Can’t argue with success? Looking for “bad math” that “gets away with it”

I’m looking for cases of invalid math operations producing (in spite of it all) correct results (aka “every math teacher’s nightmare”).

One example would be “cancelling” the 6’s in


Another one would be something like

$$\frac{9}{2} – \frac{25}{10} = \frac{9 – 25}{2 – 10} = \frac{-16}{-8} = 2 \;\;.$$

Yet another one would be

$$x^1 – 1^0 = (x – 1)^{(1 – 0)} = x – 1\;\;.$$

Note that I am specifically not interested in mathematical fallacies (aka spurious proofs). Such fallacies produce shockingly wrong ends by (seemingly) valid means, whereas what I am looking for all cases where one arrives at valid ends by (shockingly) wrong means.

Edit: fixed typo in last example.


I was quite amused when a student produced the following when cancelling a fraction:


He began by “cancelling” the $x$ and the $y$ on top and bottom, to get:


and then concluded that “two negatives make a positive”, so the final answer has to be $x+y$.

Source : Link , Question Author : Community , Answer Author : Community

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