CHAPTER 1 REAL NUMBERS
Theorem 1: Euclid’s Division Lemma
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).