Fractions as functions on subsets of natural numbers

We can consider the fraction 1/2 to be a function with the domain of even natural numbers.  At this point we don’t do signs.  So the numbers are pure natural numbers.  We consider these a type of size.

If we think of natural numbers as in factored form, then what the function 1/2 does is remove one of the factors of 2.  The function has domain limited to the even natural numbers, so the are either 0 or have a factor 2 in them.

So 1/2 as ordered pairs is (0,0), (2,1),(4,2),(8,4), etc.

The input followed by the output.

The function 1/3 has as domain the numbers with a 3 in their factorization and we include 0 as well.  The graph of a function is the set of ordered pairs. So the graph of the function 1/3 is

(0,0), (3,1),(6,2),(9,3), etc.

An operator is simply a function defined on some set of test data.  So a fraction as a function is the same as a fraction as an operator.  Here same is same, not just equal.

 

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About New Math Done Right

Author of Pre-Algebra New Math Done Right Peano Axioms. A below college level self study book on the Peano Axioms and proofs of the associative and commutative laws of addition. President of Mathematical Finance Company. Provides economic scenario generators to financial institutions.
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