Complex Numbers: Why Use Complex Conjugate?

[EngWiPy]

Complex Numbers

Hello,

Why when dealing with complex numbers, as with multiplication, we use the complex conjugate operator?

Regards

 

[Studiot]

So we can do complex 'division' and fractions.

Compare what happens in the following

$$\frac{{a + ib}}{{c + id}}*\frac{{c - id}}{{c - id}}$$

with

$$\frac{{a + ib}}{{c + id}}*\frac{{c + id}}{{c + id}}$$

 

[EngWiPy]

So we can do complex 'division' and fractions.

Compare what happens in the following

$$\frac{{a + ib}}{{c + id}}*\frac{{c - id}}{{c - id}}$$

with

$$\frac{{a + ib}}{{c + id}}*\frac{{c + id}}{{c + id}}$$

I didn't get it. I mean, in telecommunication systems, when we deal with baseband signals, we deal with complex numbers, and all the time we use the complex conjugate operator, but I don't understand why and what it is mean physically.

 

[Studiot]

I did expect you to work my examples out.

Which one contained the conjugate and which one leads to a single complex number result?

If you apply a formula in real analysis say 27*3 you want the simple answer 81, not something more difficult than you started with such as

$${\left( {\sqrt 9 } \right)^2}*{\left( {\sqrt 9 } \right)^2}$$

The same is true of complex numbers.

What does simple multiplication by a conjugate yield by the way ( a real number)?

 

[HallsofIvy]

The product of a complex number and its conjugate has the nice property that is a real number- and for any z other than 0 \(\displaystyle z*\overline{z}\) is a positive real number.