Page 19 - DMTH401_REAL ANALYSIS

Unit 1: Sets and Numbers

Notes
Figure 1.2

Through O, draw a line OM inclined to the line L. Mark the points A, A,..., A on the line OM at
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equal distances. Join P A,. Now if you draw a line through A, parallel to P P to meet the line L
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2 2
in H. Then H corresponds to the rational number i.e., OH = .
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You can do likewise for a negative rational number. Such points, then, will be to the left of O.
By having any line through O, you can see that the point P does not depend upon chosen line
OM. Thus, you have associated every rational number to a unique point on the line L.

Now arises the important question:
Have you used all the points of the line L while representing rational numbers on it?
The answer to this question is NO. But how? Let us examine this.
At the point P, draw a line perpendicular to the line and mark A such that P A = 1 unit. Cut off
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OB = OA on the line, as shown in the Figure 1.3.

Figure 1.3

Then B is a point which correspond to a number whose square is 2. You have already seen that
there is no rational number whose square is 2. In fact, the length OA = 2 by Pythagorean
Theorem. In other words, the irrational number 2 is associated with the point B on the line L.
In this way, it can be shown that every irrational number can be associated to a unique point on
the line L.
Thus, it can be shown that to every real number, there corresponds a unique point on the line L.
In other words, all the real numbers are represented as points on a line. Is the converse true? That
is to say, does every point on the line correspond to a unique real number? This is true but we are
not going to prove it here. Therefore, hence onwards, we shall say that every real number

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