# 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, i^{2} = -1, i^{3} = – i, i^{4} = 1

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

(p+qi ) ×(r+ti) = pr + pti + qir + qti^{2}

pr + (pt+qr) i + qti^{2}

Example: (6 + 3i) × (3 + 4i)

= 18 + 24i + 9i + 12i^{2}

= 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) =r ^{2}-rti + rti-t^{2} i^{2}**

**r^2-t**

^{2}(-1)**r^2+t**

^{2}**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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