I am looking for a story on continuity and limits that can be told in junior highschool and still makes sense. We would like an isomorphy between space and numbers. For some aspects, mathematical theory sends us to number theory, and for isomorph aspects, mathematical theory sends us to topology. It is awkward to have to translate similar notions, and to eliminate the overload of notions that are not directly relevant for this search for this junior highschool story.
For example, topology has rephrased results into statements on open and closed sets and boundaries, but I am wondering whether that is an effective manner of communication, when the relevant distinction is whether you are assuming a well-ordening or not. But I am not at home in number theory or topology. These comments on continuity and limits have been caused precisely because I am feeling the water.
Addendum December 16
This weblog text is a rewrite of yesterday’s text.
The framework contains both the handling of real numbers on the calculator and a development of theory.
Also in junior highschool, we want students to be aware that 0.999…. = 1.000…. so that these are the same number. You can see this by checking 3 * 1/3 = 1.
When we approximate numbers with n the number of decimals, then these basically are like the natural numbers, and there remains a well ordering. Numbers are δ[n] = 10^(-n) apart.
When we shift to the use of the infinite number of decimals then we lose this “infinitesimal”. At issue is now whether the infinitesimal can be retained in some manner.
Standard definition of density causes contradiction
Discussing the continuum and the set of real numbers R recently, I suggested (here, property (a)) that R would be a dense set, according to the standard definition of density. This definition is that for any two elements x < y there would be at least one z between, as x < z < y. This would allow you to make cuts everywhere.
Oops. I retract.
Wikipedia (no source but a portal) has:
I don’t know quite what to think about this. Elsewhere I deduced that ZFC is inconsistent. But perhaps in a revised set theory, the well order can be retained.
We would like R to have a well-order for finite intervals too. Thus every number x has a next number x’. When you select y = x’ then you couldn’t find anything between x and y. This contradicts above statement on density.
Thus, the standard definition of density doesn’t fit a well-ordered R.
Designing a new definition for density of the reals R
We can design a new definition of density.
- The standard definition is useful for the rationals Q. If we restrict your freedom to making cuts along Q, then we are safe again. In this manner, the distinction between rational and irrational numbers is useful to explain a property of R.
- R is defined as “more dense” because Q is dense w.r.t. that original definition ((a)).
- This proposal is quite similar to the Dedekind cut, with the distinction that we now allow that R might retain a well-ordering. That is, this issue on the ordering is no longer forced by the standard definition of density.
Surprise consequence as a bonus
Switching to another notion of density, generates the bonus that we have more scope to introduce the infinitesimal.
When every number x has a next x’, then we can define the infinitesimal as the difference:
δ = x’ – x
It also means that an open set (a, b) can also be seen as a closed set [a + δ, b – δ].
Wikipedia (no source but a portal) claims today:
Yet now we have (0, 1) = [δ, 1 – δ] and the least element is δ. Only the intervals with negative infinity might be excluded, check (-∞, ∞).
Properties of these infinitesimals
Some properties are:
(1) We still have 0.999…. = 1.000…..
(2) A current statement is that a line consists of points, and each point is a co-ordinate without length. We now can better express that length consists of a sum of short lengths. A sum of these infinitesimals Σδ makes sense if we regard it as the sum from x = 0 to x = 1 for x‘ – x. The trick is that the length is determined by the statement on x and not by the coefficient of δ.
(3) Using H = -1, then δ δH = 1. That is, δ ≠ 0, and thus there is no problem with division. The discussion about differentials is quite different from the discussion about these (new) infinitesimals. Much time has been spent in history in looking whether there might be a connection, but there isn’t.
Separate arithmetic for infinity and infinitesimal
Students already know that they cannot apply the rules of arithmetic to infinity. E.g. ∞ + ∞ = ∞. The same now holds for above hypothetical notion of the infinitesimal.
Property (2) carries over from δ[n] with n the number of decimals. Property (1) arises when n → ∞ . Potentially, these notions cannot be combined without some conflict.
We are accustomed to think that any real number can be divided. But e.g. δ / 2 is nonsense because it gives the distance between two numbers, and there is nothing smaller. Thus, the normal rules for arithmetic only hold for reals that are not these infinitesimals.
With δ = x’ – x we also want to consider y = (x / n). When the numbers are halved for n = 2, is the distance halved or isn’t it halved ? In the approximation δ[n] the distance can become smaller when more digits are included. For an infinite number of digits, presumably, the distance cannot be halved. Thus δ = y’ – y. Multiplication by n gives nδ = n (x / n)‘ – x. Thus x’ = n (x / n)‘ – nδ + δ for any n. This would make (most) sense by the choice δ = 0 and x’ / n = (x / n)‘. But then we are back in the classical approach again, without the well ordering. (The next number is the number itself.)
Persumably, we can argue that n * δ is as problematic as δ / n, though. The notion of Σδ namely has been solved by putting the consideration of length into the Σ sign.
I don’t know yet whether it is sufficient (consistent) to state δ = x’ – x and that the rules for arithmetic don’t apply to δ like they don’t apply to ∞. Potentially, we might write δH → ∞ (and this doesn’t mean that δ ∞ → 1).
All this depends upon whether we can develop a consistent set of definitions. Students at junior highschool might agree that they aren’t much interested in that.
Thus, it might only be in senior highschool, when we discuss the “classical” approach to the reals, that has (a, b) as an open interval only. We would be forced to this not because of the definition of density but because of the rules of arithmetic.
In itself, notions like these are not world shocking but they would tend to fit the intuitions of space and number for junior highschool.
At some point of history, the main stream in mathematics has opted for an approach to the reals so that they have no well ordering. The obstacle of the standard definition of density can be removed, as shown above. A problem still resides in arithmetic with δ = x’ – x. It is not clear to me whether this can be resolved. It is not clear to me neither whether it is okay to have benign neglect till senior highschool, and face the consequences of losing the well ordering only there.