Rational Numbers: Representation and Comparison

Representation of Rational Numbers on the Number Line

• Number Line Visualization

  • Zero at the center: separates positive and negative numbers
  • To the right: Positive numbers
  • To the left: Negative numbers

• Plotting Rational Numbers

  • Example: Representing -3/2, -1/2, 1/2, 3/2, 5/2
    • Draw the number line: Mark center as 0, right is positive, left is negative
    • Divide each unit into equal parts as needed, e.g., halves for /2 denominators
  • Repeat for different denominators by dividing units appropriately

Comparing Rational Numbers

• Concepts

  • Positive rational numbers > 0 and > all negative rational numbers
  • Negative rational numbers < 0 and < all positive rational numbers
  • Zero > any negative number and < any positive number

• Methods to Compare

  • Method 1: Using LCM
    • Equalize denominators and compare numerators
  • Method 2: Cross Multiplication
    • Cross-multiply and compare results

Examples

• Compare 5/7 and -3/5

  • Clear that 5/7 (positive) > -3/5 (negative)

• Compare 3/5 and 5/7

  • Use LCM to equalize denominators, then compare

• Compare 3/7 and -8/15

  • Cross-multiplication method shows which is greater

Inserting Rational Numbers Between Two Given Numbers

• Numerous rational numbers can be inserted between any two rational numbers
• Example: Insert numbers between 2/5 and 3/7
  • Find LCM to equalize denominators, then find numbers between them

Exercises

Exercise 2B

• Plot on the Number Line

  • Example: 2/5 and -4/5
    • Divide into 5 equal parts and mark positions
  • Example: 1/4 and -5/4
    • Divide into 4 equal parts and mark positions

• Comparing Pairs

  • Example: Compare -7/2 and 5/2
    • Directly noting that negative is always less than positive

Exercise 2C

• Addition

  • Same denominators: directly add numerators
  • Different denominators: find LCM, adjust fractions, then add

• Subtraction

  • Same denominators: directly subtract numerators
  • Different denominators: find LCM, adjust fractions, then subtract

• Word Problems

  • Determine the unknown in rational number addition and subtraction scenarios

Summary

• Representation of rational numbers on the number line
• Comparison rules and methods
• Insertion of numbers between given rational numbers
• Practical exercises in plotting, comparing, adding, and subtracting rational numbers
• Word problems to apply understanding

Note: Practice is essential for mastering rational number operations and comparisons.