PlanetMath: modulus of complex number

(more info)

Math for the people, by the people.

Main Menu

sections Encyclopædia
Papers
Books
Expositions

meta Requests (203)
Orphanage
Unclass'd
Unproven (490)
Corrections (7)
Classification

talkback Polls
Forums
Feedback
Bug Reports

downloads Snapshots
PM Book

information News
Docs
Wiki
ChangeLog
TODO List
Legalese
About

Entry average rating: No information on entry rating

modulus of complex number

(Definition)

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

$\displaystyle \vert z\vert := \sqrt{z\overline{z}}.$

There is also the notation

$\displaystyle \mod{z}$

for the modulus of

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

$\displaystyle \vert z\vert$

$\displaystyle =$

$\displaystyle \sqrt{a^2+b^2},$

(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

$\displaystyle \vert z_1+z_2\vert \le \vert z_1\vert+\vert z_2\vert,$

also

$\displaystyle \vert\Re{z}\vert \le \vert z\vert,\quad \vert\Im{z}\vert \le \vert z\vert,\quad \vert z\vert \le \vert\Re{z}\vert+\vert\Im{z}\vert.$

Modulus is multiplicative:

$\displaystyle \vert z_1z_2\vert = \vert z_1\vert\cdot\vert z_2\vert, \quad \left\vert\frac{z_1}{z_2}\right\vert = \frac{\vert z_1\vert}{\vert z_2\vert}$

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, $x = re^{i\phi}$ , , $\phi\in[0,2\pi)$ , then $\phi = 0$ or $\phi=\pi$ , so $e^{i\phi}=\pm 1$ . Thus,

$\begin{displaymath} \vert x\vert = \sqrt{x^2} = \begin{cases} x & x>0 \ 0 & x=0 \ -x & x<0 \end{cases}.\end{displaymath}$

Anyone with an account can edit this entry. Please help improve it!

"modulus of complex number" is owned by matte. [ full author list (3) ]

(view preamble)

Other names:

complex modulus, modulus, absolute value of complex number, absolute value, modulus of a complex number

This object's parent.

Log in to rate this entry.
(view current ratings)

Cross-references: triangle inequality, point, Euclidean norm, imaginary part, real part, real number, positive, polar form, complex conjugate, complex number
There are 64 references to this entry.

This is version 14 of modulus of complex number, born on 2003-05-04, modified 2007-06-19.
Object id is 4242, canonical name is ModulusOfComplexNumber.
Accessed 28048 times total.

Classification:

AMS MSC:	12D99 (Field theory and polynomials :: Real and complex fields :: Miscellaneous)
	30-00 (Functions of a complex variable :: General reference works )
	32-00 (Several complex variables and analytic spaces :: General reference works )

Pending Errata and Addenda

None.[ View all 3 ]

Discussion

forum policy

No messages.

Interact