TITLE: What is the fallacy in this proof?
QUESTION [14 upvotes]: I came across this funny proof-

$$4$$
$$=4+\frac 92-\frac 92$$
$$=\sqrt{(4-\frac 92)^2}+\frac 92$$
$$=\sqrt{16+\frac{81}{4}-36}+\frac 92$$
$$=\sqrt{25+\frac {81}{4}-45}+\frac 92$$
$$=\sqrt{(5-\frac 92)^2}+\frac 92$$
$$=5-\frac 92+\frac 92$$
$$=5$$

I suspect,that the error is in the second line where operation on negative is done before positive violating $BODMAS$ rule.But,$\sqrt{(4-\frac 92)^2}=4-\frac 92$ and the similar solving continues.So, where is the real error?
Thanks for any help!!

REPLY [2 votes]: The equality
$$\sqrt{\left(4-\frac92\right)^2} = \sqrt{\left(5-\frac92\right)^2}$$
is equivalent to
$$\sqrt{\left(-\frac12\right)^2} = \sqrt{\left(\frac12\right)^2}$$
which of course is true, but that does not imply
$$-\frac12 = \frac12$$
Adding the first square and square root is invalid, this equality:

$$4+\frac 92-\frac 92$$
  $$=\sqrt{\left(4-\frac 92\right)^2}+\frac 92$$

does not hold.