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1.4 Complex Numbers

The history of mathematics shows that man has been developing and enlarging his

concept of number according to the saying that “Necessity is the mother of invention”. In

the remote past they stared with the set of counting numbers and invented, by stages, the

negative numbers, rational numbers, irrational numbers. Since square of a positive as well

as negative number is a positive number, the square

root of a negative number does not exist in the realm of real numbers. Therefore, square

roots of negative numbers were given no attention for centuries together. However, recently,

properties of numbers involving square roots of negative numbers have also been discussed

in detail and such numbers have been found useful and have been applied in many branches

o f pure and applied mathematics. The numbers of the

form x + iy, where x, y U_ , and i = ,are called complex numbers, here x is called real

part and y is called imaginary part of the complex

number. For example, 3 + 4i, 2 - i etc. are complex numbers.

Note: Every real number is a complex number with 0 as its imaginary part.

Let us start with considering the equation.
x2 + 1 = 0 (1)

⇒ x2 = -1
⇒ x = ± -1
-1 does not belong to the set of real numbers. We, therefore, for convenience call it
imaginary number and denote it by i (read as iota).
The product of a real number and i is also an imaginary number

version: 1.1

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