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

$$modz$$

for the modulus of $z$.

If we write $z$ in polar form as  $z=r{e}^{i\varphi }$  with  $r\ge 0,\varphi \in [0,\mathrm{\hspace{0.17em}2}\pi )$,  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

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

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

$$|z_1+z_2|\le |z_1|+|z_2|,$$

also

$$|\mathrm{\Re }z|\le |z|,|\mathrm{\Im }z|\le |z|,|z|\le |\mathrm{\Re }z|+|\mathrm{\Im }z|.$$

Modulus is :

$$|z_1{z}_2|=|z_1|\cdot |z_2|,\left|\frac{z_1}{z_2}\right|=\frac{|z_1|}{|z_2|}$$

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

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