CBSE.mocha

Grade 10 - Mathematics

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 2n5m, 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 2n5m, 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 2n5m, 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