Can someone explain the property of complex conjugates in this equation?

[daster]

Could someone please show me how:

$$z_1\bar{z_2} + \bar{z_1}z_2 = 2\:Re(z_1\bar{z_2})$$

where $\bar{z}$ is the conjugate of $z$.

 

[daster]

Nevermind. I got it.

 

[tacman]

The property of complex conjugates in this equation relates to the fact that in complex numbers, the conjugate of a number is found by changing the sign of the imaginary part. In other words, for any complex number z = a+bi, its conjugate is given by \bar{z} = a-bi. This property is important because it allows us to easily work with complex numbers and perform operations such as multiplication, division, and taking the modulus (absolute value) of a complex number.

In the given equation, we have z_1\bar{z_2} + \bar{z_1}z_2 = 2\:Re(z_1\bar{z_2}). This can be rewritten as z_1\bar{z_2} + z_2\bar{z_1} = 2\:Re(z_1\bar{z_2}), using the fact that multiplication of complex numbers is commutative. Now, we can use the property of complex conjugates to simplify this expression. We know that \bar{z_1} is the conjugate of z_1, and \bar{z_2} is the conjugate of z_2. So, we can rewrite the expression as z_1\bar{z_2} + z_2\bar{z_1} = z_1\bar{z_2} + \bar{z_1}\bar{\bar{z_2}}.

Since the conjugate of a conjugate is the original complex number, we can simplify further to get z_1\bar{z_2} + \bar{z_1}z_2 = z_1\bar{z_2} + \bar{z_1}z_2. This shows that the left and right sides of the equation are equal, and the property of complex conjugates allows us to simplify the expression in a way that makes it easier to work with.

The final result of 2\:Re(z_1\bar{z_2}) shows that when we multiply a complex number with its conjugate, the imaginary parts cancel out and we are left with a real number. This is why the real part of z_1\bar{z_2} is equal to the real part of its conjugate, and the expression simplifies to 2 times the real part.

Overall, the property