Real Numbers

1. To prove that root 2, root 3 are irrationals.

2. Let x be a rational number whose decimal expansion terminates. Then we can express x in the form p/q, where p and q are coprime, and the prime factorisation of q is of the form 2^n 5 ^m, where n, m are non-negative integers.

3. Let x = p /q be a rational number, such that the prime factorisation of q is of the form 2^n 5^m, where n, m are non-negative integers. Then x has a decimal expansion which terminates.

4. Let x = p /q be a rational number, such that the prime factorisation of q is not of the form 2^n 5^m, where n, m are non-negative integers. Then x has a decimal expansion which is non-terminating repeating (recurring).

5. . Euclid’s division algorithm : This is based on Euclid’s division lemma. According to this, the HCF of any two positive integers a and b, with a > b, is obtained as follows: Step 1 : Apply the division lemma to find q and r where a = bq + r, 0 ≤ r < b. Step 2 : If r = 0, the HCF is b. If r ≠ 0, apply Euclid’s lemma to b and r. Step 3 : Continue the process till the remainder is zero. The divisor at this stage will be HCF (a, b). Also, HCF(a, b) = HCF(b, r).

6. The Fundamental Theorem of Arithmetic : Every composite number can be expressed (factorised) as a product of primes, and this factorisation is unique, apart from the order in which the prime factors occur. 4. If p is a prime and p divides a^2, then p divides a, where a is a positive integer.