Support the Monkey! Tell All your Friends and Teachers

Help / FAQ


Complex Number

Let us consider a quadratic equation say, x2 - 2x + 5 = 0. Hence the discriminant b2 - 4ac, being negative, it leads to the solution set { 1 - 2i, 1 + 2i}. The roots viz x = 1 - 2i and x = 1 + 2i is that they are partly real and partly imaginary. We now introduce a number which is a combination of real and imaginary numbers. This is called a Complex number of the type a + bi , a, b Î R. We denote this by z then

z = a + ib ...........(ii)

Here a = R (z) ...... real part z and b = I (z) ..... imaginary part z.

The same may be expressed more elegantly as an ordered pair of two real numbers a and b as,

z = (a , b) ..........(iii)

For example, z1 = (2, 3) = 2 + 3i,       z2 = (2, -3) = 2 - 3i,

            z3 = (0, 1) = 0 + i = i ,         z4 = (0, 0) = 0 + 0i = 0                 which is a real number = 0

Conjugate Complex : If z = a + bi is a complex number then z or z* = a - bi is called conjugate complex of z.

For example, z1 = 3 + 2i then 1 = 3 - 2i, 2 = - 8 - 4i then z2 = - 8 + 4i.

Basic Operations : From the point of view of an axiomatic foundation, it is advised to treat a complex as an ordered pair (a,b), a, b Î R; subject to certain operational rules.

Equality of two complex numbers : Let z1 = (a,b) and z2 = (c,d) then z1 = z2 iff a = c

and b = d.

Sum : Let z1 = (a,b) and z2 = (c,d) then z1 + z2 = (a + c, b + d )

Difference : z1 - z2 = (a - c, b - d)

Product : z1 . = (a, b) (c,d) = (ac - bd, ad + bc)

Scalar Multiplication : kz = (ka, kb) if k is a scalar and z = (a,b)

Quotient :

Product of conjugate complexes : z . z = (a + bi) (a - bi) = a2 + b2

Closer Property : From the operations between any two complex numbers is again a complex number only. As such we remark that the set 'C' of complex numbers is closed with respect to these operations.

[next page]

Index

8.1 Geometry of complex numbers
8.2 De - Moivres's theorem
8.3 Roots of complex numbers
8.4 Cirsular functions of complex angles & hyperbolic function
Supplementary Problems

Chapter 9

All Contents Copyright © All rights reserved.
Further Distribution Is Strictly Prohibited.


Search:
Keywords:
In Association with Amazon.com