number system

• Number line


  Representation of various types of numbers
• There are infinitely many numbers on the number line
• Types of numbers


  • Natural numbers
  • Whole numbers
  • Integers
  • Rational numbers
• Natural numbers


  Positive integers starting from 1
• Whole numbers


  Natural numbers including 0
• Integers


  Whole numbers including negative numbers
• Rational numbers


  Numbers that can be expressed as a ratio of two integers (p/q where q ≠ 0)
• Rational numbers include natural numbers, whole numbers, and integers
• Rational numbers do not have a unique representation in the form p/q
• Rational numbers between 1 and 2

  • 3/2
  • 5/4
  • 11/7
  • 13/8
  • 7/4
• There are infinitely many rational numbers between any two given rational numbers
• Irrational numbers


  Numbers that cannot be expressed as a ratio of two integers
• Irrational numbers were first discovered by the Pythagoreans around 400 BC
• Irrational numbers


  • √2
  • √3
  • √15
  • π
  • 0.10110111011110...
• There are infinitely many irrational numbers
• Real numbers


  The collection of all rational and irrational numbers
• Every real number is represented by a unique point on the number line, and every point on the number line represents a unique real number
• Locating √2 on the number line

  1. Draw a square with side length 1

  2. Use Pythagorean theorem to find the length of the diagonal (√2)

  3. Transfer the square onto the number line with one vertex at 0
• Real number


  A number represented by a unique point on the number line
• Every point on the number line represents a unique real number
• The number line is called the real number line
• German mathematicians Cantor and Dedekind showed that every real number corresponds to a point on the real number line, and every point on the number line corresponds to a unique real number


  1870s
• G. Cantor


  • 1845-1918
• R. Dedekind


  • 1831-1916
• Locating irrational numbers on the number line

  1. Draw a square with side length 1 unit

  2. Use Pythagorean theorem to find length of diagonal

  3. Transfer diagonal length to number line using compass
• Locating √2 on the number line

  • Draw square with side 1 unit
  • Diagonal length is √2
  • Use compass to mark √2 on number line
• Locating √3 on the number line

  • Draw square with side 1 unit
  • Construct perpendicular line of unit length
  • Use Pythagorean theorem to find diagonal length is √3
  • Use compass to mark √3 on number line
• You can locate √n for any positive integer n on the number line after locating √(n-1)
• Statements to determine if true or false

  • Every irrational number is a real number
  • Every point on the number line is of the form √m, where m is a natural number
  • Every real number is an irrational number
• Not all square roots of positive integers are irrational
• Representing √5 on the number line

  1. Draw square with side 1 unit

  2. Use Pythagorean theorem to find diagonal length is √5

  3. Use compass to mark √5 on number line
• Decimal expansions can be used to distinguish rational and irrational numbers
• Terminating decimal expansion


  Decimal expansion ends after a finite number of steps
• Non-terminating recurring decimal expansion

  Decimal expansion goes on forever with a repeating block of digits
• Decimal expansion of a rational number is either terminating or non-terminating recurring
• A number with a terminating or non-terminating recurring decimal expansion is rational
• Non-terminating non-recurring decimal expansion


  Decimal expansion goes on forever without repeating
• A number with a non-terminating non-recurring decimal expansion is irrational
• The decimal expansions of √2 and π are non-terminating non-recurring
• Rational number


  A number whose decimal expansion is either terminating or non-terminating recurring