modulus of complex number

Definition Let $z$ be a complex number, and let $\overline{z}$ be the complex conjugate of $z$. Then the modulus, or absolute value, of $z$ is defined as

 $|z|:=\sqrt{z\overline{z}}.$

There is also the notation

 $\mod{z}$

for the modulus of $z$.

If we write $z$ in polar form as  $z=re^{i\phi}$  with  $r\geq 0,\;\phi\in[0,\,2\pi)$,  then  $|z|=r$. It follows that the modulus is a positive real number or zero. Alternatively, if $a$ is the real part of $z$, and $b$ the imaginary part, then

 $\displaystyle|z|$ $\displaystyle=$ $\displaystyle\sqrt{a^{2}+b^{2}},$ (1)

which is simply the Euclidean norm of the point  $(a,\,b)\in\mathbb{R}^{2}$. It follows that the modulus satisfies the triangle inequality

 $|z_{1}+z_{2}|\leq|z_{1}|+|z_{2}|,$

also

 $|\Re{z}|\leq|z|,\quad|\Im{z}|\leq|z|,\quad|z|\leq|\Re{z}|+|\Im{z}|.$

Modulus is :

 $|z_{1}z_{2}|=|z_{1}|\cdot|z_{2}|,\quad\left|\frac{z_{1}}{z_{2}}\right|=\frac{|% z_{1}|}{|z_{2}|}$

Since $\mathbb{R}\subset\mathbb{C}$, the definition of modulus includes the real numbers. Explicitly, if we write  $x\in\mathbb{R}$  in polar form,  $x=re^{i\phi}$,  $r>0$,  $\phi\in[0,2\pi)$,  then  $\phi=0$  or  $\phi=\pi$, so  $e^{i\phi}=\pm 1$. Thus,

 $|x|=\sqrt{x^{2}}=\begin{cases}x&x>0\\ 0&x=0\\ -x&x<0\end{cases}.$
 Title modulus of complex number Canonical name ModulusOfComplexNumber Date of creation 2013-03-22 13:36:39 Last modified on 2013-03-22 13:36:39 Owner matte (1858) Last modified by matte (1858) Numerical id 17 Author matte (1858) Entry type Definition Classification msc 32-00 Classification msc 30-00 Classification msc 12D99 Synonym complex modulus Synonym modulus Synonym absolute value of complex number Synonym absolute value Synonym modulus of a complex number Related topic AbsoluteValue Related topic Subadditive Related topic SignumFunction Related topic ComplexConjugate Related topic PotentialOfHollowBall Related topic ConvergenceOfRiemannZetaSeries Related topic RealPartSeriesAndImaginaryPartSeries Related topic ArgumentOfProductAndSum Related topic ArgumentOfProductAndQuotient Related topic EqualityOfComplexNumbers