Multiplying a complex number by its conjugate

In the same way that multiplying a bracket involving a surd by the conjugate of the bracket will produce a rational number, the same is try for multiplying a complex number by the complex conjugate.

Multiplying a complex number by its conjugate will always result in a real number.

Proof

• Let z=x+yi. x and y are real.
• z^*=x-yi
• (x+yi)(x-yi)=x^2+xyi-xyi-y^2i^2
• (x+yi)(x-yi)=x^2-y^2i^2
• (x+yi)(x-yi)=x^2-y^2(-1)
• (x+yi)(x-yi)=x^2+y^2
• x and y are real, so x^2+y^2 is real.

Example: find z such that 3z+2z^*=5+2i

• Let z=x+yi
• z^*=x-yi
• 3(x+yi)+2(x-yi)=5+2i
• 3x+3yi+2x-2yi=5+2i
• 5x+yi=5+2i
• 5x=5, x=1
• 1y=2, y=2
• z=x+yi
• z=1+2i