Proving various properties of complex numbers

In summary: To prove something, you need to start with something you know to be true (axioms, theorems, etc.) and use that to show that your statement is logically equivalent to something that is already known to be true. In this case, you need to start with the definition of a complex conjugate and manipulate the expression until it looks like what you're trying to prove.
  • #1
Yosty22
185
4

Homework Statement



This problem is very easy, but I'm not sure how best to "prove" it. This part of the question just states:
Prove that (1/z)* = 1/(z*) where z* is the complex conjugate of z.

Homework Equations


The Attempt at a Solution



So the complex conjugate of z = x + iy is defined is z* = x - iy. That is, the complex conjugate of something is just making the complex part the opposite sign. So clearly, if you have 1/(x+iy) the complex conjugate would be to make the imaginary part negative, so 1/(x-iy), which clearly is equal to 1/z*.

The answer seems far too straight forward to warrant some formal "proof" but as the question is asking for a proof, I was wondering how you would go about doing it? For previous parts of this problem, such as proving that z* = re-i\theta, I could prove by graphing z and z* in the complex plane, then using Euler's identity of sines and cosines to rewrite those terms in terms of exponential functions and "prove" that way. However, I'm not sure how to "prove" this part of the problem as it seems way too straight forward.

Thanks for any advice.
 
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  • #2
Yosty22 said:

Homework Statement



This problem is very easy, but I'm not sure how best to "prove" it. This part of the question just states:
Prove that (1/z)* = 1/(z*) where z* is the complex conjugate of z.

Homework Equations


The Attempt at a Solution



So the complex conjugate of z = x + iy is defined is z* = x - iy. That is, the complex conjugate of something is just making the complex part the opposite sign. So clearly, if you have 1/(x+iy) the complex conjugate would be to make the imaginary part negative, so 1/(x-iy), which clearly is equal to 1/z*.

The answer seems far too straight forward to warrant some formal "proof" but as the question is asking for a proof, I was wondering how you would go about doing it? For previous parts of this problem, such as proving that z* = re-i\theta, I could prove by graphing z and z* in the complex plane, then using Euler's identity of sines and cosines to rewrite those terms in terms of exponential functions and "prove" that way. However, I'm not sure how to "prove" this part of the problem as it seems way too straight forward.

Thanks for any advice.

I don't think you have to graph anything. By using z and z* in rectangular form, you should be able to establish the stated identity algebraically.
 
  • #3
Yosty22 said:
Prove that (1/z)* = 1/(z*) ...So clearly, if you have 1/(x+iy) the complex conjugate would be to make the imaginary part negative, so 1/(x-iy), which clearly is equal to 1/z*.
You are assuming what you are trying to prove here.
I would recommend algebraically showing that ## \frac {1}{x-iy}=\left(\frac {1}{x+iy}\right)^* ## by separating the latter into its real and imaginary parts.
 
  • #4
To elaborate on what RUber said, all you can really assume as far as conjugation goes is ##(u+iv)^* = u-iv##. ##1/(x+iy)## isn't of the form ##u+iv##, so you can't simply replace ##iy## with ##-iy## and claim it's equal to the conjugate.
 

Related to Proving various properties of complex numbers

1. What are the basic properties of complex numbers?

Complex numbers have two components: a real part and an imaginary part. They are typically written in the form a + bi, where a is the real part and bi is the imaginary part. They follow the rules of algebra, such as addition, subtraction, multiplication, and division.

2. How do you prove that two complex numbers are equal?

To prove that two complex numbers are equal, you need to show that their real parts and imaginary parts are equal. This can be done by setting the two complex numbers equal to each other and then solving for the real and imaginary parts separately.

3. What is the conjugate of a complex number?

The conjugate of a complex number is a number with the same real part but an opposite sign on the imaginary part. This can be represented as a - bi for a complex number a + bi. The conjugate is important in complex number operations, such as division and finding the modulus.

4. How do you find the modulus of a complex number?

The modulus of a complex number is the distance of the number from the origin on the complex plane. It is also known as the absolute value or magnitude of a complex number. To find the modulus, use the Pythagorean theorem: |a + bi| = √(a² + b²). This will give you a positive real number as the result.

5. What is the polar form of a complex number?

The polar form of a complex number is another way of representing a complex number, using its modulus (r) and argument (θ). It is written as r(cosθ + isinθ), where r is the modulus and θ is the argument. This form is useful for some complex number operations, such as raising a complex number to a power.

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