I woke up this morning to see another great discussion between Alexander Bogomolny and Nassim Taleb. The problem that started the discussion is here:
and the mathematical point that caught my eye was the question -> which positive integers are close to being integer multiples of ?
One possible approach to this question uses the idea of “continued fractions.” I learned about continued fractions from my high school math teacher, Mr. Waterman, who taught them using C. D. Olds’s book.
So, today I stared off by talking about irrational numbers and reviewing a simple proof that the square root of 2 is irrational:
Next we talked about why integer multiples of irrational numbers can never be integers. This I think is an obviously step for adults, but it took the kids a second to see the idea:
Now we moved on to talk about continued fractions. I’m not trying to go into any depth here, but rather just introduce the idea. I use my high school teacher’s procedure: split, flip, and rat 🙂
We work through a simple example with and also see that the first couple of fractions we see are good approximations to .
With that background work we went up to use Mathematica to explore different aspects of continued fractions quickly. One thing we did, in particular, was use the fractions we found to find multiples of that were nearly integers.
Finally, we wrapped up by using continued fractions to find good approximations to , and a few other numbers.
Definitely a fun project, and one that makes me especially happy because of the connection to Mr. Waterman. Hopefully the boys will want to play around with this idea a bit more tomorrow.