In programming languages like C++, a class or object has its own data and procedures or methods that use its own data.
Natural numbers, signed integers as a sign and a natural number, rational numbers as a pair of signed integers are examples of objects with internal data. Procedures or methods include operators such as addition or multiplication.
Natural numbers from the Peano Axioms are not signed. They are 0, 1, 2, etc. They are not positive, they are without sign.
Exercise: Elaborate on puns and other humor. Make them help teach the lesson.
Rational numbers can be defined before sign as ordered pairs of two natural numbers with rules for manipulating a single ordered pair or two ordered pairs. We can call these unsigned or presign rational numbers.
We can then define integers as an ordered pair of a sign and a size. (+,n) and (-,n). We then define rules for these.
If n is non-zero, then
P((-n,n)) = (-,n’)
or using prime notation for predecessor
‘(-,n) = (-,n’)
The predecessor of the negative number -n, is -(n’) where n’ is the successor of the natural number n.
(-,n)’ = (-,n’) if n is non-zero.
(+,n)’ = (+,n’)
‘(+,n) = (+,’n) if n is non-zero.
Exercise: Work out the rules for the case that n is 0. So ‘(-,0) = ?, etc.
Euler has the rules for fractions in his 1765 Elements of Algebra book.
Grassmann and Dedekind developed the rules for natural numbers using ‘ or successor. Predecessor was added as a function later.
Rational numbers come before area. Area as a way to define rational numbers was attempted by Euler in 1765, but only as a throw away paragraph. It has no actual rules linked to geometry for fractions or any real definition of fraction’s properties based on geometry. Instead, Euler gives the rules for fractions using algebra without any reference back to his invoking a line segment to justify the concept of a fraction.
The same applies in Khan Academy videos or modern school education. They may invoke geometry to justify fractions, but the rules for fractions they give are not based on any geometric argument. They are thus a false clue and only confuse students. As we know, the weaker students are harmed more by this than the stronger students.