A complex number is a number of the form a + bi, where ‘a’ and ‘b’ are real numbers and ‘I’ is an indeterminate satisfying i^{2} = −1. For example, 2 + 3i is a complex number.

The equations x^{2} + 5 = 0, x^{2} + 10 = 0, x^{2} = -1 are not solvable in the real number system i.e, these equations has no real roots. For example, i is the solution of the equation x^{2} = -1 and it has two solutions i.e., x = ± i, where √-1.

The number, ‘I’ is called an imaginary number. Generally, the square root of any negative real number is called imaginary number. The concept of imaginary numbers was first introduced by mathematician “Euler”. He was the one who introduced ‘i’ (read as ‘iota’) to represent √-1. He also defined i^{2} = -1.

**Multiplication of Two Complex Numbers**

Multiplication of two complex numbers is also a complex number.

In other words, the product of two complex numbers can be expressed in the standard form A + iB where A and B are real.

Let z_{1} = p + iq and z_{2} = r + is be two complex numbers (p, q, r and s are real), then their product z_{1}z_{2} is defined as

z_{1}z_{2 }= (pr – qs) + i(ps + qr).

**Proof:**

Given z_{1} = p + iq and z_{2} = r + is

Now, z_{1}z_{2} = (p + iq)(r + is) = p(r + is) + iq(r + is) = pr + ips + iqr + i^{2}qs

We know that i^{2} = -1. Now putting i^{2} = -1 we get,

= pr + ips + iqr – qs

= pr – qs + ips + iqr

= (pr – qs) + i(ps + qr).

Thus, z_{1}z_{2} = (pr – qs) + i(ps + qr) = A + iB where A = pr – qs and B = ps + qr are real.

Therefore, product of two complex numbers is a complex number.

Note: Product of more than two complex numbers is also a complex number.

For example:

Let z_{1} = (4 + 3i) and z_{2} = (-7 + 6i), then

z_{1}z_{2} = (4 + 3i)(-7 + 6i)

= 4(-7 + 6i) + 3i(-7 + 6i)

= -28 + 24i – 21i + 18i^{2}

= -28 + 3i – 18

= -28 – 18 + 3i

= -46 + 3i

Example: Multiplication of Two Complex Numbers

**Properties of multiplication of complex numbers:**

If z_{1}, z_{2} and z_{3} are any three complex numbers, then

(i) z_{1}z_{2} = z_{2}z_{1} (commutative law)

(ii) (z_{1}z_{2})z_{3} = z_{1}(z_{2}z_{3}) (associative law)

(iii) z ∙ 1 = z = 1 ∙ z, so 1 acts as the multiplicative identity for the set of complex numbers.

(iv) Existence of multiplicative inverse

For every non-zero complex number z = p + iq, we have the complex number p / (p^{2}+q^{2}) – i q / (p^{2}+q^{2}) (denoted by z^{-1} or 1/z) such that

z ∙ 1/z = 1 = 1/z ∙ z (check it)

1/z is called the multiplicative inverse of z.

**Note:** If z = p + iq then z^{-1} = 1 / (p+iq) = 1/(p+iq) ∙ (p−iq)(p−iq) = (p−iq)/(p^{2}+q^{2})

(v) Multiplication of complex number is distributive over addition of complex numbers.

If z_{1}, z_{2} and z_{3} are any three complex numbers, then

z_{1}(z_{2} + z_{3}) = z_{1} z_{2} + z_{1}z_{3}

and (z_{1} + z_{2})z_{3} = z_{1}z_{3}+ z_{2}z_{3}

The results are known as distributive laws.

**Example**

Find the product of two complex numbers (-2 + √3i) and (-3 + 2√3i) and express the result in standard from A + iB.

Solution:

(-2 + √3i)(-3 + 2√3i)

= -2(-3 + 2√3i) + √3i(-3 + 2√3i)

= 6 – 4√3i – 3√3i + 2(√3i)^{2}

= 6 – 7√3i – 6

= 6 – 6 – 7√3i

= 0 – 7√3i, which is the required form A + iB, where A = 0 and B = – 7√3

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