# How do you find the product of 7 - 2i and its conjugate?

Dec 24, 2015

$53$

#### Explanation:

For any complex number $z = x + i y$ in rectangular form, its complex conjugate is $\overline{z} = x - i y$.
It can be shown that the product of a complex number with its complex conjugate is a real number given by
$z \overline{z} = {x}^{2} + {y}^{2}$.
Proof:
$\left(x + i y\right) \cdot \left(x - i y\right) = {x}^{2} - {i}^{2} {y}^{2} + i x y - x y i$

$= {x}^{2} + {y}^{2}$.

So in this particular case,

$\left(7 - 2 i\right) \cdot \left(7 + 2 i\right) = {7}^{2} + {2}^{2}$

$= 53 + 0 i$