Question

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

Answer 1

`58/97+15/97 i`

Explanation:

To divide a complex number by another , we require to `color(blue)"rationalise the denominator"` ie. make it a real value.

This is achieved by mutipling the numerator and denominator by the `color(blue)"complex conjugate"` of the denominator.

Given a complex number z = x ± iy then the conjugate is x ∓ iy

Note that the values of x and y remain unchanged but the 'sign' changes.

`rArr(x+iy)(x-iy)=x^2+y^2 " a real number"`

We are also making use of the fact `[i^2=(sqrt(-1))^2=-1]`

`rArr(6-i)/(9-4i)=(6-i)/(9-4i)xx(9+4i)/(9+4i)=((6-i)(9+4i))/((9-4i)(9+4i))`

now expanding numerator/denominator

`rArr(54+15i-4i^2)/(81-16i^2)=(58+15i)/(81+16)=58/97+15/97 i`