# [SOLVED] Complex Numbers

• Aug 6th 2009, 01:54 AM
Awsom Guy
[SOLVED] Complex Numbers
Hi everybody,

Question,

Find the argument of z for each of the following in the interval [0,2pie]. (Give exact answers where possible)

z=3+2i

please show the working out aswell. Thanks everyone.
• Aug 6th 2009, 03:02 AM
Plato
$\displaystyle \arg \left( {3 + 2i} \right) = \arctan \left( {\frac{2} {3}} \right)$
• Aug 6th 2009, 03:13 AM
Awsom Guy
umm..
Quote:

Originally Posted by Plato
$\displaystyle \arg \left( {3 + 2i} \right) = \arctan \left( {\frac{2} {3}} \right)$

could you explian it further
thanks
• Aug 6th 2009, 05:28 AM
mr fantastic
Quote:

Originally Posted by Awsom Guy
could you explian it further
thanks

Plot z = 3 + 2i on an Argand diagram. Simple trigonometry gives the result previously stated.
• Aug 6th 2009, 06:35 AM
Amer
Quote:

Originally Posted by Awsom Guy
Hi everybody,

Question,

Find the argument of z for each of the following in the interval [0,2pie]. (Give exact answers where possible)

z=3+2i

please show the working out aswell. Thanks everyone.

see this Complex Argument -- from Wolfram MathWorld
• Aug 6th 2009, 04:03 PM
HallsofIvy
The real question appears to be: Do you know the definition of "argument" of a complex number?
• Aug 7th 2009, 01:05 AM
Awsom Guy
Confused
Quote:

Originally Posted by Amer

thats good but could you please show me how to work this one out because I am sort of confused on this bit
Thanks
• Aug 7th 2009, 03:49 AM
mr fantastic
Quote:

Originally Posted by Awsom Guy
thats good but could you please show me how to work this one out because I am sort of confused on this bit
Thanks

You've been given all the tools necessary (and I assume you have class notes and a textbook as well). It's time you started using those tools - please show your work and clearly state where you're stuck.
• Aug 7th 2009, 09:18 AM
Amer
Quote:

Originally Posted by Awsom Guy
thats good but could you please show me how to work this one out because I am sort of confused on this bit
Thanks

complex numbers written in general as

$\displaystyle z=x+iy$ right

then the argument of z given by

$\displaystyle \tan \theta = \frac{y}{x}$ since the argument is $\displaystyle \theta$ values so to find theta values take arctan ( $\displaystyle tan^{-1}$ ) for both sides you will have

$\displaystyle \theta = \tan^{-1} \frac{y}{x}$ the argument of z denoted by arg(z)

note:- argument of z have infinitely many values

the principle argument of z denoted by Arg(z) is the unique value of $\displaystyle \theta$ such that $\displaystyle -\pi < \theta <= \pi$ as you can see the principle argument take one value

you can say that

$\displaystyle arg(z)=Arg(z)+2n\pi$ n integar number

Examples:-

Find the argument and the principle argument to the following:

$\displaystyle 1)z=\sqrt{3}+i$ y=1 and x=sqrt{3} so

$\displaystyle Arg(z)=\tan^{-1} \frac{1}{\sqrt{3}}$

$\displaystyle Arg(z)=\frac{\pi}{6}$

$\displaystyle arg(z)=\frac{\pi}{6}+2n\pi$

$\displaystyle 2)z=1+\sqrt{3}i$

$\displaystyle Arg(z)=\tan^{-1}\frac{\sqrt{3}}{1}$

$\displaystyle Arg(z)=\frac{\pi}{3}$

$\displaystyle arg(z)=\frac{\pi}{3}+2n\pi$

$\displaystyle 3)z=-2+2i$

$\displaystyle Arg(z)=\tan^{-1}\frac{2}{-2}$

$\displaystyle Arg(z)=\tan^{-1}(-1)$
the tan is negative in the second quarter and in the third quarter but the principle argument take the values in the interval $\displaystyle \left(-\pi,\pi\right]$

or to make it easier to determine the angle write it

$\displaystyle Arg(z)=-\tan^{-1}(1)$
so

$\displaystyle Arg(z)=\frac{-\pi}{4}$

$\displaystyle arg(z)=\frac{-\pi}{4}+2n\pi$

$\displaystyle 4)z=-5+7i$

$\displaystyle Arg(z)=\tan^{-1} \frac{7}{-5}$ to make it easier to determine the angle

$\displaystyle Arg(z)=-\tan^{-1} \frac{7}{5}$

$\displaystyle Arg(z)=-54.462$

$\displaystyle arg(z)=-54.462+2n\pi$

I think it is clear now