# Math Help - [SOLVED] Complex Numbers

1. ## [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

the answer is 0.588

please show the working out aswell. Thanks everyone.

2. $\arg \left( {3 + 2i} \right) = \arctan \left( {\frac{2}
{3}} \right)$

3. ## umm..

Originally Posted by Plato
$\arg \left( {3 + 2i} \right) = \arctan \left( {\frac{2}
{3}} \right)$
could you explian it further
thanks

4. 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.

5. 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

the answer is 0.588

please show the working out aswell. Thanks everyone.
see this Complex Argument -- from Wolfram MathWorld

6. The real question appears to be: Do you know the definition of "argument" of a complex number?

7. ## Confused

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

8. 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.

9. 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

$z=x+iy$ right

then the argument of z given by

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

$\theta = \tan^{-1} \frac{y}{x}$ 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 $\theta$ such that $-\pi < \theta <= \pi$ as you can see the principle argument take one value

you can say that

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

Examples:-

Find the argument and the principle argument to the following:

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

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

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

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

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

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

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

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

$3)z=-2+2i$

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

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

or to make it easier to determine the angle write it

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

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

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

$4)z=-5+7i$

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

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

$Arg(z)=-54.462$

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

I think it is clear now