Dirac Ruined Everyone’s Fun

Graham Farmelo’s biography of Paul Dirac – “The Strangest Man” – is an incredibly good book. Trust me, it’s awesome. It’s full of a whole truckload of interesting details about Dirac and his life that I’d not have expected, and it provides a compelling account of his life and the development of Quantum Mechanics which he pioneered.

Today I’m going to do a complete disservice to the book and look at the least consequential part of it. It’s a game that used to be played by mathematicians (before they invented World of Warcraft) whereby the challenge lies in restricting one’s starting conditions. To quote:

“The challenge was to express any whole number using the number 2 precisely 4 times, and using only well-known mathematical symbols. The first few numbers are easy;
1 = (2+2)/(2+2)
2 = (2/2)+(2/2)
3 = (2×2)-(2/2)
4 = 2+2+2-2”

Have a try yourself, the game gets very challenging very fast. I personally managed to get to ten before giving up. What’s most interesting is that you *can* express every number using exactly four twos. Dirac was so good at this game that he ruined everyone’s fun by being TOO good at it.
How?

Well, Dirac did something that’s at the same time very clever and incredibly obvious. He expressed an equation that allows one to express ANY number using exactly four twos, precisely within the rules of the game. Here’s the version given in the notes at the back of Farmelo’s book;

n = -log{base2} [log{base 2} (sqrt(sqrt(…2)]

Where the ellipsis indicates that the square root must be repeated n times. The “exactly four times” rule is sidestepped by writing the first square root with a 2 (like you might write a cube root). It looks rather impressive on first glance, doesn’t it? Want to know how he cheated?

It’s possible to derive this expression from first principles. Consider the equation
n = n

Pretty simple. Pretty obvious. But when you think about it, this is exactly what you want to say as simply as possible using exactly four twos. So now what we need to do is apply some operations to one side in such a way that they all cancel out. Start by raising 2 to the power of -n:
n=1/(2^n)
This obviously isn’t true, so we need something to cancel this out! Taking the log base 2 of 2 returns 1 so:
n = -log{base 2} (2^-n)
Which using the laws of logs is clearly just n=n. So we’ve used two 2’s already! Excellent! Now what?
Well we could cop out by raising all of that to the power of (2/2), but Dirac wanted something cleverer. So what he did was introduce another pair of log base 2 and 2:
n = -log{base 2} [log{base 2} (2^(2^-n))]
Since log{base 2} (2^(2^-n)) = (2^-n) this reduces to:
n = -log{base 2} (2^-n)
Which is what we had before!

So n = -log{base 2} [log{base 2} (2^(2^-n))], by simply making operations on a basic statement that n=n.

Have you spotted the problem yet? What if you want to express n=2? Then you’d have to use FIVE 2’s! Oh dear! So Dirac even found a way around this by removing the n from the right hand side and introducing this strange ellipsis notation. Then when you want to do n=2, you have no problem!

Another interesting thing about this equation is that you can extend the rules to write a number n using only the number m exactly 4 times and by replacing all 2’s with m’s, you get a perfect result! How’s that for clever!

So yeah, Dirac ruined the mathematicians fun, but I think the best thing about the anecdote is that he did it by showing them they were working on a trivial problem. That takes some balls.

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About stoove

A physicist, researcher, and gamesman. Likes to think about the mathematics and mechanics behind all sorts of different things, and writing up the thoughts for you to read. A competent programmer, enjoys public speaking and mechanical keyboards. Has opinions which might even change from time to time.
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3 Responses to Dirac Ruined Everyone’s Fun

  1. Arnold says:

    Actually, it seems to me that the book is about the human being, not the mathematician. I have heard that it an excellent book. It is being reviewed by Elaine Charles on her radio show “The Book Report” this coming weekend 21 to 22 April. Check out the website on http://www.bookreportradio.com to see how you can listen. I certainly want to hear this review in order to decide whether or not I want to read the book.

    • stoove says:

      Yes you’re right I suppose the book’s not just an account of the famous mathematical physicist, it’s an account of the man in all his aspects. On the other hand it’s clear upon reading the book that maths and physics was a huge part of what made up Dirac’s personality and contributed to his persona. What I found best about the book is that it’s a genuine emotional rollercoaster – the account of Ehrenfest’s death in context of Dirac’s opinion of him is particularly moving and I’d recommend it just for those few pages.
      If you appreciate Physics, the account of the creation of Quantum Mechanics is also just sublime.

  2. Pingback: Arithmetic Puzzle | Gaurish4Math

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