Introduction of Real Numbers

 Definitions:

* ‘Divisibility': A non-zero integer 'b' is said to divide an integer 'a' such that

    a = bc, where, 'a' is called the dividend, 'b' is called the divisor

    and 'c' is called the quotient.

* 'Lemma': Any proven statement which is used to prove other statements

    Is called a 'lemma".

* 'Euclids division lemma': If 'a' and 'b' are any two positive integers, then

    there exists unique integers 'q' and 'r', such that a = bq + r; 0 ≤

    r < b.

 Note:

 If 'b' is divided by 'a' i.e. b/a then r = 0, other wise r satisfies the stranger

 inequality 0 < r < a.

* Algorithm: A series of well defined steps which provide a procedure of

    calculations repeated successively on the results of earlier

    steps till the derived result is obtained is called an algorithm.

       (or)

    A step by step procedure to solve a problem till we get the

    solution is called an algorithm.

* Fundamental theorem of arithmetic: Every composite number can be

    expressed (factorized) as a product of primes, and this

    factorization is unique except for the order in which the prime

    factors occur.

 Note:

 1. If 'a' and 'b' are two positive integers, then the formula to find the HCF

  of 'a' and 'b' is given by HCF (a, b) = a x b LCM (a,b)

 2. If 'p', 'q', 'r' are three positive integers then the formula to find the HCF

  of 'p', 'q' and 'r' is given by HCF (p,q,r) = p x q x r x LCM (p, q, r)

  LCM(p,q) . LCM(q, r) . LCM(r,p)

 3. Conditions for a rational number to be terminating or Non terminating

  decimal expansion.

  (a) If x = a/b, be a rational number and if the denominator 'b' can be

   expressed in the form of "2" x 5", then x has terminating decimal

   expansion.

  (b) If x = a/b, be a rational number and if the denominator 'b' can't be

   expressed in the form of "2" x 5", then x has a Non terminating

   decimal expansion.