# Boxing Pythagoras

## 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.

## More on 0.999…=1

In my last post, I discussed a particular video which I found to be more than a bit misleading. The discussion centered around a simple, but extremely counterintuitive notion of mathematics: the fact that the number 0.999…, or zero-point-nine-repeating, is equal to 1.

Well, as I mentioned, the very counterintuitive nature of the result led at least one of my readers to question its validity. As such, I thought I would lay out one proof of this concept, in order to make it easier for those who do not accept the result to pinpoint exactly where they disagree. I’ll break my proof down into numbered steps, to ease in that venture.

## The Legend of Hippasus

There was once an ancient Greek geometer named Hippasus who belonged to the Pythagorean Brotherhood. The Pythagoreans were a school of philosophers who held a special reverence for numbers and proportion. To these men, mathematics was more than just a method for quantifying and describing the world around them. The Pythagoreans held that numbers, themselves, were divine things, worthy of awe and worship. Relationships between these numbers– what we would now think of as a “proportion” or “ratio” of numbers– were intensely studied, as these proportions were thought to hold the secrets of the cosmos. If one were to divide a string according to some specific ratios, he could produce beautiful music. If one compared the proportions of two legs of a triangle, he could come to understand the remaining leg. Nothing in existence was more beautiful to the Pythagoreans than the discovery of these proportions and the properties they endowed.