Imaginary numbers raised to the power

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kmalik001

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So I have this question:

If w = z1^(3000) , determine angle of w in the range [0, 2). Your answer should be correct to four decimal
places.
where z1= 3 + i

Any ideas of how I should start?
 
I don't quite remember how to raise a complex number to a power. Should I take the magnitude and then raise it to the 3000th power?
So (√10)^3000?
 
Write z in polar form \(\displaystyle z= re^{i\theta}= r(cos(\theta)+ i sin(\theta)). Then \(\displaystyle z^n= r^n e^{in\theta}= r^n(cos(n\theta)+ i sin(n\theta)\).

If z= a+ bi then \(\displaystyle r= \sqrt{a^2+ b^2}\) and \(\displaystyle \theta= arctan(\frac{b}{a})\).\)
 
kmalik001 said:
The calculator doesnt give me a value for (√10)^3000 which is r^n
Click to expand...

You got to be kidding!! You need a calculator for that!!


(√10)^3000 = [(10)1/2]3000 = 101500

That is 1 with 1500 zeroes after it!!

What is it that you exactly want to know about that?
 
kmalik001 said:
So I have this question:

If w = z1^(3000) , determine angle of w in the range [0, 2). Your answer should be correct to four decimal
places.
where z1= 3 + i

Any ideas of how I should start?
Click to expand...
Start by finding the polar angle of z1 = 3 + i. [as suggested by HallsOfIvy]
Then multiply that angle by 3000.
Take that number modulo 2pi. [I suspect the problem actually says "in the range [0,2pi)"]

You weren't asked for the modulus - just the angle.
 
so the angle is arctan(1/3) = 18.43

18.43 * 3000 = 55304

whats next take modulo?

Yes the question does say in the range (0-2pi)
 
kmalik001 said:
so the angle is arctan(1/3) = 18.43

18.43 * 3000 = 55304

whats next take modulo?

Yes the question does say in the range (0-2pi)
Click to expand...

I would abstain from degrees. \(\displaystyle \tan^{-1}(\frac{1}{3}) \approx 0.3217505544\) radians.

3000*0.3217505544 = 965.2516632

Divide this by 2pi: 965.2516632/2pi \(\displaystyle \approx\) 153.6245735

The argument is then: \(\displaystyle 965.2516632 -(153)2\pi \approx\) 3.9243
 
kmalik001 said:
965/2pi is not 153 its 1516 radians or degrees
So where did the 153 come from?
Click to expand...
153 is the integer part of 965.2516632/(2pi). Since one revolution is (2pi) radians, 153 is the number of complete revolutions around the unit circle, that you have to subtract off to find the angle in the range [0,2pi).

[If you did it in degrees, then you will have to subtract the largest multiple of 360°, which will also be 153.
3000×18.43494882° = 55304.84747°, ÷360° = 153.6..., integer part is 153
subtract 153×360° --> 224.85° = 3.924 radians]

Do you understand why we multiplied the angle by 3000?
And why we had to keep so many decimal places?
 
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