# Number systems

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## Axiomatic development

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

- Natural numbers
- Whole Numbers: All counting numbers together with 0 are called whole numbers.
- Integers
- Rational numbers
- Real numbers
- 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.