Question

How do you divide `(4-9i)/(-6i)`?

Answer 1

`3/2+2/3i`

Explanation:

We require to multiply the numerator and denominator of the fraction by the `color(blue)"complex conjugate"` of the denominator. This will eliminate the i from the denominator, leaving a real number.

If z = a ± bi then the `color(blue)"complex conjugate"` is

`color(red)(|bar(ul(color(white)(a/a)color(black)(barz=a∓bi)color(white)(a/a)|)))`
Note that the real part remains unchanged while the 'sign' of the imaginary part is reversed.

The complex number on the denominator is 0 - 6i , hence the conjugate is 0 + 6i = 6i

Multiply numerator and denominator by 6i

`(4-9i)/(-6i)xx(6i)/(6i)=(6i(4-9i))/(-36i^2)=(24i-54i^2)/(-36i^2)`

`color(orange)"Reminder " color(red)(|bar(ul(color(white)(a/a)color(black)(i^2=(sqrt(-1))^2=-1)color(white)(a/a)|)))`

`rArr(4-9i)/(-6i)=(24i+54)/36=54/36+24/36i=3/2+2/3i`