## REAL NUMBERS

INTRODUCTION

IN THIS WE WILL DISCUSS THE REAL NUMBERS.

IMPORTANT PROPERTIES OF INTEGERS.

1. Euclid's division algorithm
2. Fundamental theoremof arithmetic

ALGORITHM

An algorithm is a series of well defined steps which give a procedure for solving a problem.

LEMA

Lema is a particular type of theorem .

EUCLID'S DIVISION ALGORITHM

It is defined as

a = bq + r

Given positive integers a and b, there exist unique integers q and r satisfying

a = bq + r, where 0 ≤ r < b, q, r is whole number.

let us discuss the relation of two numbers of a pair (14, 5), (15 ,4),(8, 3),(3, 7),(30, 6).

These pair can be rewritten as

14 = 5 * 2 + 4

15 = 4 * 3 + 3

8 = 3 * 2 + 2

3 = 7 * 0 + 3

30 = 6 * 5 + 0

These relations we obtain by division process. In division Q is called quotient and r called as remainder.

HIGHEST COMMON FACTOR (H.C.F) BY EUCLID DIVISION

Example

We find out H.C.F of the integer 972 and 21.

By euclid algoritm, 972 = 21 * 46 + 6

Here the divisor is 46 and reminder is 6. We applythe division algorithm again on 21 and 6.

21 = 6 * 3 + 3

Now the divisoris 6 and the reminder is 3 . We again apply divisionalgorithm on 6 and 3.

6 = 3 * 2 + 0

Now, the remainder is 0 and H.C.F of 6 and 3 is 3 and we can say that H.C.F of 972 and 21 is 3.

here, H.C.F (972 ,21) = H.C.F(21 , 6)=H.C.F(6 ,3) = 3.

Leta and b be two positive integers if

a = bq + r, 0 ≤ r < b

H.C.F(a ,b) = H.C.F(b ,r).