Let's assume that the number we are using is [tex]a+bi[/tex], where [tex]a[/tex] and [tex]b[/tex] are real numbers. The conjugate of this would be [tex]a-bi[/tex]. Multiplying the two together would get [tex](a+bi)*(a-bi)= a^{2} + abi - abi + b^{2}=a^{2}+b^{2} [/tex]. Since [tex]a[/tex] and [tex]b[/tex] are both real numbers, the multiplication of the conjugate must also be real.
