# 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.

Method:

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)
r^2+t2

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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