1 = 2

Many people have seen “proofs” that \(1 = 2\). However there are always a rather obvious fallacy - a division by 0.

A classic such example is:

\[\begin{eqnarray*} a &=& b, \\ a^2 &=& ab, \\ a^2 - b^2 &=& ab - b^2, \\ (a+b)(a-b) &=& b(a-b), \\ (a+b) &=& b, \\ a+a &=& a, \\ 2a &=& a, \\ 2 &=& 1. \end{eqnarray*}\]

This is a decent “trick” to show people, but as I said above, the mistake simply comes from the division by 0 on the 5th line.

However, here is a different “proof”, can you see the mistake?

Consider the equation, \(2=x^{x^{x^{\ldots}}}\), with an infinite number of \(x\)’s. If we add some brackets to the equation we get, \(2=x^{\left(x^{x^{\ldots}}\right)}\) and then we can substitute the first equation in to get:

\[2=x^2.\]

So \(x=\sqrt{2}\), and hence

\[2=\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\ldots}}}.\]

Then if we repeat the exact same above process with the equation \(4=x^{x^{x^{\ldots}}}\) we see that \(4=x^4\) and so again \(x=\sqrt{2}\). So this time,

\[4=\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\ldots}}}.\]

And so \(2=4\) and thus \(1=2\)!

Where is the mistake?