While browsing I came across a weird proof which says 2 + 2 = 5. The proof is like this:

enter image description here

After going through this for almost 30 minutes, I was not able to figure out the mistake in this. What is wrong in this?

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    $\begingroup$ math.stackexchange.com/questions/457490/22-5-error-in-proof already covers this question. $\endgroup$ – JB King Apr 30 '14 at 5:26
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    $\begingroup$ By the way, it's easy to find mistakes in this sort of reasoning: just evaluate each line, one by one, and see where the expression switches from being equal to $4$ to being equal to $5$. $\endgroup$ – 6005 Apr 30 '14 at 5:28
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    $\begingroup$ To summarize the answers, the point is that $\sqrt{x^2}\neq x$. Few high school students seem to realize this fact. (Ask a high school student to solve the equation $x^2=4$ and at least half of them will say "take the square root of both sides to cancel out the square, so $x=2$.") What's true is that $\sqrt{x^2}=|x|$. The equation $\sqrt{x^2}=x$ holds only when $x\geq 0$. $\endgroup$ – symplectomorphic Apr 30 '14 at 5:32

$$ 4 - \frac92 \ne \sqrt{\left(4 - \frac92\right)^2} \text{ since } 4 - \frac92 < 0. $$


$$\sqrt{(4-\frac 9 2)^2}=|4-\frac 92|=\frac12 \neq -\frac12 $$


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