# Complex Numbers Operations

## What is Complex Numbers?

The numbers of the form a + ib are called complex numbers where a, b are real numbers. Complex numbers are generally denoted by z. Thus complex number is z = a + ib.

A complex number z = a + ib consist of two parts-

• Real part denoted by Re (z).
• Imaginary part denoted by Im (z).

Thus z = a + ib then Re (z) = a and Im (z) = b.

Purely Real Complex Number- A complex number is said to be purely real if its imaginary part is zero i.e. z is purely real if and only if the imaginary part of z = 0.

Purely Imaginary Complex Number- A complex number is said to be purely imaginary if its real part is zero i.e. z is purely imaginary if and only if the real part of z = 0.

Equality of Complex Numbers- Two complex numbers z1 = a + ib and z2 = c + id are said to be equal if an only if a = c and b = d i.e. Re (z1) = Re (z2) and Im (z1) = Im (z2).

## Operations on Complex Numbers:

(1) Addition of Complex Number- If z1 = a + ib and z2 = c + id with the two complex number, then their addition is denoted by z1 + z2 and is given as-

Properties of Addition of Complex Number:

• Commutative Property- If z1 and z2 be the two complex numbers then z1 + z2 = z2 + z1.
• Associative Property- If z1, z2, z3 be the three complex numbers then (z1 + z2) + z3 = z1 + (z2 + z3).
• Additive Identity-If z be any complex number then z + 0 = z = 0 + z, then O is called the additive identity and O = 0 + i0.
• Additive Inverse- If z1 and z2 be any two complex numbers such that z1 + z2 = 0 = z2 + z1 then z2 is the additive inverse of z1 and in general z2 = -z1.

(2) Subtraction of Complex Number- If z1 = a + ib and z2 = c + id be two complex numbers, then the subtraction of z2 from z1 is-

(3) Multiplication of Complex Number- If z1 = a + ib and z2 = c + id be the two complex numbers then their multiplication is denoted by z1.z2 and is given as-

Properties of Multiplication of Complex Number:

• Commutative Property- If z1 and z2 be two complex numbers then z1 . z2 = z2 . z1.
• Associative Property- If z1, z2, z3 be three complex numbers then (z1 . z2) . z3 = z1 (z2 . z3).
• Multiplicative Identity- If z be any complex number such that z . 1 = z = 1 . z then ‘1’ is called the Multiplicative Identity.
• Multiplicative Inverse- If z be any complex number such that z . z1 = 1 = z1 . z then z1 is called the multiplicative inverse of z. In general z = 1/z.

(4) Division of Complex Number- If z1 = a + ib and z2 = c + id are two complex numbers, then the division of z1 by z2 is defined as-

## Conjugate of a Complex Number:

If z be any complex number such that z = a + ib then its conjugate is denoted by z̄ and is given as z̄ = a – ib.

Properties of Conjugate of a Complex Number:

If z, z1, z2 be any complex number then-