**How to Multiply and Divide Complex Numbers ?**

In this section, we will see how to multiply and divide complex numbers.

**Multiplying complex numbers :**

Suppose a, b, c, and d are real numbers. Then,

- (a + bi)(c + di) = (ac − bd) + (ad + bc)i

**Division of complex numbers :**

To divide the complex number which is in the form

(a + ib)/(c + id)

we have to multiply both numerator and denominator by the conjugate of the denominator.

That is,

[ (a + ib)/(c + id) ] ⋅ [ (c - id) / (c - id) ]

= [ (a + ib) (c - id) / (c + id) (c - id) ]

**Example 1 :**

Multiply the following complex numbers

(2 + 3i) (4 - 7i)

**Solution :**

(2 + 3i) (4 - 7i) = 2(4) + 2(-7i) + 4(3i) + 3i(-7i)

= 8 - 14i + 12i - 21i^{2}

= 8 - 2i - 21(-1)

= 8 - 2i + 21

= 29 - 2i

**Example 2 :**

Multiply the following complex numbers

(4 - 2i) (3 - 5i)

**Solution :**

(4 - 2i) (3 - 5i) = 4(3) + 4(-5i) + 3(-2i) - 2i(-5i)

= 12 - 20i - 6i + 10i^{2}

= 12 - 26i + 10(-1)

= 12 - 10 - 26i

= 2 - 26i

**Example 3 :**

Multiply the following complex numbers

(-5 + 3i)(-2 + i)

**Solution :**

(-5 + 3i)(-2 + i) = -5(-2) - 5(i) + 3i(-2) + 3i(i)

= 10 - 5i - 6i + 3i^{2}

= 10 - 11i + 3(-1)

= 10 - 3 - 11i

= 7 - 11i

**Example 4 :**

Multiply the following complex numbers

(3 - i) (8 + 7i)

**Solution :**

(3 - i) (8 + 7i) = 3(8) + 3(7i) - i(8) - i(7i)

= 24 + 21i - 8i - 7i^{2}

= 24 + 13i - 7(-1)

= 24 + 13i + 7

= 31 + 13i

**Example 5 :**

Divide the complex number (3 + 2i) by (2 + 4i)

**Solution :**

(3 + 2i) by (2 + 4i) = (3 + 2i)/(2 + 4i)

Whenever we have complex numbers in the denominator, we have to multiply the numerator and denominator by the conjugate of the denominator of the given complex number.

= [(3 + 2i)/(2 + 4i)] ⋅[(2 - 4i)/(2 - 4i)]

= [(3 + 2i)(2 - 4i)/(2 + 4i) (2 - 4i)]

Multiplying the numerator, we get

(3 + 2i)(2 - 4i) = 3(2) + 3(-i) + 2i(2) + 2i(-4i)

= 6 - 3i + 4i - 8i^{2}

= 6 - 8(-1) + i

= 6 + 8 + i

= 14 + i

Multiplying the denominator, we get

(2 + 4i) (2 - 4i) = 2(2) + 2(-4i) + 4i(2) + 4i(-4i)

= 4 - 8i + 8i - 16i^{2}

= 4 - 16(-1)

= 4 + 16

= 20

(3 + 2i)/(2 + 4i) = (14 + i)/20

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