What is the definition of number system?

Answer 1

The definition of the number system starts with the 5 Peano axioms. These are:

• Zero is a number.
  
• If a is a number, the successor of a is a number.
  
• Zero is not the successor of a number.
  
• Two numbers of which the successors are equal are themselves equal.
  
• If a set S of numbers contains zero and also the successor of every number in S, then every number is in S. (The induction axiom).
  

This defined the set of Natural numbers. The set of real numbers is a mathematical structure known as a field which has the following properties:

For any three real numbers x, y and z and the operations of addition and multiplication,

• x + y belongs to R (closure under addition)
  
• (x + y) + z = x + (y + z) (associative property of addition)
  
• There is an element, 0, in R, such that x + 0 = 0 + x = x (existence of additive identity)
  
• There is an element, -x, in R, such that x + (-x) = (-x) + x = 0 (existence of additive inverse)
  
• x + y = y + x (Abelian or commutative property of addition)
  
• x * y belongs to R (closure under multiplication)
  
• (x * y) * z = x * (y * z) (associative property of multiplication)
  
• There is an element, 1, in R, such that x * 1 = 1 * x = x (existence of multiplicative identity)
  
• For every non-zero x, there is an element, 1/x, in R, such that x * 1/x = 1/x * x = 1 (existence of multiplicative inverse)
  
• x * (y + z) = x * y + x * z (distributive property of multiplication over addition)
  

Answer 2

Any system of naming or representing numbers, as the decimal system or the binary system. Also called numeral system.

Answer 3

A set of numbers, with certain operations defined for the set.