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

Question

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

 

I do not understand any of this. Please help me and show me the equation and how to solve it. It is due by tomorrow morning.

Answer 1

The conjugate of a number is one that changes the sign of the imaginary portion. More formally, the complex conjugate of a complex number is a number with an equal real part and imaginary part equal in magnitude, but opposite in sign.

 

For example

 

The conjugate of 

 

3+4i

 

will be 

 

3-4i

 

In this case, you can verify with a few steps.

 

(a+bi) + (c+di) = a+c+i(b+d)

 

The conjugate of

 

 

a + c + i(b + d) is a + c - i(b + d)

 

Now let's take the conjugate of the numbers and add them

 

a - bi + c - di = a + c - i(b + d)

 

As you can see, the conjugate of the sum is equal to the sum of the conjugates.

 

 

 

 

Answer 2

Conjugates means that we have plus/minus sign.  Recall the solution when using the quadratic formula.

 

 

First, we take the conjugate of the sum of two complex numbers. 

 

(a + bi) + (a + di) = 2a + (b + d)i

 

This is the sum of the complex numbers.  Now take the conjugate of this sum.

 

The conjugate is          2a - (b + d)i

 

 

 

Now we deal with the second part.  Take the sum of the conjugates of the numbers.

 

(a - bi) + (a - di) =

 

2a - (b + d)i

 

 

The bolded results from both parts are the same.