# Addition of Complex Numbers – Properties and Examples

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

Let’s begin –

Let $$z_1$$ = $$a_1 + ib_1$$ and $$z_2$$ = $$a_2 + ib_2$$ be two complex numbers. Then their sum $$z_1 + z_2$$ is defined as the complex number $$(a_1 + a_2)$$  + i$$(b_1 + b_2)$$.

It follows from the definition that the sum $$z_1 + z_2$$ is a complex number such that

$$Re (z_1 + z_2)$$ = $$Re (z_1)$$ + $$Re (z_2)$$ and

$$Im (z_1 + z_2)$$ = $$Im (z_1)$$ + $$Im (z_2)$$

Example : If $$z_1$$ = 2 + 3i and $$z_2$$ = 3 – 2i, then $$z_1 + z_2$$ = (2 + 3) + (3 – 2)i = 5 + i

### Properties :

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

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

(ii) Addition 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 Additive Identity : The complex number 0 = 0 + i0 is the identity element for addition i.e. z + 0 = z = 0 + z for all z $$\in$$ C.

(iv) Existence of Additive Inverse : For any complex number z = a + ib, there exists -z = (-a) + i(-b) such that z + (-z) = 0 = (-z) + z.

## Subtraction of Complex Numbers

Let $$z_1$$ = $$a_1 + ib_1$$ and $$z_2$$ = $$a_2 + ib_2$$ be two complex numbers. Then the subtraction of $$z_2$$ from $$z_1$$ is denoted by $$z_1 – z_2$$ and is defined as the addition of $$z_1$$ and $$-z_2$$.

Example : If $$z_1$$ = -2 + 3i and $$z_2$$ = 4 + 5i, then $$z_1 – z_2$$ = (-2 + 3i) + (-4 – 5i) = (-2 – 4) + i(3 – 5) = -6 – 2i