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:

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

Having not learned my lesson already, I asked my son to sketch a proof of why 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:

I’m not convinced this is a valid proof. A continued fraction could be created for any square root. sqrt(4) = 1 + 3/(2 + 3/(2 + 3/(2 + 3/(2 + …)))). If you generalize for sqrt(x), you get 1 + (x-1)/(2 + (x-1)/(2 + (x-1)/(2 + (x-1)/(2 + …)))). Just because it never ends, doesn’t mean it is not equal to a rational number. (ie. 1 = .9999…)

PS. Love your videos and am totally impressed by the level of understanding your sons exhibit. They rock!

OK. That’s probably true for a fractional portion that is “1/”. I haven’t given that much thought and am by no means a mathematician. So maybe it was just the statement that it is irrational because it never stops was what I got hung on. Thanks.

This is one of the nice features about the simple continued fraction representation: rationals always terminate, irrational square roots always have a repeating cycle, and other irrationals get crazy,

some of the key ideas in the proof for rationals terminating:
(1) The division algorithm: for a, b positive integers, there are integers d and r such that a = b*d + r and b>r>=0. This underlies the euclidean algorithm for finding a gcd that also serves as an algorithm to compute coefficients of the continued fraction expansion of a/b.

The remainder condition gives a series of decreasing terms that eventually must terminate at 0.

This shows that any rational can be expressed as a terminating continued fraction.

(2) Because each denominator is of the form a + f with a a positive integer greater than 0 and f another continued fraction, you can see that the f values must be at most 1. This allows you to see that the simple continued fraction expansion is unique (up to a final decision when you get to the last coefficient).

Note: the division algorithm is really a beautiful thing and is worth understanding deeply. For years, I didn’t appreciate it, thinking of it just as division with remainder that I’d learned in middle school and originally considered (a) obvious and (b) not useful (since we could just get a decimal expansion anyway).

## Comments

I’m not convinced this is a valid proof. A continued fraction could be created for any square root. sqrt(4) = 1 + 3/(2 + 3/(2 + 3/(2 + 3/(2 + …)))). If you generalize for sqrt(x), you get 1 + (x-1)/(2 + (x-1)/(2 + (x-1)/(2 + (x-1)/(2 + …)))). Just because it never ends, doesn’t mean it is not equal to a rational number. (ie. 1 = .9999…)

PS. Love your videos and am totally impressed by the level of understanding your sons exhibit. They rock!

I am not trying to be overly formal, but I think the idea of an infinite continued fraction of the form we are considering:

a + 1 / b + 1 / c + 1 / d + . . . . would not be equal to a rational number.

Perhaps I’m mistaken about this, but a quick look shows that Cut the Knot’s site has a similar proof:

http://www.cut-the-knot.org/proofs/SqContinuedFraction.shtml

OK. That’s probably true for a fractional portion that is “1/”. I haven’t given that much thought and am by no means a mathematician. So maybe it was just the statement that it is irrational because it never stops was what I got hung on. Thanks.

This is one of the nice features about the simple continued fraction representation: rationals always terminate, irrational square roots always have a repeating cycle, and other irrationals get crazy,

some of the key ideas in the proof for rationals terminating:

(1) The division algorithm: for a, b positive integers, there are integers d and r such that a = b*d + r and b>r>=0. This underlies the euclidean algorithm for finding a gcd that also serves as an algorithm to compute coefficients of the continued fraction expansion of a/b.

The remainder condition gives a series of decreasing terms that eventually must terminate at 0.

This shows that any rational can be expressed as a terminating continued fraction.

(2) Because each denominator is of the form a + f with a a positive integer greater than 0 and f another continued fraction, you can see that the f values must be at most 1. This allows you to see that the simple continued fraction expansion is unique (up to a final decision when you get to the last coefficient).

Note: the division algorithm is really a beautiful thing and is worth understanding deeply. For years, I didn’t appreciate it, thinking of it just as division with remainder that I’d learned in middle school and originally considered (a) obvious and (b) not useful (since we could just get a decimal expansion anyway).