Rational Numbers

• Introduction & Definition

  • Form: Expressed as $\frac{p}{q}$
    • $p$ and $q$ are Integers
    • Denominator $q \neq 0$
  • Types:
    • Positive: e.g., $\frac{2}{3}, \frac{5}{9}$
    • Negative: e.g., $-\frac{4}{7}, \frac{3}{-8}$
    • Zero: $0$ is rational ($0 = \frac{0}{1}$)
  • Standard Form: $q$ is positive; $p$ and $q$ have HCF = 1

• Properties of Rational Numbers

  • Closure Property
    • Applicable: Addition, Subtraction, Multiplication
    • Not Applicable: Division (if divided by 0)
  • Commutative Property
    • Addition: $a + b = b + a$
    • Multiplication: $a \times b = b \times a$
    • Not Applicable: Subtraction, Division
  • Associative Property
    • Addition: $a + (b + c) = (a + b) + c$
    • Multiplication: $a \times (b \times c) = (a \times b) \times c$
    • Not Applicable: Subtraction, Division
  • Distributive Property
    • Over Addition: $a(b + c) = ab + ac$
    • Over Subtraction: $a(b - c) = ab - ac$

• Identities and Inverses

  • Additive Identity
    • Value: 0
    • Rule: $a + 0 = a$
  • Multiplicative Identity
    • Value: 1
    • Rule: $a \times 1 = a$
  • Additive Inverse (Negative)
    • Inverse of $\frac{a}{b}$ is $-\frac{a}{b}$
    • Result: $\frac{a}{b} + (-\frac{a}{b}) = 0$
  • Multiplicative Inverse (Reciprocal)
    • Inverse of $\frac{a}{b}$ is $\frac{b}{a}$
    • Result: $\frac{a}{b} \times \frac{b}{a} = 1$
    • Note: 0 has no reciprocal

• Representation on Number Line

  • Reference Point: 0 (Origin)
  • Positive numbers: To the Right of 0
  • Negative numbers: To the Left of 0
  • Step: Divide each unit length into equal parts equal to the Denominator

• Rational Numbers Between Two Rational Numbers

  • Quantity: Infinitely many rational numbers exist between any two given rational numbers
  • Method 1 (Common Denominator):
    • Convert to Equivalent Fractions with same denominator
  • Method 2 (Mean Method):
    • Find average of two numbers: $\frac{a + b}{2}$