Lecture on Rational Numbers

Jul 4, 2024

Lecture on Rational Numbers

Natural Numbers and Integers

  • Introduced natural numbers and integers
  • Division leads to the concept of rational numbers

Rational Numbers

  • Cannot represent 19/5 as an integer
  • Represented as: 19/5 = 3 and 4/5
  • Rational numbers are fractions: p/q, where p and q are integers
  • Numerator: p
  • Denominator: q
  • Symbol for rational numbers: Q
    • Written with double lines on sides: đť’¬

Properties of Rational Numbers

  • Same rational number can be written in multiple ways
  • Example: 3/5 is equivalent to:
    • 6/10
    • 30/50
  • To compare or add rational numbers, convert them to have the same denominator

Adding Rational Numbers

  • Example: 3/5 + 3/4
    • 3/5 = 12/20
    • 3/4 = 15/20
    • Result: 12/20 + 15/20 = 27/20

Comparing Rational Numbers

  • Convert to common denominator for comparison
  • Example: 3/5 vs. 3/4
    • 3/5 = 12/20
    • 3/4 = 15/20
    • Since 12/20 < 15/20, 3/5 < 3/4
    • Can use larger common multiples as well
  • Reduced form of rational number:
    • No common factors between numerator and denominator
    • Example: 18/60 reduced to 3/10

Greatest Common Divisor (GCD)

  • Find gcd for reduction:
    • Example: gcd(18, 60) = 6
    • Prime factorization method

Density and Discreteness

  • Integers have a 'next' and 'previous' property
    • Example: For 22, next is 23, previous is 21
  • Rational numbers don’t have next or previous
    • Reason: Between any two rational numbers, there’s another rational number (average of the two)
    • Rational numbers are dense
  • Integers and natural numbers are discrete

Summary

  • Rational numbers (Q) come from the word 'ratio'
  • Can be represented in multiple forms
  • Reduced form is used to standardize
  • No next/previous concept like integers devido a density