Predicting Pi

Inspired by a recent post over at Foxmaths!, here’s an interesting challenge question for you to think about:

Suppose I give you the sequence of integers \lfloor \pi \rfloor, \lfloor 2 \pi \rfloor, \lfloor 3 \pi \rfloor, and so on, where \lfloor x \rfloor denotes the greatest integer less than or equal to x—in other words, it means to round down. So the first number in the sequence would be \lfloor \pi \rfloor = 3, the next number would be \lfloor 2 \pi \rfloor = 6, and so on. Given this sequence, what can you learn about \pi (assuming that you didn’t already know anything about it)?

A more general question: given the sequence of integers \lfloor k r \rfloor for k = 1,2,3…, what can you learn about r?

The answer has many interesting connections to the theory of irrational numbers and continued fractions.

About Brent

Associate Professor of Computer Science at Hendrix College. Functional programmer, mathematician, teacher, pianist, follower of Jesus.
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3 Responses to Predicting Pi

  1. Nick Johnson says:

    Well, the obvious thing one can learn given just |(10^n)r| is the first n digits of the decimal expansion of r. I presume there’s supposed to be more to learn?

  2. Brent says:

    Well done! But you can actually learn the decimal expansion of r faster than that.

  3. Pingback: Predicting pi: pretty graphs and convergents « The Math Less Traveled

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