Question

Why does multiplying a + bi by the complex conjugate a -bi eliminate I from the expression

Answer 1

EXPLANATION

The reason is that

$i^2=-1$

If we multiply $a+bi$ by its conjugate $a-bi$

We obtain;

$(a+bi)(a-bi)$

This is difference of two squares so we obtain;

$(a+bi)(a-bi)=a^2-(bi)^2$

This further gives us,

$(a+bi)(a-bi)=a^2-b^2i^2$

Since the $i^2=-1$, we substite to get;

$(a+bi)(a-bi)=a^2-b^2(-1)$

The $i$ is now eliminated and we get,

$(a+bi)(a-bi)=a^2+b^2$