Page 16 - DMTH401_REAL ANALYSIS

Real Analysis

Notes Different theories have been advanced about the origin and evolution of natural numbers. An
axiomatic approach, as evolved by G. Peano, is often used to define the natural numbers. Some
mathematicians like L. Kronecker [1823-1891] have remarked that the natural numbers are a
creation of God while all else is the work of man.
However, we shall not go into the origin of the natural numbers. In fact, we accept that the
natural numbers are a gift of nature to the mankind.
We denote the set of all natural numbers as
N = {1,2, 3, ....}.

One of the basic properties of these numbers is that there is a starting number 1. Then for each
number there is a next number. This nextness property is an important idea that you may find
fascinating with the natural numbers. You may think of any big natural number. Yet, you can
always tell its next number. What’s the next number after forty nine? After seventy seven? After
one hundred twenty three? After three thousand and ninety nine? Thus you have an endless
chain of natural numbers.
Some of the basic properties of the natural numbers are concerning the well-known fundamental
operations of addition, multiplication, subtraction and division. You know that the symbol ‘+’
is used for addition and the symbol ‘x’ is used for multiplication. If we add or multiply any two
natural numbers, we again get natural numbers. We express it by saying that the set of natural
numbers is closed with respect to these operations.

However, if you subtract 2 from 2, then what you get is not a natural number. It is a number
which we call zero denoted as ‘0’. The word, zero, in fact is a translation of the Sanskrit ‘shunya’.
It is universally accepted that the concept of the number zero was given by the ancient Hindu
mathematicians. You come across with certain concrete situations indicating the meaning of
zero. For example, the temperature of zero degree is certainly not an absence of temperature.
After having fixed the idea of the number zero, it should not be difficult for you to understand
the notion of negative natural numbers. You must have heard the weather experts saying that
the temperature on the top of the hills is minus 5 degrees written as –5°. What does it mean?
The simple and straight explanation is that –5 is the negative of 5 i.e. –5 is a number such that
5 + (–5) = 0. Hence –5 is a negative natural number. Thus for each natural n, there is a unique
number –n, called the negative of n such that

n + (–n) = 0.

1.2.2 Integers

You have seen that in the set N of natural numbers, if we subtract 2 from 2 or 3 from 2, we do not
get back natural numbers. Thus set of natural numbers is not closed with respect to the operation
of subtraction. After the operation of subtraction is introduced, the need to include 0 and negative
numbers becomes apparent. To make this operation valid, we must enlarge the system of
natural numbers, by including in it the number 0 and all the negative natural numbers. This
enlarged set consisting of all the natural numbers, zero and the negatives of natural numbers, is
called the set of integers. It is denoted as
Z = {.... –3, –2, –1, 0, 1, 2, 3 ….}.
Now you can easily verify that the set of integers is closed with respect to the operations of
addition, multiplication and subtraction.
The integers 1, 2, 3 .... are also called positive integers which are in fact natural numbers. The
integers –1, –2, –3,.... are called negative integers which are actually the negative natural numbers.

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