Riffing on mistakes

I mentioned on Twitter yesterday that one way to relieve the boredom of grading math papers is to explore mistakes. If a statement is wrong, what would it take to make it right? Is it approximately correct? Is there some different context where it is correct? Several people said they'd like to see examples, so this blog post is a sort of response.

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One famous example of this is the so-called Freshman's Dream theorem:

(a + b)^p = a^p + b^p

This is not true over the real numbers, but it is true, for example, when working with integers mod p.

(More generally, the Freshman's Dream is true in any ring of characteristic p. This is more than an amusing result; it's useful in applications of finite fields.)

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A common misunderstanding in calculus is that a series converges if its terms converge to zero. The canonical counterexample is the harmonic series. It's terms converge to zero, but the sum diverges.

But this can't happen in the p-adic numbers. There if the terms of a series converge to zero, the series converges (though maybe not absolutely).

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Here's something sorta along these lines. It looks wrong, and someone might arrive at it via a wrong understanding, but it's actually correct.

sin(x - y) sin(x + y) = (sin(x) - sin(y)) (sin(x) + sin(y))

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Odd integers end in odd digits, but that might not be true if you're not working in base 10. See Odd numbers in odd bases.

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You can misunderstand how percentages work, but still get a useful results. See Sales tax included.

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When probabilities are small, you can often get by with adding them together even when strictly speaking they don't add. See Probability mistake can make a good approximation.