Creation of fractions

Transform

Consider the transform of the set of natural numbers

$$1, 2, 3, 4, 5. \ldots$$

(called the original or primer set) by the function

$$f(x) = 2 x$$

The set of images (transforms) is the set of even numbers:

$$2, 4, 6, 8, 10, \ldots$$

Complement

The set of images (even numbers) is naturally complemented to the set $Z$ of all natural numbers:

$$Z= \{ 1, 2, 3, 4, 5, 6, 7, 8, \ldots \}$$

Inverse Transform

Now apply the inverse transform

$$f^{-1}(x) = { x \over 2}$$

to the complemented set $Z$ of images

Notice that the solution does not exist if only integers are considered. New objects - fractions - are introduced to resolve the contradiction.

Fractions!
We arrive at new (``super'' natural) numbers: These are the simplest fractions

$$f^{-1}(Z) =\{ {1 \over 2}, 1,{3 \over 2}, 2, {5 \over 2}, 3,{7 \over 2}, 4,\ldots \}$$