Rationals, Reals and Infinitesimals

A colleague and I were discussing continuous random variables the other day (yes I know), it occurred to me that the problem students have with this is understanding that the probability of any one value is zero and yet all these zeros add up to one!  This is counter intuitive for students, and is at the heart of the process of integration. In the modern world, well the last 200 years or so, we take a very prudish view of the infinite when we teach mathematics. We are seduced by Cauchy and Wierstrass and their rigorous redefinition of the calculus as being about limits. Of course, mathematically this is absolutely correct, but intuitively it is tough.

It was not always like this. There is evidence that Newton and Leibniz thought in terms of infinitesimals even though they were very suspicious of them. They used them in calculations without defining them, which led to Bishop Berkeley’s famous condemnation of “Ghosts of departed quantities”. Archimedes certainly used the idea of adding up an infinite number of infinitesimals in The Method to get the results he published elsewhere. There is of course Robinson’s wonderful Non Standard Analysis that puts all this in a rigorous setting. New systems of number such as Conway’s show how we can create infinitesimal and infinite numbers if we just wait long enough!

I would suggest that this approach to the calculus is much more intuitive than the usual stress on limits, which can happen at university. It also makes Integration more natural than Differentiation and I would always teach it first. The Fundamental Theorem of Calculus becomes proper result rather than the way integration is defined. See Elvis and his Impersonators discuss the theorem.

Now for the fun part:

Chris Binge’s intuitive demonstration of the unaccountability of the irrationals.

We know that the infinite decimal expansion of a rational number always ends in an infinite recurring series. This may be a series of zeros (as in 1/2) another number (1/3) or a longer sequence (1/7).

Take a 10 sided die, marked with the digits from 0 to 9. Write down “0.” And then start rolling the die, writing down the digits you get. What you are doing is constructing a number between 0 and 1, that is produced entirely by chance. Imagine (this is a thought experiment) the dice being thrown an infinite number of times, or thrown once each by an infinite number of monkeys.

You will produce a random real number between zero and one. Is any one real number any more likely than any other? Obviously not, so they are equally likely.

How likely is it to get ½? O.5000000…   forever? Well, if this isn’t zero then I do not know what else it could be. So the chance of any number is zero.

How likely is any infinitely recurring sequence? Intuitively, a repetition of any sequence is likely to break down eventually – with probability 1. Or at least the probability of any pattern continuing infinitely recedes towards zero. So the probability of a recurring decimal is zero. Hence the probability of a rational number is zero. Almost all reals are irrational!


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