Continued fractions have really caught my younger son’s eye the last couple of days. The link below (plus the link inside of that link) give two examples of our recent discussions:

Today we were playing around online with continued fraction calculators and other sites discussing continued fractions and found quite a surprise – there is a generalized continued fraction for that looks very similar continued fraction for . I used this little coincidence to review some basics of calculating continued fractions:

Next up – a short discussion of the generalized continued fraction that we found for . I think that kids will always find it surprising that there are relatively simple ways to describe even though just about everything you hear about is that it has no pattern.

Neat little morning – very fun to show a surprising way to see some structure in . Also quite a surprise to see the seemingly small change that changes into .

I wonder if this is surprising or if most real numbers have representations as “nice patterns.” Doing a quick google search suggests that:
(a) we are restricted to talking about computable numbers (otherwise, how do we know which number we are talking about)
(b) the finite algorithms used to compute them could be considered “nice patterns.”
(c) this is actually a small collection of numbers (only countably many)

When written in linear form, the part of the continued fraction of the square root of 10 that repeats can be indicated in the same way as with a repeating decimal. In other words, it can be written as [3; 6̅]. This seems to be the only way to (numerically) represent the square root of 10 or any other quadratic irrational in it’s entirity. Also note that all quadratic irrationals, like rational numbers, are constructable (with a ruler and compass).
Additionally, any truncation of a “simple” continued fraction yields a convergent, all of which are best rational approximations, making such expansions particularly meaningful.
But that other continued fraction expansion of pi where the numerators are squares of consecutive odd numbers doesn’t have a linear form. Additionally the sequence of partial fractions includes 22/7, but not 333/106, 355/113, or any other best rational approximation with a denominator of less than 113, be it a convergent or semi-convergent.

I wonder if this is surprising or if most real numbers have representations as “nice patterns.” Doing a quick google search suggests that:

(a) we are restricted to talking about computable numbers (otherwise, how do we know which number we are talking about)

(b) the finite algorithms used to compute them could be considered “nice patterns.”

(c) this is actually a small collection of numbers (only countably many)

When written in linear form, the part of the continued fraction of the square root of 10 that repeats can be indicated in the same way as with a repeating decimal. In other words, it can be written as [3; 6̅]. This seems to be the only way to (numerically) represent the square root of 10 or any other quadratic irrational in it’s entirity. Also note that all quadratic irrationals, like rational numbers, are constructable (with a ruler and compass).

Additionally, any truncation of a “simple” continued fraction yields a convergent, all of which are best rational approximations, making such expansions particularly meaningful.

But that other continued fraction expansion of pi where the numerators are squares of consecutive odd numbers doesn’t have a linear form. Additionally the sequence of partial fractions includes 22/7, but not 333/106, 355/113, or any other best rational approximation with a denominator of less than 113, be it a convergent or semi-convergent.