axiom field

axiom field

The real numbers, along with the operations of addition and multiplication, satisfy the following axioms. [1] [2]. Let x, y and z are members of the set of real numbers R, and the operation of x + y is the sum, and xy is the product. then:

     Axiom 1 (commutative law): x + y = y + x and xy = yx
     Axiom 2 (associative law): x + (y + z) = (x + y) + z and x (yz) = (xy) z
     Axiom 3 (distributive law): x (y + z) = (xy + xz)
     Axiom 4: Existence of identity elements. There are two distinct real numbers, denoted as 0 and 1, so that for any real number x we get 0 + x = x and 1.x = x.
     Axiom 5: The existence of a negative, or inverse of the sum. For any real numbers x, y are real numbers so that x + y = 0. We can also represent y as-x.
     Axiom 6: The existence of reciprocal, or inverse to multiplication. For any real number x is not equal to 0, there is a real number y so that xy = 1. We can represent y as 1 / x.

The set that satisfies these properties is called a field, and therefore the above axiom named as axiomatic field.