Complex Number

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 i2 = -1 , i3 = -i , i4 = 1
In general , i4n = 1 , i4n+1 = i , i4n+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.

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