Real Numbers

♦  Natural Numbers : Counting numbers are called Natural numbers. These numbers are  denoted by N = {1,2,3,..........}

♦ Whole Numbers : The collection of natural numbers along with 0 is the collection of  Whole number and is denoted by W.

♦ Integers : The collection of natural numbers, their negative along with the number zero are called Integers. This collection is denoted by Z.

♦ Rational Numbers : The numbers, which are obtained by dividing two integers, are called Rational numbers. Division by zero is not defined.

♦ Coprime : If HCF of two numbers is 1, then two numbers area called relatively prime or coprime.

1. Euclid's division lemma :

For given positive integers 'a'  and  'b'  there exist unique whole numbers  'q'  and  'r'  satisfying the relation  a = bq  + r,  0 ˂ r < b..

Theorem :  If a  and  b  are non-zero integers, the least  positive integers  which is expressible as a linear  combination  of  a  and  b is the HCF of a and  b , i.e, if d is HCF  of  a and b, then these exist integers  x1 and  y1, such that d=ax1 + by1 and d is the smallest positive ineteger which is expressible in this  form.

The HCF  of  a and b is denoted by HCF (a,b)