Question about complex numbers

In summary, the conversation is about calculating a complex number raised to a large power without using De Moivre's theorem. The suggestion is to use the Binomial theorem, but the speaker is struggling with removing the real part of the imaginary number. Another person suggests using De Moivre's theorem to easily find the twelfth power and avoid additional work. The question is then raised whether the avoidance of De Moivre's theorem is a personal choice or a requirement for the solution.
  • #1
Chuckster
20
0
Hello guys!

I have a question related to complex numbers.

How would i calculate, for example

[tex](\frac{\sqrt{3}+i}{2})^{2010}[/tex] without using the De Moivre's theorem?
 
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  • #2
Binomial expansion?
 
  • #3
Curl up in a ball and die a little inside.
 
  • #4
Use wolfram|alpha to do it for you.

Really, it would be really cumbersome to do it without De Moivre's theorem.
 
  • #5
SprucerMoose said:
Binomial expansion?

well, that was my idea originally.
using the facts that
[tex]
i^{1}=i, i^{2}=-1, i^{3}=-i, i^{4}=1
[/tex]

i tried to find a way to brake the expression given in the first post into something which could destroy the [te]i[/tex], just like i would do with, ie
[tex](1+i)^{2010}=(2i)^{1005}=2^{1005}i[/tex], but I'm having problems shaking off the real part of the imaginary number.
 
  • #6
Using De Moivre's theorem we can immediately observe that if we raise your number to the 12th power you get 1. So prove this with pencil and paper by finding the twelfth power by hand, seeing that it's one, and then you're basically done. You can do this in half the work by just raising it to the sixth power and noticing that you get negative the starting number.

The question then is why are you trying to avoid De Moivre's theorem: because you want avoid using it, or because somebody else is forbidding you from using it in your final solution
 
  • #7
Office_Shredder said:
You can do this in half the work by just raising it to the sixth power and noticing that you get negative the starting number.
You can even raise it to the 3rd power using the Binomial theorem and see what you get.
 

1. What are complex numbers?

Complex numbers are numbers that combine a real and imaginary part. The real part is a normal number that we use in everyday life, while the imaginary part is a multiple of the square root of -1, denoted by the letter "i". The general form of a complex number is a + bi, where a and b are real numbers and i is the imaginary unit.

2. How are complex numbers used in science?

Complex numbers are used in various fields of science, including physics, engineering, and mathematics. They are particularly useful in solving equations that involve oscillating or rotating phenomena, such as AC circuits, electromagnetic waves, and quantum mechanics.

3. What is the difference between real and complex numbers?

The main difference between real and complex numbers is that real numbers only have a real part, while complex numbers have both a real and imaginary part. Real numbers can be represented on a number line, while complex numbers require a two-dimensional plane called the complex plane. Additionally, real numbers can be positive or negative, while complex numbers can have different magnitudes and directions.

4. How do you perform operations on complex numbers?

To add or subtract complex numbers, you simply add or subtract the real parts and then add or subtract the imaginary parts. To multiply complex numbers, you use the distributive property and combine like terms. To divide complex numbers, you use the complex conjugate of the denominator to rationalize the fraction. You can also perform other operations such as finding the absolute value, conjugate, and powers of complex numbers.

5. Can complex numbers have real solutions?

Yes, complex numbers can have real solutions. When solving equations with complex numbers, the solutions can be real, imaginary, or complex. For example, the quadratic equation x^2 + 2x + 5 = 0 has two complex solutions: -1 + 2i and -1 - 2i. However, the equation x^2 + 4 = 0 has two purely imaginary solutions: 2i and -2i. Real solutions occur when the imaginary part of a complex number is equal to 0.

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