Editing TwoEqualsFour
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Consider the equation EQN:2=x^x^x^.^.^. and suppose we want to solve it for /x./ Because the exponential tower is infinite, we can also write it as EQN:2=x^{\(x^x^.^.^.\)}. But the part in brackets is the same as the whole, and hence is equal to 2. Thus we have EQN:2=x^2. Hence EQN:x=\sqrt{2}, and so EQN:\sqrt{2}^{\sqrt{2}}^{\sqrt{2}}^.^.^.=2. Now consider the equation EQN:4=x^x^x^.^.^. and again let's solve for /x./ As before, we can write it as EQN:4=x^{\(x^x^.^.^.\)}, and again, the part in brackets is the same as the whole, and so now we get EQN:4=x^4. But take the square root of each side and we get EQN:2=x^2, and so again EQN:x=\sqrt{2}. So now we have EQN:\sqrt{2}^{\sqrt{2}}^{\sqrt{2}}^.^.^.=4. So EQN:\sqrt{2}^{\sqrt{2}}^{\sqrt{2}}^.^.^. is 2, and it's also 4. Hence 2=4 (and halving it means 1=2).