Steps from fractions as operators to fractions as numbers.
- Natural numbers are developed with all their logical structure including successor function and functions as ordered pairs. Students are made familiar with proofs involving functions as ordered pairs including the recursion theorem which is the basis of the addition and multiplication functions on natural numbers.
- Fractions are introduced as multiplicative operators on natural numbers which are a multiple of the denominator in reduced form.
- Addition, multiplication, subtraction, and division of fractions is defined in terms of the resulting operators.
- The algebra of fractions as operators is worked out.
- Fractions are found to have some superior properties to naturals. For example division is always defined.
- Naturals are shown to be isomorphic to a subset of fractions as operators.
- The need for naturals as test data is dropped. Fractions as operators can be defined using their own internal data as ordered pairs and operations directly on them.
- The fractions as object oriented ordered pairs are recognized as suitable to be used as a concept.
- These are called rational numbers.
- Rational numbers are applied to geometric sets and seen to have better properties such as always one in between two fractions.
By working through naturals in a logical structured way, the machinery to do rationals is already set up. Functions as sets of ordered pairs sets up rationals.
It is also possible to define addition of functions as addition of the ordered pairs, and multiplication of them in various ways.
Fractions as operators can be redone using fractions as ordered pairs and thus operators as functions. This reinforces this approach by two paths leading to the same result.
This sets up other structures such as real numbers and complex numbers.
The current approach of throwing applications and partial concepts at students is greater in volume and is not well organized. We know it leaves students confused as to what fractions really mean. The same confusion was carried over from ancient times.