A game with Pi

Here’s an image of something I wrote down, took a photo of, and posted here for you. It’s a little game you can play with any irrational number. I took \pi as an example.

You just learned about an important math concept/process called continued fraction expansions.

With it, you can get very precise rational number approximations for any irrational number to whatever degree of error tolerance you wish.

As an example, if you truncate the above last expansion where the 292 appears (so you omit the “1 over 292” part) you get the rational number 335/113 which approximates \pi to 6 decimal places. (Better than 22/7.)

You can do the same thing for other irrational numbers like the square root of 2 or 3. You get their own sequences of whole numbers.

Exercise: for the square root of 2, show that the sequence you get is
1, 2, 2, 2, 2, …
(all 2’s after the 1). For the square root of 3 the continued fraction sequence is
1, 1, 2, 1, 2, 1, 2, 1, 2, …
(so it starts with 1 and then the pair “1, 2” repeat periodically forever).

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  1. November 9, 2014 at 12:19 am

    Maths and I were never going to be best friends lol I fare better with words, why I published two books 🙂

  2. November 11, 2014 at 1:28 pm

    I wished I was half as good with words as you! 😉 Hey, congrats on your publications!

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