Is the Phase of a Complex Number Always Taken with Respect to the Real Axis?

[Niles]

Homework Statement

Hi all.

Is the phase of a complex number always taken with respect to the real, positive axis? I mean, is it always the direction as shown here: http://theories.toequest.com/content_images/4/argand.gif

Thanks in advance.

 

[HallsofIvy]

Yes. Any complex number, a+ bi, can be written, in "polar coordinates", as $r (cos(\theta)+ i sin(\theta))= re^{i\theta}$ where r is the distance from (0, 0) (= 0+ i0) to (a,b) (= a+ bi) and $\theta$ is the angle the line from (0,0) to (a, b) makes with the positive x- axis.

Note that because cosine, sine and $e^{i\theta}$ are all periodic with period $2\pi$ we can add any multiple of $2\pi$ to theta: $a+ bi= r (cos(\theta+ 2n\pi)+ i sin(\theta+ 2n\pi)= re^{i(\theta+ 2n\pi)}$ for n any integer. However, that angle is still measured from the positive x-axis.

 

[Niles]

Thanks. You have helped me a lot lately.

Merry Christmas.