Complex Numbers

[Ted_Grendy]

Hi all

I was wondering if someone could help with the following complex number question:-

P = 3 + 3i
Find P^10

My solution is to convert 3 + 3i into polar form which is 3sqrt(2) < 45 Degrees
Then 3sqrt(2)^10 < 10 * 45 = 1889568 < 450 Degrees

No idea if this is correct, can anyone help?

Also I have been told that the max angle in complex number must be between 0-180 degrees - this is correct?

Thank you

 

[Harry_the_cat]

You need to convert 450 degrees to a coterminal angle.
I would also expect the answer to be converted back to a+bi form.

 

[topsquark]

Hi all

I was wondering if someone could help with the following complex number question:-

P = 3 + 3i
Find P^10

My solution is to convert 3 + 3i into polar form which is 3sqrt(2) < 45 Degrees
Then 3sqrt(2)^10 < 10 * 45 = 1889568 < 450 Degrees

No idea if this is correct, can anyone help?

Also I have been told that the max angle in complex number must be between 0-180 degrees - this is correct?

Thank you

Some advice: It's much better to use radians here, not degrees. That way the exponential form will be \(\displaystyle ae^{ib}\) instead of \(\displaystyle a^{ib(\pi / 180)}\).

-Dan

 

[Romsek]

Your idea is correct.

\(\displaystyle 3+3i = 3\sqrt{2} e^{i\pi/4}\\~\\

(3+3i)^{10} = (3\sqrt{2})^{10} e^{i10\pi/4} = 3^{10}2^5 e^{i5\pi/2} = \\~\\

3^{10}2^5 e^{i\pi/2} =1889568i\)

 

[Dr.Peterson]

Some advice: It's much better to use radians here, not degrees. That way the exponential form will be \(\displaystyle ae^{ib}\) instead of \(\displaystyle a^{ib(\pi / 180)}\).

-Dan

My impression is that the notation used here for polar form, [MATH]r\angle\theta[/MATH], is most commonly used with degrees, and perhaps not in the mathematical setting where [MATH]r e^{i\theta}[/MATH] is used, so much as in engineering. Probably many people who use this notation don't know about exponential notation for complex numbers, even though it is equivalent.

But I could be wrong.

Also I have been told that the max angle in complex number must be between 0-180 degrees - this is correct?

That may depend on what you have been taught. I would expect the angle (after normalization) to be either from 0 to 360 degrees or from -180 to +180, with r always positive. You could get away with 0 to 180 if r were allowed to be negative, but I don't think that's standard, because r is generally described as the magnitude (modulus).

Who told you this?