Rational Numbers

• Introduction & Need
  • Beyond Natural Numbers, Whole Numbers, and Integers
  • Definition: Numbers in the form $p/q$
    • $p$ and $q$ are integers
    • $q \neq 0$
  • Numerator ($p$) and Denominator ($q$)
• Types of Rational Numbers
  • Positive Rational Numbers: Both numerator and denominator have the same sign (both positive or both negative)
  • Negative Rational Numbers: Numerator and denominator have opposite signs
  • Zero ($0$): Neither positive nor negative, but a rational number ($0/1$)
• Standard Form (Lowest Form)
  • Condition 1: Denominator $q$ is positive
  • Condition 2: HCF of $|p|$ and $q$ is $1$ (no common factor other than 1)
  • Method: Divide numerator and denominator by their HCF
• Representation on a Number Line
  • Line extends infinitely in both directions
  • Positive numbers to the right of zero
  • Negative numbers to the left of zero
  • Dividing segments based on the denominator
• Rational Numbers Between Two Rational Numbers
  • Infinite rational numbers exist between any two rational numbers
  • Method: Make denominators equal (using LCM), then find numbers between numerators
• Operations on Rational Numbers
  • Addition
    • With same denominators: Add numerators, keep denominator same
    • With different denominators: Take LCM, convert to like fractions, then add
  • Subtraction
    • Add the additive inverse of the second rational number to the first
  • Multiplication
    • Multiply numerator by numerator
    • Multiply denominator by denominator
  • Division
    • Multiply the first rational number by the reciprocal (multiplicative inverse) of the second
• Important Properties
  • Additive Inverse: For $p/q$, it is $-p/q$ (Sum is $0$)
  • Multiplicative Inverse (Reciprocal): For $p/q$, it is $q/p$ (Product is $1$)