The rational number 4/1 as a set

One way to define a rational number (a-b)/c is to define it as the (infinite) set of all 3-tuples of natural numbers (a₁, b₁, c₁) for which (a-b)/c = (a₁-b₁)/c₁ (b is needed for negative numbers).

Klick on a 3-tuple to see how it may be defined as a set.

The equation (a-b)/c = (a₁-b₁)/c₁ is equivalent to a·c₁ + b₁·c = a₁·c + b·c₁ - so only addition and multiplication of natural numbers are needed to define the rational numbers.

For rational numbers Q, Q₁ as defined above, Q < Q₁ is defined as a·c₁ + b₁·c < a₁·c + b·c₁ for one/all (a, b, c) ∈ Q, (a₁, b₁, c₁) ∈ Q₁.

Q + Q₁ is defined as (a₂-b₂)/c₂, where a₂ = a·c₁ + a₁·c, b₂ = b·c₁ + b₁·c, c₂ = c·c₁ for one/all (a, b, c) ∈ Q, (a₁, b₁, c₁) ∈ Q₁.

Be aware that (a₂-b₂)/c₂ is simply a notation for the set determined by a₂, b₂ and c₂ here - not an expression using subtraction and division.

The definition for Q + Q₁ above simply is a transformation of the expression (a-b)/c + (a₁-b₁)/c₁.

Assuming that a,c is minimal for a positive rational number a/c or (a-0)/c, we can enumerate all members of the set by doing this:

Let n be 1
Repeat:
  For all n1 from 0 to n-1:
    Let a1 be n·a
    Let b1 be n1·a
    Let c1 be (n-n1)·c
    Enumerate (a1,b1,c1)
  Increase n by 1

The enumeration as Python function with a limiting parameter k which will cause the function to enumerate (k·(k+1))/2 elements of a/c:

def print_rational_number(a,c,k):
    print str(a)+'/'+str(c)+' = ('+str(a)+'-0)/'+str(c)+' = {'
    for n in range(1,k+1):
        for n1 in range(n):
            a1=n*a
            b1=n1*a
            c1=(n-n1)*c
            print '( '+str(a1)+', '+str(b1)+', '+str(c1)+' ),'
        print
    print "..."
    print "}"

(back to √2)