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The function of a real variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085100/s0851001.png" /> which is equal to 1 if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085100/s0851002.png" /> is positive, equal to 0 if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085100/s0851003.png" /> is zero and equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085100/s0851004.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085100/s0851005.png" /> is negative. Notation: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085100/s0851006.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085100/s0851007.png" />. Thus,
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085100/s0851008.png" /></td> </tr></table>
 
 
 
  
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The function of a real variable $x$ which is equal to $1$ if $x$ is positive, equal to 0 if $x$ is zero and equal to $-1$ if $x$ is negative. Notation: $\operatorname{sgn} x$ or $\operatorname{sign} x$. Thus,
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\begin{equation*}
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\operatorname{sgn} x =
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\begin{cases}
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\ \ \,\,1\quad \text{if } x>0,\\
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\ \ \,\,0\quad \text{if } x=0,\\
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-1\quad \text{if } x<0.\\
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\end{cases}
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\end{equation*}
  
 
====Comments====
 
====Comments====
The signum function is usually extended to the complex plane by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085100/s0851009.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085100/s08510010.png" /> (and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085100/s08510011.png" />). Thus, it measures the angle of the ray from the origin on which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085100/s08510012.png" /> lies.
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The signum function is usually extended to the complex plane by $\operatorname{sgn} z = z / |z|$ if $z\ne0$ (and $\operatorname{sgn} 0=0$). Thus, it measures the angle of the ray from the origin on which $z$ lies.

Latest revision as of 15:41, 23 November 2012


The function of a real variable $x$ which is equal to $1$ if $x$ is positive, equal to 0 if $x$ is zero and equal to $-1$ if $x$ is negative. Notation: $\operatorname{sgn} x$ or $\operatorname{sign} x$. Thus, \begin{equation*} \operatorname{sgn} x = \begin{cases} \ \ \,\,1\quad \text{if } x>0,\\ \ \ \,\,0\quad \text{if } x=0,\\ -1\quad \text{if } x<0.\\ \end{cases} \end{equation*}

Comments

The signum function is usually extended to the complex plane by $\operatorname{sgn} z = z / |z|$ if $z\ne0$ (and $\operatorname{sgn} 0=0$). Thus, it measures the angle of the ray from the origin on which $z$ lies.

How to Cite This Entry:
Signum. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Signum&oldid=14138
This article was adapted from an original article by Yu.A. Gor'kov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article