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I find this slightly funny. I saw this on a meme:$$\begin{align}a=x\\ a+a=a+x\\ 2a=a+x\\ 2a-2x=a+x-2x\\ 2(a-x)=a+x-2x\\ 2(a-x)=a-x\\ 2=1\end{align}$$ How can these strange algebraic manipulations not be true? I feel that this is a stupid question but it's just weird how people come up with these things. Is there some kind of theorem or something that would make this untrue?

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The problem in many of these false proofs becomes more apparent if you choose numbers for the variables. – Jair Taylor Dec 6 '12 at 5:35
By all these responses I'm feeling kind of dumb... I just thought it was funny that someone actually took the time to do this. – TheHopefulActuary Dec 6 '12 at 5:54
up vote 8 down vote accepted

Yes there is:

Theorem: If you divide by $0$ you can prove anything ;)

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And by Godel, you can't prove everything. Take the contrapositive, and voila! – Jebruho Dec 6 '12 at 5:33

you can't devide two side by $(a-x)$, beacuse $a-x=0$.

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As soon as you get to the fourth step, your equations are all equivalent to $0 = 0$, and you need to be careful about dividing.

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You first assumed $a=x$ so the line before the last one is $2(a-a)=a-a$.

Indeed $2\cdot 0 = 0$ but you can not deduce $2=1$ from it since you divided by $a-a=0$.

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