REAL NUMBERS

INTRODUCTION

since our childhood we have been using four fundamental operations of addition, subtraction, multiplication and division. We have applied these operations on natural numbers, integers, rational and irrational numbers. In this chapter, we will begin with a brief recall of 'divisibility on integers and will state some important properties of integers, namely, Euclid's division Lemma, Euclid's division algorithm and the Fundamental Theorem of Arithmetic which will be used in the remaining part of this chapter to learn more about integers and real numbers.

NATURAL NUMBER - {1, 2, 3, 4, .....................}. It is also called counting numbers.

WHOLE NUMBER - { 0, 1, 2, 3, ....................................}. It is also called zero number.

INTEGER NUMBER - { .......................-3, -2, -1, 0, 1, 2, 3, .......................................} Representing  on the number line.

RATIONAL NUMBER -    
 Where P,Q are integers.

IRRATIONAL NUMBER - Non-terminating and Non-Repeating numbers

REAL NUMBER - Sum of Rational and Irrational numbers.

Euclid's division lemma tells us about divisibility of integers. It is quite easy to state and understand. It states that any positive integer $a$ can be divided by any other positive integer $b$ in such a way that it leaves a remainder $r$ that is smaller than $b$. This is nothing but the usual long division process. Euclid's division lemma provides us a step-wise procedure to compute the H.C.F. of two positive integers. This step-wise procedure is known as Euclid's algorithm. We will use the same for finding the H.C.F. of positive integers.

The Fundamental Theorem of Arithmetic tells us about expressing positive integers as the product of prime integers. It states that every positive integer is either prime or it can be factorised (expressed) as a product of powers of prime integers. This theorem has many significant applications in mathematics and in other fields. We have learnt how to find the H.C.F. and LCM of positive integers by using the Fundamental Theorem of Arithmetic in earlier classes.  we will apply this theorem to prove the irrationality of many numbers such as   etc. We know that the decimal representation of a rational number is either terminating or if it is non-terminating then it is repeating. The prime factorisation of the denominator of a rational number completely reveals the nature of its decimal representation. In fact, by looking at the prime factorisation of the denominator of a rational number one can easily tell about its decimal representation whether it is terminating or non-terminating repeating. We will also use the Fundamental Theorem of Arithmetic to determine the nature of the decimal expansion of rational numbers.