Overview of Number Construction
Overview of Number Construction
The goal is sequential construction of number systems in School arithmetic,
- Natural to Integer to Rational ,
in a reasoned process involving only numbers themselves, and then understanding the architecture of that process.
A large fraction of American students emerges from School arithmetic not understanding negative numbers and fractions. Generally they have been taught to look for “real world” objects that are intrinsically “negative” or “partial”, and then to understand numbers in terms of such objects.
I assume this approach is taken over from understanding natural numbers as enumerators of discrete objects. This works because we all agree on what discreteness/oneness/individual is, and we can base our understanding of natural numbers on that. This is not the case for “negative” or “fraction”.
To use negative numbers or fractions we must understand the numbers themselves. Only in accommodating academic environments can we get away with knowing numbers as associations with objects.
Calculation can ultimately rely on rules for negative numbers and fractions, but a real problem requires formulation. Formulation relies on knowing properties of numbers because those properties must be matched to roles of numbers assigned by the solver to the problem.
The properties of negative numbers and fractions are installed in Integers and Rationals during their construction.
The signed property of integers arises from the two sorts of difference between two natural numbers; there is a “positive” difference, and there is a “negative” difference,
-6 = (0,6), and a “positive” difference, +6 = (6,0).
Natural numbers are ordered, this renders the differences 0-to-6 and 6-to-0 distinguishable: the difference 0-to-6(6-to-0) is positive(negative) because 6(0) is greater-than(less-than) 0(6).
Since 0 in the pair notation is common to all the Integers, it can can be dropped leaving us with the usual notation: +6 and -6. When the positive sign, “+”, is dropped, this can suggest a positive Integer is a Natural number, and a negative number is “special”. This is not the case.
It is here that School arithmetic, including Common Core Standards, makes the mistake leading to incorrect understanding of negative numbers: it asks students to believe there is an intrinsic “negativeness”, like there is an an intrinsic “discreteness” associated with Natural Numbers. Negative Integers were discovered within mathematics itself and then “negativeness” was sought. It has not been found, although from the 17th century on some have thought it would be.
Negative numbers are not Natural numbers with a sign prepended, and Natural numbers are not positive, they are just unsigned. Integers, positive and negative, are signed differences from, or to, zero. The Integer Number System is its own separate mathematical object with its own properties. So is the Natural Number System.
Signed numbers open a new frontier; with them addition and multiplication can be “subtractive”, in fact “additive” and “subtractive” become the duals of one another, all the mystery of “negative” vanishes.With Integers we have a system of signed numbers without fractional parts.
Rational numbers are the signed numbers with (that can accommodate) fractional parts. A Rational number is a fraction of an Integer number, or, what is the same thing, proportional to an Integer number with a positive proportionality less than one. That is all there is to it, and it is easy to see: for R a Rational number, Z an Integer, and N+ a non-zero Natural number:
- R = Z / N+
= Z x ( 1 / N+ )
= Z x ( N+)-1
Since N+ 1, ( 1 / N+ ) and ( N+)-1 are one of the fractions 1 , 1 / 2, 1 / 3, 1 / 4, ….
Example: Z = -6, and N = 5, then R = -6 / 5; ie. R is one-fifth of minus six, or R is proportional to minus six with one fifth the coefficient of proportionality; these are facts of the construction, but object constructed is a number, -6/5.
The Rational Number System supplies all the numbers, and the arithmetic for them, to value measurements and evaluate problems.
A Fraction can be calculated as the inverse of a non-zero Natural number (or better from the positive Rational N/1 with N a positive Integer). Inversion can be instantiated as a physical process that relies on geometry, measurement, and proportion; not on assumed intrinsic physical properties.