Why do all numbers in the complex plane define a unit circle radius?

[rustynail]

Hi, I have recently started learning about complex numbers and the complex plane, and there is something I don't quite understand. Why is it that for every number z \in C

e^{i\theta} = z

we have

\sqrt { [Re(z)]² + [Im(z)]² } = 1 ?


Except for \theta = \frac {\pi}{2} or \frac {3\pi}{2} of course. In other words, why do all numbers z define a unit circle's radius ?

Any help would be very much appreciated

 

[slider142]

That is not true. Only complex numbers of the form z = e^(iθ) lie on the unit circle and satisfy that equation. Complex numbers of the form z = Re^(iθ) lie on circles of radius R in the complex plane, as you can verify.

 

[rustynail]

I have stated that z must satisfy

e^{i\theta} = z

 

[slider142]

In that case, apply Euler's formula: e^(iθ) = cos(θ) + i*sin(θ), from which the result is immediate. The derivation of Euler's formula is implied by the Taylor series for e^x, cos(x) and sin(x), but its full justification requires a little more rigor.

 

[rustynail]

Ok so adding the taylor series for cos(θ) and for i sin(θ) should give the taylor series for e^(iθ) ? I will definitely try that out. Thank you!

 

[SteveL27]

Ok so adding the taylor series for cos(θ) and for i sin(θ) should give the taylor series for e^(iθ) ? I will definitely try that out. Thank you!

Yes, that's one of the proofs. Definitely worth going through the steps. Euler's formula falls right out of the power series for e^(iθ).

 

[chiro]

Also if you apply the fact that the complex conjugate of e^(iθ) = e^(-iθ) then multiply the two gives you e^(iθ - iθ) = e^(0) = 1.

You can prove this using the propery that cos(-x) = cos(x) and -sin(x) = sin(-x) using properties of odd and even functions.