Complex Numbers: Solving Equations with z and zeta?

In summary, the conversation discusses finding all solutions for the equation (z^2 + 1)^4 = 1, where z is a complex number. The speaker provides tips on approaching the problem by using the equation 1=e^{ik2\pi} and taking the fourth root. They also mention using previous experience solving equations of the type z^n=a where a is complex. The conversation also suggests setting \zeta = z^2+1 and finding all 4 solutions for \zeta^4 = 1, then finding 2 solutions for each \zeta to get a total of 8 solutions for z.
  • #1
lektor
56
0
Hi, I have no clue how to approach this question, was in my last years final exams.

(z^2 + 1)^4 = 1

Find all solutions, where z is a complex number.

Tips please?
 
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  • #2
First off, write [itex]1=e^{ik2\pi}[/itex]
where k is an integer.
Now, take the fourth root of the equation:
[tex]z^{2}+1=e^{i\pi\frac{k}{2}}[/tex]
For how many choices of k will the right-hand side represent DISTINCT complex numbers?
 
  • #3
Supposing you have some experience solving equations of the type [itex]z^n=a[/itex] where a is complex...

Set [itex]\zeta = z^2+1[/itex]. Then find all 4 solutions of [itex]\zeta^4 = 1[/itex]. Then go back to [itex]\zeta -1 = z^2[/itex] and for all four [itex]\zeta[/itex] found, find the 2 solutions of z associated, for a total of 8.
 
Last edited:

1. What are complex numbers?

Complex numbers are numbers that contain both a real part and an imaginary part. They are represented in the form a + bi, where a is the real part and bi is the imaginary part.

2. How do you add or subtract complex numbers?

To add or subtract complex numbers, you simply add or subtract the real and imaginary parts separately. For example, (3 + 2i) + (1 + 5i) = (3 + 1) + (2i + 5i) = 4 + 7i.

3. What is the difference between a real number and a complex number?

A real number is a number that can be represented on a number line, while a complex number contains both a real and imaginary part and cannot be represented on a number line.

4. How do you multiply complex numbers?

To multiply complex numbers, you use the distributive property and the fact that i^2 = -1. For example, (3 + 2i)(1 + 5i) = 3 + 15i + 2i + 10i^2 = 3 + 17i - 10 = -7 + 17i.

5. What are some real-life applications of complex numbers?

Complex numbers are used in fields such as engineering, physics, and computer science. They are used to represent quantities that have both magnitude and direction, such as electrical currents and forces. They are also used in signal processing and control systems.

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