Page 17 - DMTH401_REAL ANALYSIS

Unit 1: Sets and Numbers

The number 0, however, is neither a positive integer nor a negative one. The set consisting of all Notes
the positive integers and 0 is called the set of non-negative integers. Similarly we talk of the set
of non-positive integers. Can you describe it?

1.2.3 Rational Numbers

If you add or multiply the integers 2 and 3, then the result is, of course, an integer in each case.
Also if you subtract 2 from 2 or 2 from 3, the result once again in each case, is an integer. What
do you get, when you divide 2 by 3? Obviously, the result is not an integer. Thus if you divide
an integer by a non-zero integer, you may not get an integer always. You may get the numbers
of the form
1 1  2  4 5
, , , ,  so on.
2 3 3 5 6
Such numbers are called rational numbers.
Thus the set Z of integers is inadequate when the operation of division is introduced. Therefore,
we enlarge the set Z to that of all rational numbers. Accordingly, we get a bigger set which
includes all integers and in which division by non-zero integers is possible. Such a set is called
p
the set of rational numbers. Thus a rational number is a number of the form , q  0, where
q
p and q are integers. We shall denote the set of all rational numbers by Q. Thus,
p
Q = {x = , P  Z, q  Z, q  0).
q
Now if you add or multiply any two rational numbers you again get a rational number. Also if
you subtract one rational number from another or if you divide one rational number by a non-
zero rational, you again get a rational numbers in each case. Thus the set Q of rational numbers
looks to be a highly satisfactory system of numbers in the sense that the basic operations of
addition, multiplication, subtraction and division are defined on it. However, Q is also inadequate
in many ways. Let us now examine this aspect of Q.
2
Consider the equation x = 2. We shall show that there is no rational number which satisfies this
equation. In other words, we have to show that there is no rational number whose square is 2.
We discuss this question in the form of the following example:

Example: Prove that there is no rational number whose square is 2.
Solution: If possible, suppose that there is a rational number x such that x = 2. Since x is a rational
2
number, therefore x must be of the form
p
x = , p  Z, q  Z, q  0.
q
Note that the integers p and q may or may not have a common factor. We assume that p and q
have no common factor except 1.

Squaring both sides, we get
p 2
= 2.
q 2
Then we have
2
2
p = 2q .

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