Page 15 - DMTH401_REAL ANALYSIS

Unit 1: Sets and Numbers

This raises another question: Under what conditions a function is as an inverse? If a function Notes
f: ST is one-one and onto, then it is invertible * conversely, if f is invertible, then f is both
one-one and onto. Thus if a function is one-one and onto, then it must have an Inverse.

1.2 System of Real Numbers

You are quite familiar with some number systems and some of their properties. You will,
perhaps recall the following properties:

(i) Any number multiplied by zero is equal to zero,
(ii) The product of a positive number with a negative number is negative,
(iii) The product of a negative number with a negative number is positive among takers.

To illustrate these properties, you will most likely use the natural numbers or integers or even
rational numbers. The questions, which begin to arise are: What are these various types of
numbers? What properties characterise the distinction between these various sets of numbers?

In this section, we shall try to provide answers to these and many other related questions. Since
we are dealing with the course on Real Analysis, therefore we confine our discussion to the
system of real numbers. Nevertheless, we shall make you peep into the realm of a still larger
class of numbers, the so called complex numbers.

The system of real numbers has been evolved in different ways by different mathematicians. In
the late 19th Century, the two famous German mathematicians Richard Dedekind [1815-1897]
and George Cantor [1845-1918] gave two independent approaches for the construction of real
numbers. During the same time, an Italian mathematician, G. Peano [1858-1932] defined the
natural numbers by the well-known Peano Axioms. However, we start with the set of natural
numbers as the foundation and build up the integers. From integers, we construct the rational
numbers and then through the set of rational numbers, we reach the stage of real numbers. This
development of number system culminates into the set of complex numbers. A detailed study of
the system of numbers leads us to a beautiful branch of Mathematics namely. The Number Theory,
which is beyond the scope of this course. However, we shall outline the general development of
the system of the real numbers in this section. This is crucial to understand the characterization
of the real numbers in terms of the algebraic structure to be discussed in Unit 2. Let us start our
discussion with the natural numbers.
1.2.1 Natural Numbers

The notion of a number and its counting is so old that it is difficult to trace its origin. It developed
much before the time of even the recorded history that its manner of development is based on
conjectures and guesses. The mankind, even in the most primitive times, had some number
sense. The man, at least, had the sense of recognizing ‘more’ and ‘less’, when some objects were
added to or taken out from a small collection. Studies have shown that even some animals
possess such a sense. With the gradual evolution of society, simple counting became imperative.
A tribe had to count how many members it had, how many enemies and how many friends. A
shepherd or a cowboy found it necessary to know if his flock of sheep or cows was decreasing or
increasing in size. Various ways were evolved to keep such a count. Stones, pebbles, scratches on
the ground, notches on a big piece of wood, small sticks, knots in a string or the fingers of hands
were used for this purpose. As a result of several refinements of these counting methods, the
numbers were expressed in the written symbols of various types called the digits. These digits
were written differently according to the different languages and cultures of the societies. In the
ultimate development, the numbers denoted by the digits 1, 2, 3, .... became universally acceptable
and were named as natural numbers.

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