Notes on Rational Numbers

RATIONAL NUMBERS

1.1 Introduction

  • In mathematics, equations often need to be solved using different types of numbers.
  • Example Equations:
    • Equation (1): x + 2 = 13; Here, x = 11 (natural number)
    • Equation (2): x + 5 = 5; Here, x = 0 (whole number), but not solvable by natural numbers.
    • Equation (3): x + 18 = 5 requires x = -13 (not a whole number).
  • To solve such equations, we expand our number system to integers (including both positive and negative numbers).
  • Example Equations (4) and (5) highlight the need for rational numbers:
    • Equation (4): 2x = 3 requires x = 3/2
    • Equation (5): 5x + 7 = 0 requires x = -7/5
  • This leads us to rational numbers.

1.2 Properties of Rational Numbers

1.2.1 Closure
(i) Whole Numbers
  • Addition: Whole numbers are closed: a + b is a whole number (0 + 5 = 5).
  • Subtraction: Not closed (5 - 7 = -2, not a whole number).
  • Multiplication: Closed (0 x 3 = 0).
  • Division: Not closed (5 ÷ 8 = 5/8, not a whole number).
(ii) Integers
  • Addition: Closed (−6 + 5 = −1).
  • Subtraction: Closed (7 - 5 = 2, but 5 - 7 = -2, still an integer).
  • Multiplication: Closed (5 × 8 = 40).
  • Division: Not closed (5 ÷ 8 = 5/8, not an integer).
(iii) Rational Numbers
  • Defined as: p/q where p and q are integers, and q ≠ 0.
  • Example: 2/3, 6/7, -5/9 are rational.
  • Addition: Closed (sum of two rational numbers is rational).
  • Subtraction: Closed (difference of two rational numbers is rational).
  • Multiplication: Closed (product is rational).
  • Division: Not closed (a ÷ 0 is undefined but is closed otherwise).
1.2.2 Commutativity
(i) Whole Numbers
  • Addition: a + b = b + a (commutative)
  • Subtraction: Not commutative.
  • Multiplication: a × b = b × a (commutative).
  • Division: Not commutative.
(ii) Integers
  • Addition: Commutative
  • Subtraction: Not commutative
  • Multiplication: Commutative
  • Division: Not commutative
(iii) Rational Numbers
  • Addition: Commutative (a + b = b + a).
  • Subtraction: Not commutative (a - b ≠ b - a).
  • Multiplication: Commutative (a × b = b × a).
  • Division: Not commutative.
1.2.3 Associativity
(i) Whole Numbers
  • Addition: Associative (a + (b + c) = (a + b) + c).
  • Subtraction: Not associative.
  • Multiplication: Associative.
  • Division: Not associative.
(ii) Integers
  • Similar explorations confirm associative properties.
(iii) Rational Numbers
  • Addition: Associative
  • Subtraction: Not associative
  • Multiplication: Associative
  • Division: Not associative
1.2.4 The Role of Zero (0)
  • Zero acts as an identity for addition: a + 0 = a for all types of numbers.
1.2.5 The Role of 1
  • One acts as an identity for multiplication: a × 1 = a for all types of numbers.
1.2.6 Distributivity of Multiplication Over Addition
  • For all rational numbers a, b, and c:
    • a(b + c) = ab + ac
    • Example with rational numbers demonstrates this.

Summary

  1. Closure under addition, subtraction, and multiplication for rational numbers.
  2. Addition and multiplication are both commutative and associative for rational numbers.
  3. Zero is the additive identity, while one is the multiplicative identity for rational numbers.
  4. Distributive property holds for rational numbers: a(b + c) = ab + ac.
  5. Infinite rational numbers exist between any two rational numbers.