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Axiomatic development

For a more extensive list, please see Number systems/Axiomatic development.

  1. Natural numbers
  2. Whole Numbers: All counting numbers together with 0 are called whole numbers.
  3. Integers
  4. Rational numbers
  5. Real numbers
  6. Complex numbers



  • 0 is the smallest whole number and there is no largest whole number.
  • Our number system is based on counting in tens i.e. it has base 10. Every whole number can be written by using the symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 called digits.
  • The place value of a (non-zero) digit depends upon the place it occupies in the number; the place value of the digit 0 is always 0 regardless of the place it occupies in the number.
  • The face value of a digit in a number is the digit itself, regardless of the place it occupies in the number.
  • Addition properties of whole numbers
    Closure property: If a and b are any whole numbers then a +b is also a whole number.
    Commutative property: If a and b are any whole numbers then a+b= b+a.
    Associative law: If a, b, c are any whole numbers then (a+b)+c = a+(b+c).
  • Multiplication properties of whole numbers
    Closure property: If a and b are any whole numbers then a × b is also a whole number.
    Commutative property: If a and b are any whole numbers then a × b = b × a.
    Associative law: If a, b, c are any whole numbers then (a × b) × c = a × (b × c).
    Distributive law: If a, b, c are any whole numbers then a × (b+c)=a × b + a × c.
  • Division algorithm
    If a is any whole number and b is another smaller non-zero whole number then there exist unique whole numbers q and r such that
    a = b × q + r where 0 r < b.

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