# Multiplication of Complex Numbers – Properties and Examples

Here you will learn multiplication of complex numbers and its properties with examples.

Let’s begin –

## Multiplication of Complex Numbers

Let $$z_1$$ = $$a_1 + ib_1$$ and $$z_2$$ = $$a_2 + ib_2$$ be two complex numbers. Then the multiplication of $$z_1$$ with $$z_2$$ is denoted by $$z_1 z_2$$ and is defined as the complex number

$$(a_1a_2 – b_1b_2)$$ + i$$(a_1b_2 + a_2b_1)$$.

Thus,  $$z_1$$$$z_2$$ = $$a_1 + ib_1$$$$a_2 + ib_2$$

$$\implies$$  $$z_1$$$$z_2$$ = $$(a_1a_2 – b_1b_2)$$ + i$$(a_1b_2 + a_2b_1)$$

$$\implies$$  $$(a_1a_2 – b_1b_2)$$ = [$$Re(z_1) Re(z_2) – Im(z_1) Im(z_2)$$] + i [$$Re(z_1) Im(z_2) + Re(z_2) Im(z_1)$$]

Example : If $$z_1$$ = 3 + 2i and $$z_2$$ = 2 – 3i, then

$$z_1 z_2$$ = (3 + 2i)(2 – 3i)

= $$(3 \times 2 – 2 \times (-3)) + i(3 \times -3 + 2 \times 2)$$ = 12 – 5i

Note : The product $$z_1z_2$$ can also be obtained if we actually carry out the multiplication ($$a_1 + ib_1$$)($$a_2 + ib_2$$) as given below :

($$a_1 + ib_1$$)($$a_2 + ib_2$$) = $$a_1a_2$$ + $$ia_1b_2$$ + $$ib_1a_2$$ + $$i^2b_1b_2$$

= $$(a_1a_2 – b_1b_2)$$ + i$$(a_1b_2 + a_2b_1)$$           [$$because$$  $$i^2$$ = -1]

### Properties of Multiplication

(i) Multiplication is Commutative : For any two complex numbers $$z_1$$ and $$z_2$$, we have

$$z_1 z_2$$ = $$z_2 z_1$$

(ii) Multiplication is Associative : For any three complex numbers $$z_1$$, $$z_2$$, $$z_3$$, we have

($$z_1$$ $$z_2$$) $$z_3$$  = $$z_1$$ ($$z_2$$ $$z_3$$)

(iii) Existence of Identity Element for Multiplication : The complex number 1 = 1 + i0 is the identity element for multiplication i.e. for every complex number z, we have

z.1 = z = 1.z

(iv) Existence of Multiplicative Inverse : Corresponding to every non-zero complex number z = a + ib there exists a complex number $$z_1$$ = x + iy such that

$$z.z_1$$ = 1 = $$z_1.z$$

(v) Multiplications of complex numbers is distributive over addition of complex numbers : For any three complex numbers $$z_1$$, $$z_2$$, $$z_3$$, we have

(i) $$z_1(z_2 + z_3)$$ = $$z_1z_2 + z_1z_3$$

(ii) $$(z_2 + z_3)z_1$$ = $$z_2z_1 + z_3z_1$$