}\right)$

$\sqrt{2}{\left|u\right|}^2\left|v\right|$

$\mathrm{assume}\left(u\,\mathrm{positive}\right)\:$$\mathrm{simplify}\left(\mathrm{expr}\right)$

$\sqrt{2}{\mathrm{u~}}^2\left|v\right|$

$\mathrm{expr2}≔\mathrm{abs}\left(b^4{c}^2{d}^3\right)$

$\mathrm{expr2}≔{\left|b\right|}^4{\left|c\right|}^2{\left|d\right|}^3$

$\mathrm{assume}\left(0<b\&comma;0<c\right)\&colon;$$\mathrm{simplify}\left(\mathrm{expr2}\right)$

${\mathrm{b~}}^4{\mathrm{c~}}^2{\left|d\right|}^3$

$\mathrm{int}\left(\mathrm{abs}\left(y\right)\&comma;y\right)$

$\left\{\begin{array}{cc}-\frac{y^2}{2}& y\le 0\\ \frac{y^2}{2}& 0<y\end{array}\right.$

$\mathrm{diff}\left(\mathrm{abs}\left(y\right)\&comma;y\right)$

$\mathrm{abs}\left(1\&comma;y\right)$

$\mathrm{diff}\left(\mathrm{abs}\left(y\right)\&comma;y\&comma;y\right)$

$\mathrm{signum}\left(1\&comma;y\right)$

$\mathrm{abs}\left(2\&comma;y\right)$

$\mathrm{signum}\left(1\&comma;y\right)$

$\mathrm{abs}\left(\mathrm{Array}\left(1..2\&comma;1..2\&comma;-2\right)\right)$

$\left[\begin{array}{cc}2& 2\\ 2& 2\end{array}\right]$

$\mathrm{abs}\left(⟨⟨-1\&comma;2\&comma;3⟩|⟨-4\&comma;5\&comma;-6⟩⟩\right)$

$\left[\begin{array}{cc}1& 4\\ 2& 5\\ 3& 6\end{array}\right]$

$\mathrm{abs}\left(\mathrm{Matrix}\left(3\&comma;\mathrm{fill}=-1\right)\right)$

$\left[\begin{array}{ccc}1& 1& 1\\ 1& 1& 1\\ 1& 1& 1\end{array}\right]$

$\mathrm{abs}\left(1\&comma;\mathrm{rtable}\left(1..2\&comma;-1\right)\right)$

Error, invalid input: `simpl/abs` expects its 2nd argument, a2, to be of type algebraic, but received Array(1..2, [-1,-1])