1. ## Complex Numbers

Hi there, first time post. I wasn't quite sure what category to put this under so apologies if its in the wrong section. Basically, part of my uni course is electronics and some questions involve complex numbers, mainly the use of j and other things. I would get the answer as the solution gets it down to 4-j7 and then it says this equals 8.0623<-60.2551° .................................................. ...........< is supposed to be like/_ but couldnt find the proper symbol. I just cant understand the meaning of j and such things. Any help appreciated.

Thanks, Andy

2. The electrical engineers use $j\equiv\sqrt{-1},$ because $i$ is already used for current. So you can write a complex number as $z=a+jb.$ Here $a$ is the real part of $z,$ written $\text{Re}(z)=a,$ and $b$ is the imaginary part of $z$, written $\text{Im}(z)=b.$ This is called the cartesian representation of a complex number.

You can also write complex numbers as a magnitude and an angle - this is called the polar representation of a complex number. That is, $z=re^{j\theta},$ where $r$ is the magnitude, and $\theta$ is the angle, or argument. Thus you can write $|z|=r,$ and $\text{Arg}(z)=\theta.$ You'll also see this notation, as you've already alluded to: $z=r\angle\theta.$

The transformation from cartesian to polar and back goes like this:

$a=r\cos(\theta),$
$b=r\sin(\theta),$ and

$r=\sqrt{a^{2}+b^{2}},$
$\theta=\tan^{-1}(b/a).$

All of these equations come from elementary trigonometry. So, with this information, can you see how they obtained the polar representation from the cartesian one?

3. Thank you very much, this is exactly what I needed. Greatly appreciated

4. You're very welcome. Have a good one!

5. Oh, those engineers and their jmaginary numbers!