Continuing our look at continued fractions

Yesterday we did revisited continued fractions:

A short continued fraction project for kids

Today I wanted to boys to explore a bit more. The plan was to explore one basic property together and then for them to play a bit on the computer individually.

Here’s the first part -> Looking at what happens when you compute the continued fraction for a rational number:

Next I had the boys go the computer and just play around.

Here’s what my younger son found. One thing that made me very happy was that he stumbled on to the Fibonacci numbers!

Here’s what my older son found. The neat thing for me was that he decided to explore what continued fractions looked like when you looked at multiples of a specific number.

So, a fun project overall. Continued fractions, I think, are a terrific advanced math topic to share with kids.

A short continued fraction project for kids

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 \pi?

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 \sqrt{2} and also see that the first couple of fractions we see are good approximations to \sqrt{2}.

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 \sqrt{2} that were nearly integers.

Finally, we wrapped up by using continued fractions to find good approximations to \pi, e 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.

3 proofs that the square root of 2 is irrational

My younger son has been learning a little bit about square roots over the last couple of weeks and I thought it would be fun to show him some proofs that the square root of 2 is irrational. Because this conversation was going to explore some ideas in math that are both important and pretty neat, I asked my older son to join it.

I wasn’t super happy with how this little project went – it felt a bit rushed while we were going through it. Hopefully a few of the ideas stuck.

We started by talking about the square root of 2 and what basic properties the boys already knew about it:

After that short introduction we moved on to the first proof that the square root of 2 is irrational – I think this is probably the most well-known proof. The proof is by contradiction and starts by assuming that \sqrt{2} = A / B where A and B are integers with no common factors.

The next proof is a geometric proof that I learned a few years ago from Alexander Bogomolny’s wonderful site Cut The Knot. It is proof 8”’ here:

Proof 8”’ that the square root of 2 is irrational on Cut the Knot’s site

If you like this proof, we have also explored some geometric infinite descent proofs in a slightly different setting previously inspired by a really neat post from Jim Propp:

An infinite descent problem with pentagons

Finally, we looked at a proof that uses continued fractions. It has been a while since I talked about continued fractions with the boys, and will probably actually revisit the topic soon. It is one of my favorite topics and always reminds me of how lucky I was to have Mr. Waterman for my math teacher in high school. He loved exploring fun and non-standard topics like continued fractions.

So, although I don’t go deeply into all of the continued fraction ideas here – hopefully there’s enough here to show you that the continued fraction for the \sqrt{2} goes on forever.

So, although this one didn’t go quite as well as I was hoping, I still loved showing the boys these ideas. We’ll explore them more deeply as we study some basic ideas in proof over the next year.

A continued fraction experiment

I’m a big fan of continued fractions – especially the many different ways that you can use them to help kids learn elementary math. Right now I’m studying square roots with my younger son and he’s taken quite a liking to continued fractions, too. See yesterday’s project, for example:

A surprise square root of 2 discussion

I intended for the focus of that discussion to be the standard proof of why \sqrt{2} is irrational. Instead, though, a large part of the discussion was about how you could use the continued fraction for \sqrt{2} to prove that it was irrational.

Having not learned my lesson already, I asked my son to sketch a proof of why \sqrt{5} is irrational, and he went down the continued fraction path again.

Even though this project is pretty difficult and many of the parts are really over my son’s head, I think this was a useful exercise. I also think that it all pretty much stands on its own, so I’ll present the four steps below without much comment.

Following this project, my son asked me if we could study more about continued fractions this week rather than just studying the current chapter in our book about square roots. Something about this topic has really caught his attention!

The continued fraction calculator we are using in the last video is here:

Celebrating Pi day with Evelyn Lamb’s idea

Last week Evelyn Lamb wrote a nice piece about \pi and continued fractions.

Since we’ve talked a little bit about continued fractions in the past, this seemed like a great way to celebrate \pi day. We started with a quick reminder about continued fractions:


After the quick introduction, we used my high school teacher’s fun continued fraction technique – Split, Flip, and Rat – to calculate the continued fraction for \sqrt{2}. This exercise gives you a great opportunity to talk with kids about fractions and decimals.


Next up was today’s activity – the continued fraction for \pi! Unfortunately, for this continued fraction split, flip, and rat doesn’t work so well. Nonetheless, we do get to have a good discussion about decimals while calculating the first two pieces of the continued fraction for \pi.


To calculate a few more parts of the continued fraction we went to Wolfram Alpha. Turned out to be a pretty neat way (and obviously a much quicker way) to see the next few numbers in the continued fraction. Again, we got to have a great discussion about decimals and reciprocals.


Now, having found a few terms in the continued fraction, we went and looked at what fractions other than 22/7 were good approximations to \pi. Happy 333/106 day everyone 🙂


Finaly (and sorry for the camera screw up on this one), I wanted to show a different continued fraction for \pi. In a previous video my younger son thought that we’d find a pattern in the continued fraction for \pi. We didn’t in the first one that we looked at, but there are indeed continued fractions for \pi that do have amazingly simple patterns.


So, a fun little project for pi day. A great opportunity to review lots of arithmetic in the context of learning about continued fractions and \pi.

Fun math that I saw this week

At the beginning of 2014 Numberphile published an incredible interview with Ed Frenkel with the provocative title:  “Why do people hate mathematics?””

There is a particularly interesting exchange around the 5:00 mark:

Numberphile:  “Let’s apportion some blame.  Let’s blame someone.  Sounds to me like you are blaming high school teachers . . . . back in our school days they were making us paint fences instead of showing us Picasso.”

Frenkel:  “Well, if I really were to assign blame, I would assign the blame to myself and to my colleagues – professional mathematicians.  We don’t do nearly enough of exposing these ideas to the public in an accessible way.  Often times we aren’t willing to come up with good metaphors and analogies.

That exchange has really stayed with me.

Later in the year, actually about a month ago, David Coffey wrote a nice piece in which he answered a similar question – Whose fault is it that you aren’t good at math?:

Whose Fault is it that you aren’t good at math?

I like his answer a lot -> you aren’t good at math because you didn’t have the experiences that you needed to be good at math.  I wrote a little too long of a response to Coffey’s piece here where I suggested some places where students (and, well, everyone) might find fun math experiences:

Responding to Cafid Coffey’s Challenge

Despite writing way too much last time, I want to write more because I’ve seen so many great math experiences  just in the last few days.

(1)  Let’s start with this incredible public lecture from Terry Tao at New York’s Museum of Mathematics:



This lecture seems to be almost exactly what Frenkel was taking about in the piece of his interview that I quoted above.  Tao shows some wonderful ideas of mathematics and how those ideas helped us understand how to measure distance in the universe.  A beautiful and accessible lecture from one of the world’s top mathematicians.   It has been online since the beginning of June and as of today doesn’t even have 700 views!!   Ugh, though to be fair I try to look out for stuff like this and didn’t even see it until yesterday.   I hope more people find out about this lecture and are able to watch it.

(2) Speaking of the Museum of Math, this weekend they finished their portion of the MegaMenger project: .  My family went down to participate in the build and it was really fun to see all of the kids helping out and playing around in the Museum.  Hopefully there will be more projects like this one that can show kids that math is more than their 20 question algebra assignment.  Here’s a picture of my son standing inside of the finished product:

(3) Fawn Nguyen’s digit puzzle.

Last week Fawn Nguyen shared a wonderful digit problem that she did with her class.  A few days later the online math world was buzzing left and right about her problem.  On Sunday morning 5 different people had shared it on my twitter feed.  So fun and I’m so happy for Fawn’s incredible work is being recognized by an ever-growing audience.  But don’t take my word that this is an “utterly kick-ass” exercise.  Try it out, too:

(4) Continued Fractions

Continued fractions have a special place in my heart because my high school math teacher, Mr. Waterman, loved them.  He taught us out of the book pictured below (sorry I only have the picture side by side with Geometry Revisited, but trust me, that’s a great book as well!):

Book Pic

Amazingly I saw two ideas in the last week where continued fractions either have or could have played a role.  It really is a beautiful subject and it is a shame that it no longer has much of a place in high school or college math programs.

In any case, here’s a really neat blog post from Sam Shah showing how he incorporated continued fractions into a lesson in his class:

Substitution and Continued Fractions

and here’s a fun tweet from Steven Strogatz from this morning showing a problem in which continued fractions could help students make a fun connection:

Since the picture from Gilbert Strang’s book doesn’t come through, let me expand a little on the continued fraction connection.  On page 36 of the book Strang mentions the pattern he’s showing comes from a connection that 2\pi has to the fractions 44/7, 25/4, and 19/3.  Those three fractions just so happen to be the first three “convergents” of the continued fraction for 2 \pi.  The next one is 333/53 which might be fun to look for in Strang’s pattern.

Anyway, it has been great to see all of this fun math online (and in person) during the last week.   Hopefully there will be many more weeks like this one to come.

Happy Square Root of 10 day!!

\pi day was a just a few days ago.  Maybe I was paying more attention this year, but it seemed like there was quite a backlash against it this year.  Oh well, we had fun with it anyway, and no one eating the pie seemed to object!


But yes, some of the objections about \pi day are valid and inspire fun little tweets like these:

Actually, wait, that’s a good idea – let’s celebrate \sqrt{10} day, too!!

First, the square root of 10 is irrational:

That was fun and full of giggles, what’s next?  How about representing \sqrt{10} in an easier way with continued fractions:

Maybe we can make every day a fun math day!!