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I stumbled across the following computation proving $2+2=5$

calculation proving 2+2=5

Clearly it doesn't, but where is the mistake? I expect that it's a simple one, but I'm even simpler and don't really understand the application of the binomial form to this...

Michael Rozenberg
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Paul Uszak
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    There are many such posts with "fake proofs" that are fun to review, for example; http://math.stackexchange.com/questions/438/why-sqrt-1-times-1-neq-sqrt-12 – Moo Jan 02 '17 at 00:54
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    $(-1)^2 = 1^2 \implies -1 = 1$! –  Jan 02 '17 at 01:49
  • Don't think that the edit is appropriate. Someone is suggesting that I replace the actual question with the answer from a previous question, which would kinda make my not knowing the answer initially to my question moot. Perhaps it sits better on philosophy.SE :-) – Paul Uszak Jan 02 '17 at 02:21
  • Why do you believe $\sqrt{a^2} = a$? It doesn't. Example: $\sqrt{(4-9/2)^2} = 1/2 != 4 - 9/2$. It doesn't. Why did you think it does? – fleablood Jan 02 '17 at 03:47
  • @fleablood It's really down to an abusive childhood and very poor education interspersed with long periods in borstal. On the plus side, if we all knew all the answers and were generally omniscient there wouldn't be any point in having forums like Stack Exchange to bring joy and meaning to our lives would there? I'll try to think up a cleverer stupid question next time. – Paul Uszak Jan 02 '17 at 04:14
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    One way to find where the error is, would be to see on which line does it stop saying true things and on what line does it start to say false things. Lines 1-4 are all true. Lines 5-7 are all false. So 4 must somehow not imply 5. 4 says $ (-1/2)^2 = (1/2)^2$ and line 5 says $-1/2 = 1/2$. So can you see 4 does not imply 5? – fleablood Jan 02 '17 at 07:25
  • The binomial is slight of hand to distract you. But what it says is. $16 -36 + 81/4 = 4^2 -249/2+(9/2)^2 = (4-9/2)^2$ and $25-45+81/4=5^2-259/2+(9/2)^2=(5-9/2)^2$ so $(4-9/2)^2= (5-9/2)^2$. Which is true as they are squares of terms that are opposite signs of each other. – fleablood Jan 02 '17 at 07:34

3 Answers3

26

Because $x^2=y^2\Leftrightarrow x=y$ is not true.

Also $\sqrt{x^2}=x$ is wrong: actually, $\sqrt{x^2}=|x|$.

πr8
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Michael Rozenberg
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The error is in the step where the derivation goes from $$\left(4-\frac{9}{2}\right)^2 = \left(5-\frac{9}{2}\right)^2$$ to $$\left(4-\frac{9}{2}\right) = \left(5-\frac{9}{2}\right)$$

In general, if $a^2=b^2$ it is not necessarily true that $a=b$; all you can conclude is that either $a=b$ or $a=-b$. In this case, the latter is true, because $\left(4-\frac{9}{2}\right) = -\frac{1}{2}$ and $\left(5-\frac{9}{2}\right) = \frac{1}{2}$. Once you have written down the (false) equation $-\frac{1}{2} = \frac{1}{2}$ it is easy to derive any false conclusion you want.

mweiss
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  • I am curious why this was downvoted. Does the downvoter have any feedback or comment on how it could be improved? – mweiss Jan 02 '17 at 02:38
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$9^2 = (-9)^2$
But $9 \ne -9$

If we had taken as $\left|4 - \dfrac{9}{2} \right|=\left|5 - \dfrac{9}{2} \right|$
$0.5=0.5$

Kiran
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