What Interesting Pattern Do Complex Numbers Reveal in Matlab?

[MathAmateur]

I was playing around with complex numbers in Matlab this evening and noticed this interesting pattern:

Given:

$$a = (e^{x})^{i \pi/2}$$

When x is incremented an integer power (0,1,2,3), the a is rotated $${\pi/2}$$ radians in the complex plane. It started out at 0 radians with x = 0 and then rotated to $${\pi/2}$$ radians with x= 1 (the familiar Euler result) and then then to $${\pi}$$, etc, around and around the unit circle.

I found this very interesting and just wanted to share it and ask if there were any comments on why this may be so.

 

[olivermsun]

You might find this Wiki article on http://en.wikipedia.org/wiki/Euler%27s_formula_in_complex_analysis" interesting.

 

[MathAmateur]

Thank you for pointing out the interesting article. I now see that what I found out above was:

$$e^{a}e^{a}=e^{a+a}$$ for complex numbers.

This maybe isn't Earth shaking but it does give me a more intuitive feel for what is going on to discover it on my own.