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Archive for the tag “calculus”

17 Aug 2019

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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17 Oct 2017

Infinitesimal Calculus 2: The Changes in Change

The mathematics of change are quite interesting. In a naive sense, we can often describe a change by a simple collection of data points. For example, let’s think about a little boy rolling a ball across the floor. The boy pushes the ball, and four seconds later, the ball has come to be 2 meters away from him. Given these data points, we may attempt to connect them in some meaningful analytical manner– perhaps by saying that the ball rolled at a speed of half a meter per second. But even this is a somewhat naive bit of information, as it only really tells us something about the completed journey. Mathematicians are greedy, however; they want to be able to know about every point of the ball’s travel, at any arbitrary moment in time.

We can use a function for just such a purpose. A function is a specific mathematical tool which allows us to describe an entire set of data points all at once which we symbolize as f(x) (read “f of x“). We encode the data by means of a mathematical formula. For example, our exemplary rolling ball might well have been encoded by the function f(x)=\frac{1}{2}x, where the x represents the time, in seconds, that the ball has been rolling, and the value of the function, f(x) tells us the distance in meters which the ball has traveled in that time. In this particular function, the coefficient of x tells us the rate at which distance changes as time passes– that is, \frac{1}{2} a meter per second. When the boy first rolls it, the ball is traveling at \frac{1}{2} a meter per second; when it finishes it had been traveling at \frac{1}{2} a meter per second; and at any single point during the journey the ball is traveling at \frac{1}{2} a meter per second.

However, this is a very simple example. It describes a situation involving a constant velocity. Things become a bit more muddied when the rate at which a change occurs is, itself, changing.

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30 Sep 2017
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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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24 Jan 2017
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On Teaching Calculus

Almost universally, when Calculus is taught to modern students, we preface the entire subject by introducing those students to a concept known as a “limit.” The reason for this, historically, was to ensure that mathematics was taught in a rigorous and well-defined manner. When Leibniz (and, independently, Newton) first developed methods for performing calculus, the concept of a limit was nowhere to be found. However, the tool which these men did utilize in their work was something which they had not rigorously defined, at the time. Newton called it a “fluxion” and Leibniz called it a “differential,” but the concept was the same: a number which was not zero, but which was so small that adding it to any Real number did not yield a different Real number.

Many other mathematicians and philosophers of the time rightfully balked at the notion. It seemed entirely ludicrous. Bishop George Berkeley famously scoffed at Newton, asking if his fluxions were “the ghosts of departed quantities.” However, it was quite plain that the mathematics which Leibniz and Newton presented worked. When the results which could be found from the methods of Calculus were able to be confirmed using other methods, they were found to be accurate and true. Indeed, the Calculus was such a powerful tool that even most mathematicians and philosophers who recognized its flaws continued to utilize it in their work. Many began searching for some way to make the Calculus just as rigorous as the rest of mathematics. These efforts culminated in the work of Karl Weierstrass, who found a way to base Calculus upon a different tool. Instead of the Newtonian “fluxion” or the Leibnizian “differential,” Weierstrass gave mathematics a well-defined notion of the limit.

It is Weierstrass’ method of limits which is still taught, even to this day, in nearly every Calculus textbook in the world; but perhaps it is time to abandon this notion and return to the concept which Newton and Leibniz pioneered.

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