What are the Powers of Complex Numbers?

In summary, the conversation revolves around a complex number problem and finding the magnitude of the ratio of two complex numbers. Several methods are discussed, including using polar form and simplifying, the fact that the numerator and denominator are complex conjugates, and the general property that |zn| = |z|n and |a/b| = |a|/|b|. It is eventually revealed that the solution is quite simple and only requires knowledge of the modulus, not the argument, of the complex numbers involved.
  • #1
cepheid
Staff Emeritus
Science Advisor
Gold Member
5,199
38
Hello,

I'm having trouble with this problem:

[tex] \left| \frac{(\pi + i)^{100}}{(\pi - i)^{100}} \right| = \ \ ? [/tex]

My first thought was, "put it in polar form and simpify," but that is not helping.

For the numerator pi + i :
[tex] r = \sqrt{\pi^2 + 1} [/tex]

[tex] \theta = \arctan{ \frac{1}{\pi} } [/tex] = ?

I don't see how this will help, it's not an easy one to put in polar form

I can also see that the numerator and denominator are complex conjugates, so maybe that is the starting point. But I can't see how to proceed
 
Physics news on Phys.org
  • #2
Hint : they're only asking for the magnitude of the ratio of the two. Since, the numerator and denominator are complex conjugates, what can you say about their magnitudes ? Don't they cancel out ? What will be left when the magnitudes cancel out ?

This is a trick question, with a trivial solution.
 
  • #3
Sorry to break into your thread, but I suddenly have an irrepressible need to know if this is allowed in the world of complex numbers:

[tex] \left | \frac{(pi + i)^{100}}{(pi - i)^{100}}\right | { }^?_=
\left ( \frac{|(pi + i)|}{|(pi - i)|} \right )^{100} [/tex]

Is it?
 
  • #4
Yes, that works out. In fact, the numerator and demoninator can be any old complex numbers, not necessarily conjugates.
 
  • #5
In other words, you don't NEED to know the argument. All you are asked about is the modulus so that's all you need to know!

In general, |zn|= |z|n and |a/b|= |a|/|b|.
 
  • #6
Curious3141 said:
This is a trick question, with a trivial solution.

I can't believe I didn't see that! Even though I realized the numerator and denominator were conjugates.

Thanks for your help, everyone.

Halls:

Yeah, we also had to prove |z1z2| = |z1||z2| in this homework assignment
so I can see where those come from.

Thanks again.
 

1. What is a complex number?

A complex number is a number that can be written in the form a + bi, where a and b are real numbers and i is the imaginary unit, defined as the square root of -1.

2. What are the properties of complex numbers?

Complex numbers have several properties, including commutativity, associativity, distributivity, and the existence of additive and multiplicative identities. They also have a conjugate property, where the conjugate of a complex number a + bi is a - bi.

3. How do you add and subtract complex numbers?

To add or subtract complex numbers, you simply add or subtract the real and imaginary parts separately. For example, to add 3 + 4i and 5 + 2i, you would add 3 and 5 to get 8, and add 4 and 2 to get 6, resulting in 8 + 6i.

4. How do you multiply and divide complex numbers?

To multiply complex numbers, you use the FOIL method (First, Outer, Inner, Last). For example, to multiply 3 + 4i and 5 + 2i, you would multiply 3 and 5 to get 15, multiply 3 and 2i to get 6i, multiply 4i and 5 to get 20i, and multiply 4i and 2i to get -8. Then you combine like terms to get the final answer, 15 + 26i. To divide complex numbers, you use the conjugate property and the distributive property to simplify the expression.

5. How are complex numbers used in real life?

Complex numbers have many applications in real life, such as in electrical engineering, signal processing, and quantum mechanics. They are also used in various mathematical models and equations to represent and solve problems involving real-world phenomena.

Similar threads

Replies
7
Views
1K
  • Precalculus Mathematics Homework Help
Replies
14
Views
156
  • Introductory Physics Homework Help
Replies
10
Views
1K
  • Precalculus Mathematics Homework Help
Replies
12
Views
884
  • Introductory Physics Homework Help
Replies
8
Views
586
  • Introductory Physics Homework Help
Replies
1
Views
1K
  • Introductory Physics Homework Help
Replies
11
Views
623
  • Introductory Physics Homework Help
Replies
8
Views
498
  • Linear and Abstract Algebra
Replies
1
Views
708
  • Calculus and Beyond Homework Help
Replies
28
Views
1K
Back
Top