One of the examples in my son’s Prealgebra book today was prove that is less than 2. We were having a pretty good discussion about the ideas in this example, so I thought it would be fun to see if we could go a little deeper. Since we just talked about continued fractions last weekend, I was hoping that end up being able to find something to say that was much more accurate than just “less than 2.”
Our initial discussion of the problem is here:
Next up was the beginning of looking at as a continued fraction. We’ve spent very little time on this subject, so it is still new to him and we had to go slowly through the process. Luckily the continued fraction starts to repeat fairly quickly.
We finished up by figuring out some of the fractions that approximate . This exercise was why I wanted to go down the path of calculating the continued fraction. First off, we’ll see some of the fractions that we saw already in part 1. Second, we’ll find a couple better approximations, which is neat. Third, we’ll get to see directly that these fractions are nearly equal to 2 when you square them. AND, we get lots of good fraction practice in the process. Yes!
Have you seen the “magic box” array for calculating convergents of a continued fraction? it is a nice algorithm that lends visual support to induction proofs relating to continued fractions. Take a look at page 12 table 1 of this paper for an example. For now, only look at the line a_n (the cont’d fraction coefficients) and the lines p_n and q_n. The p_n/q_n are the convergents. There is a nice recursion algorithm for stepping left to right through the array.
Mike's Math Page 
What about 1/root(2)? More or less than 1/2 ?
We played around with that in the lead up to the project.
Have you seen the “magic box” array for calculating convergents of a continued fraction? it is a nice algorithm that lends visual support to induction proofs relating to continued fractions. Take a look at page 12 table 1 of this paper for an example. For now, only look at the line a_n (the cont’d fraction coefficients) and the lines p_n and q_n. The p_n/q_n are the convergents. There is a nice recursion algorithm for stepping left to right through the array.
I actually talked about Pell’s equation with my older son a few years ago – just for fun. The first in that series is here: