Two Equals Four

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Two Equals Four - 2011/04/14

I'm amazed that I need to say this, but it appears that I do ...

I know this is fallacious, and I do know what's wrong with it. Please don't assume that I really do think that 2=4, or that 1=2.


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Here's a cool puzzle.

Consider the equation $2=x^{x^{x^{x^{\ldots}}}}$ and suppose we want to solve it for x.

Because the exponential tower is infinite, we can also write it as $2=x^{\left({x^{x^{x^{\ldots}}}}\right)}$

But the part in brackets is the same as the whole, and hence is equal to 2. Thus we have $2=x^2.$

Hence $x=\sqrt{2}$ and so

Now consider the equation and again let's solve for x.

As before, we can write it as and again, the part in brackets is the same as the whole, and so now we get 4=x4

But take the square root of each side and we get 2=x2 and so again

So now we have

So is 2, and it's also 4.

Hence 2=4 (and halving it means 1=2).

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In an email, Iain Murray has pointed to exercise 4.20 on page 86 of the book:

  • Information Theory, Inference, and Learning Algorithms
    • by David J.C. MacKay FRS

A copy of the book can be found online here: The exercise itself is in this PDF: and the solution/discussion is on page 89:



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