**Basic Concepts :**

A number in the form of a + ib, where a, b are real numbers and i = √-1 is called a complex number.

A complex number can also be defined as an ordered pair of real numbers a and b and may be written as (a, b), where the first number denotes the real part and the second number denotes the imaginary part.

If z = a + ib, then the real part of z is denoted by Re (z) and the imaginary part by Im (z).

**A complex number is said to be purely real** if Im(z) = 0, and is said to be purely imaginary if Re(z) = 0.

The complex number 0 = 0 + i0 is both purely real and purely imaginary.

**Two complex numbers are said to be equal** if and only if their real parts and imaginary parts are separately equal i.e.

a + ib = c + id implies a = c and b = d.

However, there is no order relation between complex numbers and the expressions of the type a + ib < (or >) c + id are meaningless.

**Remark:**

⋄ Clearly i^{2} = -1 , i^{3} = -i , i^{4} = 1

**In general ,** i^{4n} = 1 , i^{4n+1} = i , i^{4n+2 }= -1 for an integer n.

__Geomertical Representation Of Complex Number__

A complex number z = x + iy, written as an ordered pair (x, y), can be represented by a point P whose Cartesian coordinates are (x, y) referred to axes OX and OY, usually called the **real and the imaginary axes.**

The plane of OX and OY is called the **Argand diagram or the complex plane.**

Since the origin O lies on both OX and OY, the corresponding complex number z = 0 is both purely real and purely imaginary.