Page 22 - DMTH401_REAL ANALYSIS

Real Analysis

Notes z + z = (a + a ) + i (b + b )
1 2 1 2 1 2
z .z = (a a – b b ) + i (a b – a b ).
1 2 1 2 1 2 1 1 2 2
The real numbers represent points on a line while complex numbers are identified as points on
the plane.
Before concluding this section, we would like to mention yet another classification of numbers
as enunciated by some mathematicians. Consider the number 2 . This is an example of what is
called an Algebraic Number because it satisfies the equation
2
x – 2 = 0.
A number is called an Algebraic Number if it satisfies a polynomial equation

a x + a x t .... + a x + a,, + a = 0
n
n–l
0 1 n–1 n
where the coefficients a , a , a ,.... a, are integers, a,  0 and n > 1. The rational numbers are always
0 1 2
algebraic numbers. The numbers defined in terms of the square root etc., are also treated as
algebraic numbers. But there are some real numbers which are not algebraic. Such numbers are
called the Transcendental numbers. The numbers  and  are transcendental numbers.
You may think that the operations of algebraic operations viz. addition, multiplication, etc. are
the only aspects to be discussed about the set of real numbers. But certainly there are some more
important aspects of the set of real numbers as points on the real line. We shall discuss these
aspects in Unit 3 namely the point sets of the real line called also the topology of the real line. But
prior to that, we shall discuss the structure of real numbers in Unit 2.
We conclude this unit by talking briefly about an important hypothesis-closely linked with the
system of natural numbers. This is called the Principle of Induction.

1.5 Mathematical Induction

The Principle of Induction and the natural numbers are inseparable. In Mathematics, we often
deal with the proofs of various theorems and formulas. Some of these are derived by the direct
proofs, while some others can be proved by certain indirect methods. Consider, for example, the
validity of the following two statements:
n
(i) The number 4 divides 5 –1 for every natural number n.
n(n 1)
+
(ii) The sum of the first n natural numbers is i.e.
2
n(n 1)
+
1 + 2 + 3 + ... + n = .
2
In fact, you can provide most of the verifications for both statements in the following way:
n
For (i), if n = 1, then 5 –1 = 5 –1 = 4 which is obviously divisible by,
2
if n = 2, then 5 –1 = 24, which is also divisible by 4;
6
if n = 6, then 5 –1 = 15624, which is indeed divisible by 4.
Similarly for (ii) if n = 10 then 1 + 2 + .... 4 – 10 = 55, while the formula

n(n 1)
+
= 55 when n = 10.
2
Again, if n = 100; then also you can verify that in each way, the sum of the first hundred natural
numbers is 5050 i.e.

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