I recently gave a math talk to a group of high school students about something called the p-adic numbers. This is something I’ve been wanting to do for a while now. It is a topic that almost no one ever hears of unless he/she intends to study math at the graduate level. I believe the idea is simple enough for even high school students, albeit very smart ones, or undergraduates without a strong math background to understand. Here’s how it goes: We want to draw a picture of all the rational numbers (i.e. fractions), and this is almost always done with some idea of distance in mind that is often based on our physical understanding of the world. We make a dot for an origin, usually called name it ‘0′, and one measurement to the right (a predetermined unit), we place a dot and label it ‘1′. Two units to the left, we place another dot and label it ‘-2′, etc. With enough dots for each fraction, a picture of a line emerges. However, this line is incomplete – it has holes – since some distances from the origin, like the square root of 2, cannot be written as a fraction. (This fact is not at all obvious. The ancient Greeks didn’t really believe in such ‘irrational’ numbers – numbers that cannot be written as a ratio – to the point that the word ‘irrational’ came to mean ‘crazy’). If you fill in the holes you get the ‘number line’, which everyone has seen since grade school. Almost all the math we do afterwards – including all of calculus- depends fundamentally on this picture of the real line.
What will happen if our idea of distance was different? Kurt Hensel discovered that measuring how many times a prime number, say 2, divides another number gives a perfectly good way to measure distance. For example, the more times 2 divides the numerator of a fraction will mean that the fraction is closer to the origin. All this can be spelled out mathematically, and we get the 2-adic distance (or absolute value) and a very different picture of the rational numbers (it is not a line!). Now the picture of the rational numbers under the 2-adic distance, like the case with the usual distance, has holes and we need to complete it (‘completion’ is actually the technical term used). Filling in the holes gives the 2-adic numbers, and it’s picture often looks like some crazy fractal. This new world has some exotic properties which can make it very useful. For example, every point in a circle is it’s center and every triangle is isosceles. Even calculus is oftentimes much easier to do. So now instead of having the ‘real’ world, for every prime p, we get a p-adic world and a more complete picture of our universe (that is, our mathematical universe).
This was pretty much the gist of the introductory talk I gave. I am going to do a follow up talk a week from Monday.
