__CHAPTER 1 REAL NUMBERS__Theorem 1: Euclid’s Division Lemma

- Given positive integers a and b, there exists a unique integers q and r satisfying a = bq + r, 0 ≤ r < b

**Euclid’s division algorithm**

To obtain the HCF of two positive integers, say c and d, with c > d, follow the steps below:

Step 1: Apply Euclid’s division lemma, to c and d. So, we find whole numbers, q and r such that c= dq + r, 0 ≤ r < d.

Step 2: If r=0, d is the HCF of c and d. If r ≠ 0, apply the division lemma to d and r.

Step 3: Continue the process till the remainder is zero. The divisor at this stage will be the required HCF.

Theorem 2: 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.

Theorem 3:

- Let p be a prime number. If ‘p’ divides ‘a^2’, the ‘p’ divides ‘a’, where ‘a’ is a positive integer.

Theorem 4:

- √2 is irrational.

Theorem 5:

- Let x be a rational number whose decimal expansion terminates. Then x can be expressed in the form of p/q, where p and q are co-prime, and the prime factorisation of q is of the form of 2^n5^m, where n and m are non-negative integers.

Theorem 6:

- Let x = p/q be a rational number, such that the prime factorisation of q is of the form of 2^n5^m, where n and m are non-negative integers. Then x has a decimal expansion which terminates.

Theorem 7:

- Let x = p/q be a rational number, such that the prime factorisation of q is not of the form of 2^n5^m, where n and m are non-negative integers. Then x has a decimal expansion which is non-terminating repeating (recurring).