CHAPTER 1 : Real Numbers

Revision Checklist

1. Introduction
2. Euclid's Division Lemma
• Theorem 1.1 : Euclid's Division Lemma
  Given positive integers a and b, there exists unique integers q and r satisfying
  a = bq + r, 0 ≤ r ≤ b.
• Definitions:
  Algorithm : An algorithm is a series of well defined steps which gives a procedure for solving a type of problem.
  Lemma : A lemma is a proven statement used for proving another statement.
• Euclid's Division Algorithm
• 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.
• Exercise 1.1
3. The Fundamental Theorem of Arithmetic
• Theorem 1.2 : 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.
  OR
  The prime factorisation of a natural number is unique, except for the order of its factors.
• For any two positive integers a and b,
  HCF (a, b) × LCM (a, b) = a × b.
• Exercise 1.2
4. Revisiting Irrational Numbers
• Definitions:
  Irrational Number : A number s is called irrational if it cannot be written in the form, p⁄q where p and q are integers and q ≠ 0.
  Examples: √2, √3, π, √2⁄3, 0.1011011101110…, etc.
• Theorem 1.3 : Let p be a prime number. If p divides a², then p divides a, where a is a positive integer.
  Check out the explanation and proof in the linked video.
• Theorem 1.4 : √2 is irrational.
  Check out the explanation and proof in the linked video.
• The sum or difference of a rational and an irrational number is irrational.
• The product and quotient of a non-zero rational and irrational number is irrational.
• Exercise 1.3
5. Revisiting Rational Numbers and Their Decimal Expansions
• Theorem 1.5 : Let x be a rational number whose decimal expansion terminates. Then x can be expressed in the form p⁄q, where p and q are coprime, and the prime factorisation of q is of the form 2ⁿ5^m, where n, m are non-negative integers.
• Theorem 1.6 : Let x = p⁄q be a rational number, such that the prime factorisation of q is of the form 2ⁿ5^m, where n, m are non-negative integers. Then x has a decimal expansion which terminates.
• Theorem 1.7 : Let x = p⁄q be a rational number, such that the prime factorisation of q is not of the form 2ⁿ5^m, where n, m are non-negative integers. Then x has a decimal expansion which is non-terminating repeating (recurring).
• The decimal expansion of every rational number is either terminating or non-terminating repeating
• Exercise 1.4