# Complex Numbers

• September 25th 2008, 06:01 AM
shinhidora
Complex Numbers
I assume this goes here ;)

Been a long while since I posted a problem :p the difficulty level has probably gone up loads since last year xD (which happens if your forced to take 2 years of math in a week :p) Anyway, I had theoretical math this morning and suddenly the professor started babbling something like...

$\omega = \rho^{j\varphi}$ is a root out of $z = r(\cos\theta + j \sin\theta) = re^{j\theta}$ if

$(\rho e ^{j\varphi})^n = r e^{j\theta}$

I had the basics of complex numbers taught at a very fast pace this summer, but I haven't seen roots of complex numbers... So I asked the professor if she could elaborate on the meanings of the various symbols that suddenly popped up and she wouldn't because I'm supposed to know basic math. So could anyone elaborate a bit on the meaning of $\omega, \rho, \varphi$? As I can't find anything about them in my high school notes... Thanks ^^
• September 25th 2008, 07:48 AM
wisterville
Hello,

I don't know if my guess is right... It seems to refer to the following fact:
The roots of the equation $z^n=re^{j\theta}$ ( $r\in\mathbb{R}, r>0, \theta\in\mathbb{R}$)are given by $z=\rho e^{j\varphi}$ where $\rho$ is the unique (positive) real number satisfying $\rho^n=r$ and $\varphi=\frac{\theta+2k\pi}{n}$ for some integer $k$.

Bye
• September 26th 2008, 07:19 AM
CaptainBlack
Quote:

Originally Posted by shinhidora
I assume this goes here ;)

Been a long while since I posted a problem :p the difficulty level has probably gone up loads since last year xD (which happens if your forced to take 2 years of math in a week :p) Anyway, I had theoretical math this morning and suddenly the professor started babbling something like...

$\omega = \rho^{j\varphi}$ is a root out of $z = r(\cos\theta + j \sin\theta) = re^{j\theta}$ if

$(\rho e ^{j\varphi})^n = r e^{j\theta}$

I had the basics of complex numbers taught at a very fast pace this summer, but I haven't seen roots of complex numbers... So I asked the professor if she could elaborate on the meanings of the various symbols that suddenly popped up and she wouldn't because I'm supposed to know basic math. So could anyone elaborate a bit on the meaning of $\omega, \rho, \varphi$? As I can't find anything about them in my high school notes... Thanks ^^

You are supposed to know the polar form of a complex number.

Let $z=a+jb$,

then we can write:

$
z=|z|(\cos(\theta)+j \sin(\theta))
$

where $|z|=\sqrt{a^2+b^2}$ , and $\cos(\theta)=a/|z|$ and $\sin(\theta)+b/|z|$.

Then as:

$e^{j \theta}= \cos(\theta)+j \sin(\theta)$

we have:

$
z=r \ e^{j \theta}
$

where $r=|z|$.

Now we have $w$ is an $n$th root of $z$ then:

$w^n=z$

and if $w=\rho\ e^{j \phi}$ is the polar representation of $w$, and $z=r\ e^{j\theta}$ is the polar representation of $z$ then:

$
(\rho \ e^{j \phi})^n=r \ e^{j \theta}
$

(there is nothing going on in what your teacher wrote on the board than writing the condition for a complex number to be an n-th root of another in polar form)

RonL
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