My biggest bane in college was the library, it sucked in most of my free time. One of my favorite things to do was browse the mathematics and physics stacks looking for interesting books. It was rare when I didn’t have a stack of 10 or 20 library books sitting on my bedroom floor. That’s also why I graduated with $300 in library fines and I didn’t get my diplomas until years later when I finally paid them off.
One of my favorite discoveries in the PMA library at UT was I. M. Yaglom, a Russian mathematician who authored a series of books aimed at advanced high school students or beginning college students focusing mostly on geometry. In my next few posts I plan to cover some of the topics from some of his books. They combine the algebra of complex numbers, the geometry of special relativity and string vibrations.
What I find even more interesting is that it provides a mathematical model for motion on a string which has some strong parallels with the free particle one dimensional Schroedinger equation.
To begin, I’d like to start with some basic properties of numbers and geometry. This is based on the Foundations of Geometry by David Hilbert. We first learn about numbers by counting. If we have an apple in our left hand and an apple in our right hand, then we have two apples. From counting we advance to addition and subtraction. This is modelled by adding and removing items from a group. If we have a basket with 10 apples and we eat 2 then we have 8 apples left.
Moving on we learn how to add groups of items. If we have 3 baskets containing 5 apples each, then we have 3 times 5 or 15 apples total. Multiplication is just addition of groups of smaller items. This model works fine for the positive model of addition and multiplication, but begins to break down for subtraction or division. Attempting to create a logical model of subtraction leads to the concept of negative numbers which don’t make much sense in terms of apples. Nobody has ever seen a negative apple. Reversing multiplication leads to division and the introduction of fractions which might or might not make sense for the objects we are dealing with.
In Foundations of Geometry Hilbert describes the construction of an algebra of line segments based on geometric operations between lines and line segments. In this view addition is simple, just join two line segments together and line them up. Subtraction is just as natural, all you have to do is reverse the direction of one of the segments. A negative number is just a line going in the opposite direction.
Multiplication is more complicated and I’ll cover that in the next post. In fact there are a couple of ways to define it, and these have to deal the two multiplication operations of vector algebra. If you don’t already know, think of how you would multiply two line segments together with just a ruler and compass. It sounds much easier then it is.