The conjugate of the product of the two complex numbers is equal to the product of the conjugates of the numbers.

Question

Here is the rest of the problem: The conjugate of the product of the two complex numbers is equal to the product of the conjugates of the numbers. Using a+bi and c+di to represent two complex numbers. Verify this.

 

I do not understand any of this. Please help me to get the answer. It is due tomorrow morning!

 

Thanks!

Answer 1

Conjugates have the plus/minus signs.  Recall the solution when using the quadratic formula to solve a quadratic equation.

 

For this one, we use the fact that    i²=-1

 

 

First, work the first part of the proof.

 

(a + bi)(c + di) = ac + adi + bci + bdi²

 

                       = (ac - bd) + (ad + bc)i

 

This is the product of the numbers.  Now take the conjugate of this product.

 

(ac - bd) - (ad + bc)i

 

 

Now, we work the second part of the proof. Take the product of the conjugates.

 

(a - bi)(c - di) = ac - adi - bci + bdi²

 

                     = (ac - bd) - (ad + bc)i

 

 

The two bolded results from both parts are the same.