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I have found something really weird which compelled me to put up this post. I wish I could resolve this problem but was unsuccessful.

Consider a positive integer say 7 , now is it possible to prove that it is equal to 8 and 8 is equal to 9 and so on? Quite puzzled! Have a look at this picture.

enter image description here

\begin{align*} 7 &= 7-\frac{15}2+\frac{15}2 = \sqrt{\left(7-\frac{15}2\right)^2}+\frac{15}2 \\ &= \sqrt{7^2-2\times7\times\frac{15}2+\left(\frac{15}2\right)^2}+\frac{15}2 \\ &= \sqrt{49-105+\left(\frac{15}2\right)^2}+\frac{15}2 \\ &= \sqrt{-56+\left(\frac{15}2\right)^2}+\frac{15}2 \\ &= \sqrt{64-120+\left(\frac{15}2\right)^2}+\frac{15}2 \\ &= \sqrt{8^2-2\times8\times\frac{15}2+\left(\frac{15}2\right)^2}+\frac{15}2 \\ &= \sqrt{\left(8-\frac{15}2\right)^2}+\frac{15}2 \\ &= \left(8-\frac{15}2\right)+\frac{15}2 = 8 \\ \end{align*}

If this is correct, we would be able to show that all positive integers are equal.

I am sure that it is not correct but I would appreciate if someone can point out what the mistake is? Or is there something to be pondered about?

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    please don't post picturs but use Latex – 2017-02-01
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    the problem is in the top line, $\frac{15}{2}$ is greater than 7, you have created a negative value, squared it making it positive, then taken a square root - the equality is only true if you consider the negative square root. – 2017-02-01
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    $\sqrt{x^2}=|x|$. – 2017-02-01
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    Yep, you took the negative sqrt branch at the top and the positive at the bottom. – 2017-02-01
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    I don't see why it needs to be marked down, he tried his own working, better than some questions that get transcribed here – 2017-02-01
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    Not a bad question, actually. I've had department heads tell me "requiring students to know $\sqrt{x^2}=|x|$" (not "x") is "nitpicking". – 2017-02-01
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    Probably a handful of these proofs hide in http://math.stackexchange.com/questions/tagged/fake-proofs – 2017-02-01
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    To add to Asaf's comment, here are a few examples of posts which are basically duplicates: [What is the fallacy in this proof?](http://math.stackexchange.com/q/1543111), [$4=5$. Is this possible?](http://math.stackexchange.com/q/2021178) or [$2+2 = 5$? error in proof](http://math.stackexchange.com/q/457490). – 2017-02-13

2 Answers 2

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There is a mistake in the second equality. Note that $7-\frac{15}2=-\frac12$ is negative, so

$$7-\frac{15}{2} + \frac{15}{2}= {\bf\color{red}{-}}\sqrt{\Big(7-\frac{15}2\Big)^2} + \frac{15}2$$

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The problem is that $7-\frac{15}{2}$ is negative.

Hence, $-0.5=7-\frac{15}{2}\neq \sqrt{(7-\frac{15}{2})^2}=|7-\frac{15}{2}|=0.5$