## 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 +

*i*b.

A complex number z = a + *i*b consist of two parts-

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

Thus z = a + *i*b 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 z_{1} = a + *i*b and z_{2} = c + *i*d are said to be equal if an only if a = c and b = d i.e. Re (z_{1}) = Re (z_{2}) and Im (z_{1}) = Im (z_{2}).

## Operations on Complex Numbers:

(1) **Addition of Complex Number-** If z_{1} = a + *i*b and z_{2} = c + *i*d with the two complex number, then their addition is denoted by z_{1} + z_{2} and is given as-

z_{1} + z_{2} = (a + c) + i(b + d)⇒ z _{1} + z_{2} = [Re (z_{1}) + Re (z_{2})] + i[Im (z_{1}) + Im (z_{2})] |

**Properties of Addition of Complex Number:**

**Commutative Property-**If z_{1}and z_{2}be the two complex numbers then z_{1}+ z_{2}= z_{2}+ z_{1}.**Associative Property-**If z_{1}, z_{2}, z_{3 }be the three complex numbers then (z_{1}+ z_{2}) + z_{3}= z_{1}+ (z_{2}+ z_{3}).**Additive Identity-**If z be any complex number then z + 0 = z = 0 + z, then O is called the additive identity and O = 0 +*i*0.**Additive Inverse-**If z_{1}and z_{2}be any two complex numbers such that z_{1}+ z_{2}= 0 = z_{2}+ z_{1}then z_{2}is the additive inverse of z_{1}and in general z_{2}= -z_{1}.

(2) **Subtraction** **of Complex Number-** If z_{1} = a + *i*b and z_{2} = c + *i*d be two complex numbers, then the subtraction of z_{2} from z_{1} is-

z_{1} – z_{2} = z_{1} + (-z_{2}) = (a + ib) + (-c – id) = (a – c) + i(b – d) |

(3) **Multiplication of Complex Number-** If z_{1} = a + *i*b and z_{2} = c + *i*d be the two complex numbers then their multiplication is denoted by z_{1}.z_{2} and is given as-

z_{1} . z_{2} = (a + ib) (c + id)⇒ z _{1} . z_{2} = ac + i(ad + bc) – bd⇒ z _{1} . z_{2} = (ac – bd) + i(ad + bc)⇒ z _{1} . z_{2} = [Re (z_{1}) . Re (z_{2}) – Im (z_{1}) . Im (z_{2})] + i[Re (z_{1}) . Im (z_{2}) + Im (z_{1}) . Re (z_{2})] |

**Properties of Multiplication of Complex Number:**

**Commutative Property-**If z_{1}and z_{2}be two complex numbers then z_{1}. z_{2}= z_{2}. z_{1}.**Associative Property-**If z_{1}, z_{2}, z_{3}be three complex numbers then (z_{1}. z_{2}) . z_{3}= z_{1}(z_{2}. z_{3}).**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 . z_{1}= 1 = z_{1}. z then z_{1}is called the multiplicative inverse of z. In general z = 1/z.

Example- Find the Multiplicative Inverse of a + ib.Solution: Let z = a + ib then multiplicative inverse of z is 1/(a + ib)⇒ z = 1/(a + ib) x (a – ib)/(a – ib)⇒ z = (a – ib)/[a^{2} – (ib)^{2}]⇒ z = (a – ib)/(a^{2} + b^{2})⇒ z = z̄/| z | ^{2} |

(4) **Division** **of Complex Number-** If z_{1} = a + *i*b and z_{2} = c + *i*d are two complex numbers, then the division of z_{1} by z_{2} is defined as-

z_{1}/z_{2} = (a + ib)/(c + id)⇒ z _{1}/z_{2} = (a + ib).(c – id)/(c + id).(c – id)⇒ z _{1}/z_{2} = [(ac + bd) + i(bc – ad)]/(c^{2} + d^{2})⇒ z _{1}/z_{2} = (ac + bd)/(c^{2} + d^{2}) + i (bc – ad)/(c^{2} + d^{2}) |

## Conjugate of a Complex Number:

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

**Properties of Conjugate of a Complex Number:**

If z, z_{1}, z_{2} be any complex number then-