Geometry, Continued Fractions, and the square root of 2

Of all the interesting and non-standard topics that Mr. Waterman covered with us in high school, continued fractions was definitely my favorite.  I don’t have any great reason, it was just the beauty of the topic really appealed to me.  Each time that I covered fractions with my sons I made sure that we spent a little time talking about continued fractions just for fun.   That exercise with my youngest son happened last week.

I don’t know why continued fractions seem to have vanished from the math ed. landscape.  My working assumption is that for high school the topic was  just one of many casualties of curriculum decisions, and for college the topic was probably viewed as being too elementary.  The actual reasons are probably more nuanced.

The topic was previously held in pretty high regard in mathematics, though.  Take this famous story about Hardy seeing Ramanujan’s work for the first time (from Wikipedia) for example:

“After [Hardy] saw Ramanujan’s theorems on continued fractions on the last page of the manuscripts, Hardy commented that “they [theorems] defeated me completely; I had never seen anything in the least like them before.”

Beyond simply introducing a fun topic, one other reason that I’ve enjoyed playing around with continued fractions with my kids is that  it gives me an easy opportunity to give them a little more practice doing arithmetic with fractions.  The book Mr. Waterman used to teach the topic to us is filled with tons of amazing examples, and it is the same one I use today – C. D. Olds’s “Continued Fractions” from the MAA’s New Mathematical Library.    Here’s a pic of the book along with the only other book in that series that I refer to more frequently (neither quite in mint condition these days) :

So, after a brief discussion about the basics of continued fractions on Thursday, on Friday morning we looked at the continued fraction for :

Then on Friday evening there was a fun little coincidence on twitter.  First, Steven Strogatz posted a link to a neat proof that the square root of 2 is irrational:

Elegant proof that sort(2) is irrational, by Caltech's Tom Apostol http://t.co/lewU80Me4H (ht Chris Adams)

— Steven Strogatz (@stevenstrogatz) February 1, 2014

Patrick Honner responded with a link to a second geometric proof:

.@stevenstrogatz Here's a similarly clever proof of the irrationality of sqrt(2), via @divbyzero http://t.co/D8IVlVHdGI

— Patrick Honner (@MrHonner) February 1, 2014

At the end of this proof was a neat little surprise – the approximation that that we’d found earlier in the day shows up in the geometry.  In fact, the entire series of approximations that you get from the continued fractions could show up.  I thought that the boys would really like to see this particular connection between geometry and arithmetic, so we went through it this morning:

I’m always happy when the neat math that people are sharing on social media sites happens to overlap with the stuff I’m doing with the boys    Given this particular topic already has a special place in my math heart, this little coincidence yesterday was especially nice.