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Concept
Sign function
Summary
In mathematics, the sign function or signum function (from signum, Latin for "sign") is a function that returns the sign of a real number. In mathematical notation the sign function is often represented as \sgn (x).
Definition
The signum function of a real number x is a piecewise function which is defined as follows:
\sgn x :=\begin{cases}
-1 & \text{if } x < 0, \
0 & \text{if } x = 0, \
1 & \text{if } x > 0. \end{cases}
Properties
Any real number can be expressed as the product of its absolute value and its sign function:
x = |x| \sgn x.
It follows that whenever x is not equal to 0 we have
\sgn x = \frac{x}{|x|} = \frac{|x|}{x},.
Similarly, for any real number x,
|x| = x\sgn x.
We can also ascertain that:
\sgn x^n=(\sgn x)^n.
The signum func
Official source
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