Question

How do you simplify `(4 + 4i ) div(5 + 4i )`?

Answer 1

`36/41 +4/41 i`

Explanation:

To simplify the fraction , require to make the denominator real.
This is achieved by multiplying the complex number on the denominator by it's `color(blue)" complex conjugate "`

If (a + bi ) is a complex number then it's conjugate is (a - bi )

Note that the 'real part' remains unchanged , while the 'imaginary' part becomes negative.

and (a+bi)(a-bi) `= a^2-bi+bi-bi^2 = a^2+b^2" a real number "`

using `[ i^2 = (sqrt(-1))^2 = -1 ]`

now multiply numerator and denominator by (5 - 4i )

`rArr( (4+4i)(5-4i))/((5+4i)(5-4i)) = (20-16i+20i-16i^2)/(25-16i^2)`

which simplifies to

`(36+4i)/41 = 36/41 + 4/41 i`