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Online Algebra Tutoring: Complex Numbers (Division)

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Learn Multiplication &  Division of Complex  Numbers  from Certified  Online Algebra Tutor

3. Multiplication

We use FOIL Method (which we use to multiply two binomials) to multiply two complex numbers.

F = Firsts
O = Outers
I = Inners
L = Lasts

i = √-1, i2 = -1, i3 = – i, i4 = 1

p+qi and r+ti are two complex numbers.

(p+qi ) ×(r+ti) = pr + pti + qir + qti2

pr + (pt+qr) i + qti2

Example: (6 + 3i) × (3 + 4i)

= 18 + 24i + 9i + 12i2
= 18 + (24 + 9) i + 12(-1) ———— (because i2 = -1)
= 18 + 33i – 12
= 6+ 33


4. Division

p+qi is a complex number. p-qi is the conjugate of p+qi. We can use conjugates to divide two Complex Numbers.


To divide two Complex Numbers, multiply numerator and denominator by the conjugate of the denominator.


The numerator and denominator can be solved by using FOIL.

(r+ti)×(r-ti) =r2-rti + rti-t2 i2
r^2-t2 (-1)

Example: (2+4i)÷ (3-4i)

Conjugate of 3 – 4i is 3 + 4i, so multiplying the numerator and denominator by this conjugate.


Check point:

1. (3 – 3i) × (2 + 4i)
2. (6 – 3i)÷(2 + i)
3. (7 + 3i) ÷ (2 + 5i)
4. (4 + 5i) ÷ (3 + 2i)
5. (2 – 3i) × (4 + 7i)

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