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Creation of fractions

Transform

Consider the transform of the set of natural numbers

\begin{displaymath}1, 2, 3, 4, 5. \ldots \end{displaymath}

(called the original or primer set) by the function

\begin{displaymath}
f(x) = 2 x \end{displaymath}

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

\begin{displaymath}2, 4, 6, 8, 10, \ldots \end{displaymath}

Complement

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

\begin{displaymath}Z= \{ 1, 2, 3, 4, 5, 6, 7, 8, \ldots \} \end{displaymath}

Inverse Transform

Now apply the inverse transform

\begin{displaymath}
f^{-1}(x) = { x \over 2} \end{displaymath}

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

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


next up previous
Next: Creation of negative numbers Up: Expansion by inversion Previous: Expansion by inversion
Andre Cherkaev
2001-11-16