Complex Number Question - Euler's Formula?

In summary, the conversation discussed how to express (-2 + 2i)^10 in the form re^(iθ) using Euler's Formula. The correct solution is (2√2)^10 e^(i(15pi/2)), where r = (2√2)^10 and theta = 3pi/2. The conversation also mentioned simplifying the angle and the standard form for writing angles in polar form.
  • #1
KAISER91
27
0
Complex Number Question - Euler's Formula?

Homework Statement



Express (-2 + 2i)^10 in the form re^(iθ)

Homework Equations



Euler's Formula, I think?

The Answer given is: (2√2)^10 e^[i(15pi/2)]

The Attempt at a Solution

I don't know what I'm doing wrong here;

r = √[(-2)^2 + 2^2] = √8 = 2√2theta = pi - arctan (y/x)

= pi - arctan (2/-2)

= pi - 45

= pi - (pi/4)

= 3pi / 4Help will be appreciated. Thanks
 
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  • #2


Well for one thing you didn't raise your r and theta to the power of 10.

(re^(ix))^10= r^10(e^(i(10x)). Your next step should be to reduce 10x to a smaller angle since angles repeat after 2pi.

Hope that helps.

If you need more explainatiom post again. :-):-)
 
  • #3


╔(σ_σ)╝ said:
Well for one thing you didn't raise your r and theta to the power of 10.

(re^(ix))^10= r^10(e^(i(10x)). Your next step should be to reduce 10x to a smaller angle since angles repeat after 2pi.

Hope that helps.

If you need more explainatiom post again. :-):-)
Thanks for the reply mate.

So I have to raise r and theta to ^10.

That makes r = (2√2)^10

However how did theta become 15pi/2 ?
 
  • #4


I think I got it.

Just bring down 10 as the power??

e^[10 x i(3pi/4)]
 
  • #5


Yes and then simplify the angle. For example 7pi is the same as pi + 2(3)pi.

The answer you gave is correct but was not put in the "best" way.

So 10/4(3pi) = 15pi/2 = 7pi + pi/2 = 6pi + 3pi/2

Angles in polar form are usually written as i( theta + 2npi) :-)

We usually do not put the 2npi part since it is "implied" thus, your answer should be

e^(i 3/2(pi))
 
  • #6


╔(σ_σ)╝ said:
Yes and then simplify the angle. For example 7pi is the same as pi + 2(3)pi.

The answer you gave is correct but was not put in the "best" way.

So 10/4(3pi) = 15pi/2 = 7pi + pi/2 = 6pi + 3pi/2

Angles in polar form are usually written as i( theta + 2npi) :-)

We usually do not put the 2npi part since it is "implied" thus, your answer should be

e^(i 3/2(pi))
OH!

I understand now. Thanks heeps for the help.
 

Related to Complex Number Question - Euler's Formula?

What is Euler's formula?

Euler's formula is a mathematical equation that relates complex numbers to trigonometric functions. It is written as e^(ix) = cos(x) + i*sin(x), where e is the base of the natural logarithm, i is the imaginary unit, and x is the angle in radians.

What is a complex number?

A complex number is a number that contains both a real part and an imaginary part. It is written in the form a + bi, where a is the real part and bi is the imaginary part, and i is the imaginary unit (equal to the square root of -1).

How is Euler's formula derived?

Euler's formula can be derived using Taylor series for the exponential function and trigonometric functions, along with the definition of imaginary unit. It is a result of the deep connection between complex numbers and trigonometry.

What is the significance of Euler's formula?

Euler's formula is significant in mathematics because it provides a powerful tool for solving problems involving complex numbers and trigonometric functions. It is also important in physics and engineering for its applications in wave and signal analysis.

How is Euler's formula used in real life?

Euler's formula is used in a variety of fields such as electrical engineering, signal processing, and quantum mechanics. It is also used in graphics and animation to create complex curves and shapes. Additionally, it has applications in finance and economics for modeling exponential growth and decay.

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