Partly Paradoxes, Part 4

Ahh, here we are at last!  We have arrived at the final five memes in our Partly Paradoxes miniseries.  Perhaps you’d like to check out Part 1, Part 2, and Part 3 before proceeding.

As before, we’re still working our way through the Cracked.com article called 20 Paradoxes Most Human Minds Can’t Wrap Themselves Around.  It’s not that the meme-based article is particularly bad; it’s just that the person who compiled the memes doesn’t seem to fully understand what a paradox is.  I’ll admit, it can be difficult to determine whether a proposition is truly paradoxical or not, but that’s why I’m doing this.  I thought it would be an educational and challenging mental exercise to examine each of the article’s memes and determine whether they depict paradoxes or something else.  And you know what?  It has been!  I feel that I’ve learned a lot over the past week, and if you’ve been following along, I hope you did too.  So without further ado, let’s finish this!

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Partly Paradoxes, Part 3

Let’s continue our whirlwind tour of Cracked’s roundup of 20 Paradoxes Most Human Minds Can’t Wrap Themselves Around.  If you haven’t already read them, you might want to check out Part 1 and Part 2.

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A Slice of Infinity Pi

Pi Worlds

Remember pi from your high school math classes? Well just in case you’ve forgotten, pi is the ratio between the distance around a circle (its circumference) and the distance across a circle (its diameter). A perfectly round, perfectly planar circle is about 3.14 times farther around than it is across. But it’s not exactly 3.14 times longer. In fact, there’s no way to write the exact relationship, because pi is irrational.

I don’t mean to say that pi isn’t logical or reasonable; I mean that pi cannot be expressed as the ratio of two integers. The numbers 18, 5/9, and -54.5 are all rational because they can all be written as fractions with whole numbers (positive or negative) in the numerator and the denominator. All rational numbers have a few traits in common: when written in decimal form, the digits after the decimal either come to an end (as in -54.5) or repeat the same pattern infinitely (as in 5/9, which is the same as 0.555555…).

Pi isn’t like that. There are no two integers you can think of that, when written as a fraction, will be exactly equal to pi. The fraction 22/7 is fairly close to pi, but not exactly equal.

And it’s not just that mathematicians have not yet been successful in finding the magic fraction that precisely captures the value of pi; it is utterly impossible to do so. The absolute irrationality of pi was first proved in 1761 by Johann Heinrich Lambert. But I digress.

What about the rest of this meme? Is it really possible to find every picture, every word, every social security number buried within the digits of pi? To be honest…nobody really knows.

Imagine that you had a random number generator – I mean a truly random number generator (you get them at the same place you buy frictionless pulleys and absolute zero meat lockers). Let’s say you set the generator to run day and night, churning out digits to fill the decimal places of a number that is to be infinitely long. In other words, eighty squillion years from now, when the last proton decays into a burst of energy and all other matter is gone, this random number generator will still be cranking out digits (having somehow avoided decaying itself…I’m still working out the logistics on that one). Invoke your God-like powers to travel to the end of space and time, where existence is an abstraction, and examine the infinite sequence of random digits. Yes, in this infinitely long sequence of randomly generated digits, you will truly find every conceivable series of digits. Want to find the telephone number of your first crush? Got it. All of Shakespeare’s plays, translated into Klingon and then represented in binary? Got it. A digital recreation of the Mona Lisa as it would look if it were painted by Andy Warhol? Got it. It’s all there, hidden somewhere within the infinite possibilities, if you know where to look.

Although the digits in pi are not randomly generated, the digits are expected to turn up in about the same manner as if they were randomly generated. In other words, in the first million digits of pi, you’d expect to see approximately 100,000 ones, 100,000 twos, 100,000 threes, and so on. And so far, that has held true, but we cannot claim to know that it will always be true. That’s the nature of infinity; even if we know one googol of pi’s digits, we still know essentially zero percent of them. There is always some possibility that, for reasons yet unknown (and perhaps unknowable), the digit seven will suddenly stop appearing in pi somewhere after the 44 quadrillionth digit. And if it does, then any sequence containing the digit seven that has not yet appeared…will never appear. In fact, it’s possible that all but two digits could stop occurring in pi, and pi could still go on forever without repeating. And although these may sound like mathematical abstractions – outcomes that will never occur – we cannot say for sure.

I would like this meme better if it contained some hedge words like “probably” or “maybe”, but it doesn’t. It’s presented as a sexy math fact – a quotably nifty truth that might not actually be true. It has that “gee whiz” appeal, but it does little to prepare one for the brain-busting unknowableness of the infinite, with its endless possibilities. And for my money, it’s aimed in the wrong direction. To me, it’s a far less interesting possibility that pi might contain every imaginable number sequence, and a far more interesting possibility that it might not.