This week I had the honor of being on Go’kväll, a national Swedish tv program, to talk about beauty in mathematics. How cool is it that the program has started to take on topics in mathematics and science: Very cool. We discussed what kind of example to give to give viewers a sense of what beauty in mathematics is like. In the end, even though we had simpler examples, we went with the one that really turned me on to mathematics. It is an argument, formulated by Goerg Cantor in the late 1800s, which shows that there are different orders (sizes) of infinity. Some sets of numbers, like counting numbers (1, 2, 3, 4… ) and rational numbers (including 1/2, 3/18, 9/49) can be shown to have the same size. There is a clever way or ordering the rational numbers when you count them. But if you try to do the same with real numbers (which includes irrational numbers like π and e, and all the rational numbers) you simply can’t do it. The way to show this is to make a list of all the real numbers, as if you could do it. Just write them all down, doesn’t matter in what order, but make sure you have got them all. Once you finish the list, which is supposed to have all the numbers, you can still construct another number that is not on the list! You do it by writing down a number that differs in the first decimal place of the first number, in the second decimal place of the second number, etc.
Here is a link to the show: GoKvall
For more info on Fibonacci numbers: http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibnat.html