Background

Peano Postulates 29.1:139 (refers to Dodge, section 29.1, page 139)

  [1] 1 N
  [2] Each x N has a unique sucessor x'N
  [3] If x, y N and x' = y' then x = y
  [4] 1 is not the successor of any element of N
  [5] The only natural numbers are those given by [1] and [2]
        [5] is due to S. C. Kleene and is equivalent to Dodge's [6].
  [6] If S N such that 1 S, and k S (k+1) S, then S = N     N8:110
  [7] Definition of Addition: Let a + 1 = a' and a + b' = (a + b)'
  [8] Definition of Multiplication: Let a1 = a and ab' = ab + a     29.3:140
  [9] CLOSURE +     a, b N, a + b is defined     29.4:140
[10] CLOSURE     a, b N, ab is defined     29.5:140
[11] INDEPENDENCE     The Peano postulate set is independent     29.6:140
[12] TRICHOTOMY     For given a, b N, exactly one of the following three statements is true:
          a + x = b       a = b       a = b + y       for some x N or some y N     N7:110

The following properties of Natural Numbers can also be proved:

[13] COMMUTATIVE +     If a, b N, then a + b = b + a
[14] COMMUTATIVE     If a, b N, then ab = ba
[15] ASSOCIATIVE +     If a, b, c N, then (a + b) + c = a + (b + c)
[16] ASSOCIATIVE     If a, b, c N, then (ab)c = a(bc)
[17] DISTRIBUTIVE     if a, b, c N, then a(b + c) = ab + ac
[18] IDENTITY     1 N such that n N, n1 = 1n = n

Extension to integers (Z):

[19] Definition: Let Z' denote the set of all ordered pairs (a,b) of natural numbers a and b     32.1:154
[20] Definition: We define two elements (a,b) and (c,d) of Z' to be equivalent, and we write (a,b) (c,d),
      iff a + d = b + c     32.2:154
Dodge proves the preceding definition satisfies reflexive, symmetric, and transitive properties and is therefore an equivalence relation.     32.3:154
[21] Definition: The sum of members (a,b) and (c,d) of Z' is defined to be the ordered pair (a + c, b + d)     32.4:155
[22] Definition: The product of members (a,b) and (c,d) of Z' is defined to be the ordered pair (ac + bd, ad + bc)     32.5:155
[23] Definition: Let the set of all equivalence classes of Z' be called integers and be denoted by Z     32.10:156

Dodge proves the following theorems:

[24] CLOSURE, COMMUTATIVE, ASSOCIATIVE +
      Z is closed under addition, which is commutative and associative.     33.1:158
[25] CLOSURE, COMMUTATIVE, ASSOCIATIVE
      Z is closed under multiplication, which is commutative and associative.     33.2:158
[26] IDENTITY +     The equivalence class of (m,m) is the additive identity and is unique.
      Denote this equivalence class by "0".     33.3-5:159
[27] INVERSE +     Each z Z has a unique additive inverse, -z, such that z + (-z) = 0     33.12-6:160

Extension to Rational Numbers (Q)

[28] Definition: Denote by Q' the set of all ordered pairs [a,b] of integers a and b with b 0     38.1:179
[29] Definition: We define two elements [a,b] and [c,d] of Q' to be equivalent, and write:
      [a,b] [c,d] iff ad = bc     38.2:179
      Dodge shows this is an equivalence relation.     38.3:179
[30] Definition: The sum of members [a,b] and [c,d] of Q' is defined to be the ordered pair [ad + bc, bd].     38.4:180
[31] Definition: The product of the members [a,b] and [c,d] of Q' is defined to be the ordered pair [ac,bd]     38.5:180
[32] Definition: Let the set of all equivalence classes of Q' be called rational numbers and be denoted by Q     38.9:180

      Dodge shows that Q satisfies all properties of a field:

[33] CLOSURE +
[34] CLOSURE
[35] COMMUTATIVE +
[36] COMMUTATIVE
[37] ASSOCIATIVE +
[38] ASSOCIATIVE
[39] IDENTITY +
[40] IDENTITY
[41] INVERSES +
[42] INVERSES
[43] DISTRIBUTIVE PROPERTY

[44] Definition: For given x, y Q, if there is a unique d Q, such that x = y + d, then we define d = x - y to be the difference of x and y. The operation is called subtraction.     40.2:184
Dodge shows this is equivalent to the definition:
[45] x - y = x + (-y)
[46] Definition: For given x, y Q, if there is a unique q Q, such that x =yq, then we define q = x y to be the quotient of x and y. We also write q = x/y. The operation is called division.
Dodge shows this is equivalent to the definition:
[47] If y 0, then x y = xy*, where y* is the multiplicative inverse of y.

Extension to real numbers ()

Q can be extended to the field of real numbers, , in many ways. The exact method, whether by Cauchy sequences, Dedekind Cuts, or other method, is not of importance here. The important thing to note is that satisfies all field properties [33] - [43] as did Q. The definitions of subtraction [45] and division [47] are also relevant.