Sunday, May 31, 2015

The Random Things My Head Comes Up With: Pi

I'm just lying in bed the other night, having woken up at a random hour and unable to sleep, a random thought pops into my head. This isn't the time I usually have random thoughts; rather I get them all the time. If I'm awake, my mind is spinning.

Hopefully, you all know the basics about about pi, the number that represents the ratio between the distance across a circle (the diameter) and the distance around it (the circumference); and that it's an irrational number (one that cannot be expressed by a fraction) and has no known pattern to its digits.

For people who like to make jokes about irrational numbers, and often pi in particular, is that they'll suggest that somewhere inside its digits, you can find any sequence of numbers.

That's just the preface, not what was spinning around my head. What I was actually thinking follows:

You can't find any sequence of numbers in pi. Or if you can, then it's not truly non-repeatingly endless.

Sure, maybe it's child's play (and reasonable) to suggest you could find someone's phone number or social security code within the digits of pi. That's fine, because they're relatively short series of numbers, and the shorter the series, the greater the likelihood of finding it, and the shallower (closer to the decimal point) you would find it.

However, if it were perfectly and completely true that you could find any sequence of digits within pi, then you would be able to find the complete series of decimal places of pi itself within its own digits (aside from the initial series, immediately following the decimal point). If 14159.... (and continuing onward, all the way) existed within pi, then the point at which these digits appeared, pi would repeat. And repeat. And repeat. Being a self-referential sequence, it would have to include the sequence of it quoting itself within the sequence of it quoting itself.

If this were true, then pi would no longer be non-repeating. It would be stuck in a loop of repeating itself, and thusly, have a sequence of numbers that repeats. Now, unlike some decimals which might repeat a short series of digits (0.33333... or 0.4545454545...) the one contained within pi's decimal would be considerably more complex (taking into account how many digits have been calculated and still no pattern or repetition becoming apparent), but it would still be repeating nonetheless.