# Elementary algebra of complex variables problem

• jdinatale
In summary, the conversation discusses a mathematical problem involving deducing the value of Re(z) = 0. Various methods are suggested, including multiplying by (1-w*) and using the conjugate method. Ultimately, it is determined that z is pure imaginary, therefore proving that Re(z) = 0. There is some confusion and difficulty in understanding the calculations, but it is eventually resolved.

#### jdinatale

I'm having difficulty deducing that Re z = 0.

Try multiplying the top and bottom by (1-w*).

vela said:
Try multiplying the top and bottom by (1-w*).

Thank you, that seems to work. Can you confirm that this is correct?

http://i45.tinypic.com/2wc2xl4.png

Last edited by a moderator:
You made a minor error in calculating the denominator when you canceled the ones.

Sorry for butting in, but although I can see where jdinatale went wrong in his last post, I still can't see how Re(z) can be proved to be zero.

Of course it's correct, but I can't quite see it.

oay said:
Sorry for butting in, but although I can see where jdinatale went wrong in his last post, I still can't see how Re(z) can be proved to be zero.

Of course it's correct, but I can't quite see it.

The calculation shows z is pure imaginary and that's still true after you make the correction. Doesn't that show Re(z)=0?

Dick said:
The calculation shows z is pure imaginary
How does it?

Sorry, I'm probably being very thick here and will live to regret it.

How does

$$z = (1 + \omega) / (1 - \omega)$$
where $\omega = e^{ik\pi/50}$

imply that $z$ is purely imaginary?

Like I say, I know it's correct to say so, but I'm lost on how to prove it - even after trying using the conjugate method as offered above.

It'll probably be a face-palm moment when I find out...

oay said:
How does it?

Sorry, I'm probably being very thick here and will live to regret it.

How does

$$z = (1 + \omega) / (1 - \omega)$$
where $\omega = e^{ik\pi/50}$

imply that $z$ is purely imaginary?

Like I say, I know it's correct to say so, but I'm lost on how to prove it - even after trying using the conjugate method as offered above.

It'll probably be a face-palm moment when I find out...

Look at what happens when jdinatale multiplies by (1+w*). The results has w-w* (which is pure imaginary) in the numerator and w+w* (which is pure real) in the denominator. What kind of a number is imaginary/real?

Dick said:
Look at what happens when jdinatale multiplies by (1+w*). The results has w-w* (which is pure imaginary) in the numerator and w+w* (which is pure real) in the denominator. What kind of a number is imaginary/real?
Ha-ha! I was right!

A complete face-palm...

Thanks for that. I blame my incompetence on my getting on a bit.

## 1. What is elementary algebra of complex variables?

Elementary algebra of complex variables is the study of mathematical operations and equations involving complex numbers. It includes topics such as addition, subtraction, multiplication, division, and solving equations with complex variables.

## 2. What are complex numbers?

Complex numbers are numbers that have both a real and imaginary component. They are written in the form a + bi, where a is the real part and bi is the imaginary part, with i representing the imaginary unit (√-1).

## 3. How is elementary algebra of complex variables used in science?

Elementary algebra of complex variables is used in various fields of science, such as physics, engineering, and mathematics. It helps in solving problems involving electrical circuits, fluid dynamics, and quantum mechanics, among others.

## 4. What are the fundamental operations in elementary algebra of complex variables?

The fundamental operations in elementary algebra of complex variables are addition, subtraction, multiplication, and division. These operations follow similar rules as in real numbers, with the addition of rules for dealing with imaginary components.

## 5. How do you solve equations with complex variables?

To solve equations with complex variables, you can use the same techniques as solving equations with real numbers, such as isolating the variable and using inverse operations. You can also use graphical methods or specialized techniques such as the quadratic formula for complex numbers.