Page 18 - DMTH401_REAL ANALYSIS

Real Analysis

Notes This means that p is even and hence p is even (verify it). Therefore, we can write p = 2k for some
2
integer k. Accordingly, we will have
p = 4k = 2q
2
2 2
or
2
2
q = 2k .
Thus p and q are both even. In other words, p and q have 2 as a common factor. This contradicts
our supposition that p and q have no common factor.
Hence there is no rational number whose square is 2.

Why don’t you try the following similar exercises?
Thus you have seen that there are numbers which are not rationals. Such numbers are called
irrational. In other words, a number is irrational if it cannot be expressed as p/q, p  Z, q  Z,
q  0. In this way, 2, 3, 5, etc. are irrational numbers. In fact, such numbers are infinite.
Rather, you will see in Unit 2 that such numbers are even uncountable. Also you should not
conclude that all irrational numbers can be obtained in this way. For example, the irrational
numbers e and  are not of this form. We denote by I, the set of all irrational numbers.
Thus, we have seen that the set Q is inadequate in the sense that there are number which are not
rationals.
A number which is either rational or irrational is called a real number. The set of real numbers
is denoted by R. Thus the set R is the disjoint union of the sets of rational and irrational numbers
i.e. R = Q  I, Q  I = O.
Now in order to visualise a clear picture of the relationship between the rationals and irrationals,
their geometrical representation as points on a line is of great help. We discuss this in the next
section.

1.3 The Real Line

Draw a straight line L as shown in the Figure 1.1.
Figure 1.1

Choose a point O on L and another point P, to the right of O. Associate the number O (zero) to the
point O and the number 1 to the point P . We take the distance between the points P and P as a
1 1
unit length. We mark a succession of points P , P , …… to the right of P each at a unit distance
2 3 1
from the preceding one. Then associate the integers 2, 3, .... to the points P , P , ..., respectively.
2 3
Similarly, mark the points P , P ,..., to the left of the point O, Associate the integers –1, –2,… to
–1 –2
the points P , P ,…. Thus corresponding to each integer, we have associated a unique point of
–1 –2
the line L.
Now associate every rational number to a unique point of L. Suppose you want to associate the
2 2 1
rational number to a point on the line L. Then = 2 × i.e., one unit is divided into seven
7 7 7
parts, out of which 2 are to be taken. Let us see how you do it geometrically.

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