Page 20 - DMTH401_REAL ANALYSIS

Real Analysis

Notes corresponds to a unique point on the line and conversely every point on the line corresponds to
a unique real number. In this sense, the line L is called the Real Line.
Now let L be the real line.

Figure 1.4

We may define addition (+) and multiplication (.) of real numbers geometrically as follows:

Suppose A represents a real number r and B represents a real number s so that OA = r and OB = s.
Shift OB so that O coincides with A. The point C which is the new position of B is defined to
represent r + s. See the Figure 1.4.
The construction is valid for positive as well as negative values of r and s. A real number r is said
to be positive if r corresponds to a point on the line L on the right of the point O. It is written as
r > 0. Similarly, r is said to be negative if it corresponds to a point on the left of the point O and
is written as r < 0. Thus if r is a real number then either r is zero or r is positive or r is negative
i.e. either r = 0 or r > 0 or r < 0. You should try the following exercise:
What about the product r.s of two real numbers r and s? We shall consider the case when r and
s are both positive real numbers.

Figure 1.5

Though O draw some other line OM. On L, let A represent the real number s. On OM take a point
D so that OD = r. Let Q be a point on I, so that OQ = 1 unit. Join QD. Through A draw a straight
line parallel to QD to meet OM at C. Cut off OP on the line equal to OC. Then F represents the
real number r.s.

14 LOVELY PROFESSIONAL UNIVERSITY