Page 21 - DMTH401_REAL ANALYSIS

Unit 1: Sets and Numbers

Suppose s is a positive real number and r is a negative real number. Then, there exists a number Notes
r such that r = –r where r is a positive real numbers. Therefore, the product rs can be defined on
L as
rs = (– r)s = –(rs).
–
Similarly you can state that rs = r(–s) = (rs) where s is negative and s = –s for some positive s,
while r is positive.
If both r and s are negative and r = –r and s = –s where r and s are positive real numbers, then
we define

rs = rs = (–r) (–s).
We can also similarly define 0, r = r! 0 = 0 for each real number r.

1.4 Complex Numbers

So far, we have discussed the system of real numbers. We have yet, another system of numbers.
For example, if you have to find the square root of a negative real number say –5, then you will
write at as  1, 5. You know that 5 is a real number but what about  1? Again you can
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verify that a simple equation x + 1 = 0 does not have a solution in the set of real numbers because
the solution involves the square root of a negative real number. As a matter of fact, the problem
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is to investigate the nature of the number  1 which we denote by such that i = –1. Let us
discuss the following example to know the nature of i.

Example: Show that i is not a real number.
We claim that i is not a real number. If it is so, then either i = 0 or i > 0 or I < 0.
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If i = 0, then i = 0. This implies that –1 = 0 which is absurd. If i > 0, then i > 0 which implies that
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–1 > 0. This is also absurd. Finally, if i < 0, then again i > 0 which implies that –1 > 0. This again
is certainly absurd. Thus i is not a real number. This number ‘i’ is called the imaginary number.
The symbol ‘i’ is called ‘iota’ in Greek language. This generates another class of numbers, the so
called complex numbers.
The basic idea of extending the system of real numbers to the system of complex numbers arose
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due to the necessity of finding the solutions of the equations, like x + 1 = 0 or x + 2 = 0 and so on.
The first contribution to the notion of such a number was made by the most celebrated Indian
Mathematician of the 9th century, Mahavira, who showed that a negative real number does not
have a square root in the set of real numbers. But it was an Italian mathematician, G. Cardon
[1501-1576] who used imaginary numbers in his work without bothering about their existence.
Due to notable contributions made by a large number of mathematicians, the system of complex
numbers came into existence in the 18th century. Since we are dealing with real numbers,
therefore, we shall not go into the detailed discussion of complex numbers. However, we shall
give a brief introduction to the system of complex numbers. We denote the set of complex
numbers as
C = {z = a + i b, a and b real numbers}
In a complex number, z = a + i b, a is called its real part and b is called its imaginary part.
Any two complex numbers z = a + i b and z = a + i b are equal if only their corresponding real
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and imaginary parts are equal.
If z = a + i b and z = a + i b are any two complex numbers, then we define addition (+) and
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multiplication (.) as follows:

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