History and Motivation

Once upon a time, Isaac Newton wanted to define the speed of an object, whose speed changes continuously. He erected a weird tower of concepts like “infinitesimals” to analyze this concept, and came up with something that, surprisingly enough, worked. However, Newton’s notation and concepts were hard to understand and, for anyone who was not Newton, to convince themselves that this was not a tower built on sand. When mathematicians started to figure out how to properly define these things so anyone can understand them, they realized that before even delving into the question of speed, they needed to understand the things that measure speed – numbers.

Natural numbers hearken back to the prehistory of man, but a proper understanding of them in Europe, where calculus was invented, was a 13th century discovery. Still, even then, a proper understanding of zero, negative numbers and fractions would need a while. However, to properly understand those numbers Pythagoras and his students hated so much they denigrated them as “irrational” (how much vitriol does someone need to call a number, literally, a “crazy” number?) required still more machinery. But proper calculus could not be done without those numbers – those “crazy” numbers kept cropping up anywhere where instantaneous change was hiding – even in calculating the answers like “if your bank decided to charge interest based on APR-to-EAR calculation based on very small periods, what is the maximum they can charge?” the “crazy” number \(e\) makes an appearance!

In this text, we try to work through the work of these 18th and 19th century mathematicians, to understand what we mean by numbers, limits and, finally, speeds. Tying all this together is the quest to tell a story using the language of math. Imagine a 100 meter dash course. At the beginning, you are at the starting position. You want to spend some time at the end position, say a minute, chatting with someone, and then get back to the starting position. The catch? the points in between are boring, and you want to spend as little time as possible in them. Our quest will be to find a function that captures that story while being amenable to questions such as “what is the speed? what is the acceleration?” at each point.

For this, we will need to understand what numbers are, how to define speed and how to relate the distance travelled to speed. It’s a long journey, with many \(N\), \(\epsilon\) and \(\delta\) – but at the end, the concepts of calculus will seem so obvious that it will be hard to believe it took a Newton to invent them.