Boxing Pythagoras

Philosophy from the mind of a fighter

Archive for the tag “Abraham Robinson”

Classical Limits vs. Non-Standard Limits

One of the most important and fundamental concepts taught in modern Calculus classes is that of the Limit. I have discussed this idea once before, but I thought I would revisit it, here. In that first article, I noted that the classical definition for a Limit is fairly complex and that we can utilize a more intuitive notion of infinitesimals to accomplish the same task, insofar as derivatives are concerned. However, there are other uses and purposes for limits, in mathematics, so we would not want to simply omit them entirely, even using a non-standard approach to Calculus lessons.

Thankfully, even the very difficult and complex definition of “limit” can be simplified and made easier to understand by use of non-standard infinitesimals.

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Infinitesimal Calculus 1: The Numbers Between Numbers

If I were to ask a person to name a number which comes between 1 and 3, everyone from a three-year-old child to a white-bearded great-grandfather is likely to respond by saying, “2.” If I rephrase the question to ask about a number between 1 and 2, then the young child might be confused, but a fourth-grader might be able to respond with 1\frac{1}{2}. We have to extend our understanding of what we mean by “number” to include some concepts which are not quite so intuitive. That is to say, in between the Integers, there are other numbers which are known as Rational numbers. In fact, given any Integer, n, there are an infinite number of Rational numbers which are greater than n and yet less than any other Integer which is greater than n.

There are numbers in between the Rational numbers, too. We can define some number, r, which is not equal to any Rational number. There are Rational numbers which are greater than r, and those which are less than r, but somehow our number r squeezes itself into a gap in between the Rational numbers. In order to find such a number, we need to further extend our understanding of “number” to include the Real numbers. This should all be very familiar to the average high-school student.

Now, what happens if we extend this idea one step further? Are there more numbers which are in between the Real numbers?

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