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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 6s in

$$\frac{64}{16}.$$

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 are cases where one arrives at valid ends by (shockingly) wrong means.

Edit: fixed typo in last example.

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    Firstly, it seems like the result being right or wrong is rather unimportant if the means are wrong to begin with. Secondly, "shockingly wrong" and "seemingly valid" are totally subjective. What is seemingly valid to one person (but wrong) is probably shockingly invalid to another. Mathematical fallicies are probably the *only* interesting thing to discuss, here. Both of your examples illustrate fallacies that one could hold for fraction addition and "power addition".2012-12-17
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    @rschwieb: I don't understand your comment at all. The OP is asking for examples where, by wrong means, you end up with a *correct* result, and not where by "seemingly" valid means, you end up with a *wrong* result. It does not matter how you interpret "seemingly"; any "proof" that ends up with a wrong conclusion should not be posted as an answer here. That's the OP's choice.2012-12-17
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    This is reminding me of an anecdote about a physicist from the early 20th century with a reputation for making arithmetic errors who as a joke intentionally made a huge order of magnitude error (10^10???) in a published paper; and then published a correction the next month noting that the error didn't affect the results of the computation. Unfortunately I'm failing to Google it so I can't see if what he did would be relevant to this question or not.2012-12-17
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    The *College Mathematics Journal* used to have a section entitled "Fallacies, Flaws, and Flimflam." It regularly featured exactly these kinds of things.2012-12-17
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    @TMM I just meant that I think the OP is making a meaningless distinction between "wrong ends by invalid methods" and "right ends by invalid methods." There is no reason to set aside fallacies: fallacies are the most interesting thing to look at! Also, "seemingly" and "shockingly" are mostly subjective, so it adds little to the question.2012-12-17
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    Highly related: http://math.stackexchange.com/questions/1236332012-12-17
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    Why is this *on topic*?2012-12-17
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    There is a whole book about this "Mathematical Fallacies, Flaws, and Flimflam". This book is exactly what you are looking for. It has scores of such examples in a whole bunch of categories like algebra, calculus, multi-variable calculus, and so on. Love this book!2012-12-17
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    Also somewhat related: [Successful absurd formalities](http://math.stackexchange.com/questions/168051/successful-absurd-formalities)2012-12-17
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    @Dan: It also reminds me of another anecdote about a physicist from the early 20th century. He had a horseshoe on his wall. Someone asked, "Surely you, a world-renowned physicist, don't believe that horseshoes bring good luck?!" He replied: "Of course not, but they say it works even if you don't believe in it." (The somewhat stretched analogy being that he got the right result (good luck) despite applying the wrong method (not believing))2012-12-18
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    You might find that "Howler" is a good search term for this, which I've seen used before.2012-12-18
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    @DanNeely: There was something like this in the Journal of Irreproducible Results.2012-12-18
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    @Anko: (a) Some ‘successful’ errors of this type lead to at least mildly interesting mathematics, e.g., when does this particular error yield the right answer? (b) Good examples are potentially useful for teachers of mathematics.2012-12-23
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    I see no good reason to have closed this question and have therefore voted to reopen it.2012-12-23
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    Try searching for "Lucky Larry" sometime...2013-04-03
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    @JM: searching for "Lucky Larry" was exceptionally irrelevant to this question, AFAICT.2013-04-04
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    @kjo, I had [this](http://www.amatyc.org/publications/AMATYC-Review/Fall2008/) in mind. Search that site for other issues, all featuring Lucky Larry. (For some reason that message did not ping me; I only came upon this thread again by accident.)2013-05-18
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    i guess a *lesson* here is that if one is *consistent* in the *inconsistencies* one uses, then indeed it can be just as valid (remember Poincare and conventionalism??) :)2014-06-16
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    He means when young studens who don't get at all the basic operations have mistakes and do something wrong but the answer is right, I saw this all the time.2015-05-10

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