Question

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

Answer 1

`53`

Explanation:

For any complex number `z=x+iy` in rectangular form, its complex conjugate is `barz=x-iy`.
It can be shown that the product of a complex number with its complex conjugate is a real number given by
`zbarz=x^2+y^2`.
Proof:
`(x+iy)*(x-iy)=x^2-i^2y^2+ixy-xyi`

`=x^2+y^2`.

So in this particular case,

`(7-2i)*(7+2i)=7^2+2^2`

`=53+0i`