# Divide 8 + 0i and 3 + 4i

a = bi = <-- Enter a and bi piece
c = di = <-- Enter c and di piece (not needed for square root or absolute value or conjugate)

Perform the complex number division below:
 8 3 + 4i

Complex number division involves multiplying numerator and denominator by the conjugate of the denominator.
If the denominator is c + di, the conjugate is c - di. Multiplying top and bottom by the conjugate (3 - 4i), we get:
 (8)(3 - 4i) (3 + 4i)(3 - 4i)

Expand the denominator (3 + 4i)(3 - 4i)
The formula for this using the FOIL method is: (a * c) + (b * c) + (a * d) + (b * d) where:
a = 3, b = 4, c = 3, and d = -4

Now plug these values into our formula and evaluate:
(3 + 4i)(3 - 4i) = (3 * 3) + (4i * 3) + (3 * -4i) + (4i * -4i)
(3 + 4i)(3 - 4i) = 9 + 12i - 12i - 16i2

Group the like terms that contain i:
(3 + 4i)(3 - 4i) = 9 + (12 - 12)i - 16i2
(3 + 4i)(3 - 4i) = 9 - 16i2

Simplify our last term:
i2 = √-1 * √-1 = -1, so our last term becomes:
(3 + 4i)(3 - 4i) = 9 - 16* (-1)
(3 + 4i)(3 - 4i) = 9 + 16

Now group the 2 constants and finalize our answer
(3 + 4i)(3 - 4i) = (9 + 16)
(3 + 4i)(3 - 4i) = 25

Expand the numerator (8)(3 - 4i)
The formula for this using the FOIL method is: (a * c) + (b * c) + (a * d) + (b * d) where:
a = 8, b = 0, c = 3, and d = -4

Now plug these values into our formula and evaluate:
(8)(3 - 4i) = (8 * 3) + (8 * -4i)
(8)(3 - 4i) = 24 - 32i

(8)(3 - 4i) = 24 - 32i

After expanding and simplifying numerator and denominator, we are left with:
 8 3 + 4i
 =
 24 - 32i 25

This fraction cannot be reduced down anymore, so we have our answer