Some places to start a journey …
We spend a lot of time in school mathematics learning how to work with numbers, but what are they? What is a number and how can we construct our collection of these very abstract things that we use every day in conversation and planning?
These are called ‘Series’ … a very important concept in dealing with motion, physics and much more.
Can we possibly get a sensible, finite, answer when we try to add up an infinite number of positive numbers all greater than zero?
This is a very ancient problem in mathematics — the different ways of resolving (or banishing) it have been profoundly important to mathematics, physics and wider ideas of philosophy. And to the history of the development of modern science, technology and industry along with modern economies and societies.
We start exploring calculus (just like they did in the 17th century, while inventing it) by trying to describe the way objects move. Seconds and metres are both measurements that are not restricted to whole numbers. Length and time (at least on any scale we are concerned with here!!) can take any real-number value. Speed is a number saying how far an object goes during each unit of time — a sprinter running at \(\small10\,\)m/s moves \(\small10\) metres every second. Acceleration is a measure of how quickly speed is changing. We discovered that this last property is much more regular, that there are very specific rules that natural objects seem to obey concerning acceleration.
Try Euclidea, a modern version of the classical compass-and-straight-edge problems that kept European mathematicians and students busy for many centuries.