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

1 Answer
Feb 2, 2016

Answer:

# 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 #