Question

How do you divide `(5 - i) / (5 + i)`?

Answer 1

`12/13 - 5/13 i`

Explanation:

To divide 2 complex numbers , multiply the numerator and

denominator by the' complex conjugate' of the denominator.

If a + bi is a complex number then`color(red)(" a - bi is the conjugate")`

This ensures that the denominator is real , as shown below.

multiplying a complex number and it's conjugate.

`(a + bi )(a - bi ) = a^2 + abi - abi - bi^2 = a^2 + b^2`

which is real . [ remember `i^2 =( sqrt-1)^2 = -1]`

hence question becomes: `((5-i)(5-i))/((5+i)(5-i))`

`= (25 - 10i + i^2)/(25 - i^2) =( 24 - 10i)/26 = 24/26 - 10/26 i`

which simplifies to : `12/13 - 5/13 i`