## Complex Numbers Division

Consider two complex numbers z1, z2

z1 = a1 + ib1

z2 = a2 + ib2

Multiplying the numerator and denominator by the complex conjugate of z2, that is

= a2 - ib2

we obtain

#### Example 1

Divide 4 - 6i by 2 - 3i

Let z1 = 4 - 6i,            a1 = 4,      b1 = -6

z2 = 2 - 3i,             a2 = 2,      b2 = -3

#### Example 2

Divide 7i - 5 by 10 + 2i

Let z1 = 7i-5,           a1 = -5,                   b1 = 7

z2 = 10+ 2i,         a2 = 10,                  b2 = 2

Comparing with

#### Example 3

Divide -12-5i by 1-4i

Let Z1 =-12-5i           a1 = -12                  b1 = -5

Z2 = 1+ 4i            a2 = 1                     b2 = -4

#### Example 4

Divide -1 -i by -2-4i

Let z1 = -1 -i,          a1 = -1                  b1 = -1

z2 = -2- 4i,         a2 = -2,                 b2 = -4

#### Try these questions

Divide the following complex numbers

1. Divide 8 + 3i by 5 + 6i
Answer: let z1 = 8+3i,          a1 = 8,           b1 = 3
z2 = 5+6i,          a2 = 5,           b2 = 6
Comparing with

2. Divide -3 + 4i by 10-2i
Answer: Let z1 = -3+4i,         a1=-3,           b1=4
z2=10-2i,         a2=10,          b2=-2

3. Divide -1 -i by (-4 -3i )
Answer:  Let z1 =-1-i         a1=-1,          b1=-1
z2 =-4-3i            a2= -4,         b2=-3

4. Divide 12i by 3 - 4i
Answer:  Let z1 = 12i = 0+12i,         a1= 0,          b1= 12
z2 =-3-4i,                          a2=3,           b2=-4

5. Divide -5 by 7 + 5i
Answer:  Let  z1 = -5 + 0i         a1= -5,          b1= 0
z2 = 5 + 6i                a2= 5,            b2= 6

6. Divide 9 + 11i by 2
Answer:  Let z1 =9+11i         a1=9           b1=11
z2 = 2=2+0i           a2=2           b2=0

On reducing to the lowest terms for each fraction

7. Divide (-3 -18i ) by 12i
Answer:    Let z1 = -3-18i                   a1=-3          b1= -18
z2 = 12i =0+12i                 a2=0           b2=12

8. Divide 4 - 6i by 4 + 6i
Answer:  Let z1 = 4 - 6i         a1= 4          b1= -0
z2 = 4 + 6i            a2= 4          b2= 6

On reducing each fraction to the lowest term

9. Divide -5 -12i by -5 +12i
Answer:    Let z1 =-5- 12i         a1=-5          b1=-12
z2 =-5 + 12i            a2=-5            b2=12

On reducing to the lowest terms for each fraction