Question

How do you divide `i/(2-3i)`?

Answer 1

`-3/13+2/13 i`

Explanation:

The aim here is to obtain a real value on the denominator of the fraction.
This is achieved by multiplying the numerator and denominator by the `color(blue)"complex conjugate"` of the denominator.

Given a complex number #z=x ± yi then the complex conjugate is.

`bar(z)=x∓yi`

Note that the real part remains unchanged while the `color(red)"sign"` of the imaginary part is reversed.

The conjugate of `2-3i" is " 2+3i`

and `(2-3i)(2+3i)=4-9i^2=4+9=13" a real number"`

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

We must multiply numerator/denominator by 2+3i

`rArri/(2-3i)xx(2+3i)/(2+3i)=(i(2+3i))/((2-3i)(2+3i))`

`=(2i+3i^2)/13=(-3+2i)/13=-3/13+2/13i`