# How do you find the product of 7i + 4 and its conjugate?

Jun 28, 2016

It turns out that the product of a complex number and its conjugate is the square of its modulus:
Let a random complex number be given by $a + b i$
Therefore it's conjugate is $a - b i$
(This is why I've written it as $a + b i$ not $a i + b$ otherwise the conjugate would begin with a negative since it always attaches itself to the imaginary part
Their product: $\left(a + b i\right) \left(a - b i\right) = {a}^{2} + a b i - a b i - {b}^{2} {i}^{2}$
Since by definition ${i}^{2} = - 1$, this is equivalent to ${a}^{2} + {b}^{2}$
(The modulus of a complex number is given by $\sqrt{{a}^{2} + {b}^{2}}$ so the above is the square of this)

Therefore, we could go through and multiply our particular complex number $7 i + 4$ by its conjugate (which I'll omit here because it's just done in the same way as above, just with values not letters), but we may as well use the little formula we just generated:
${7}^{2} + {4}^{2} = 49 + 16 = 65$