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)