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Symmetry in Mathematics

Mark Dominus describes trying to convince some people that, while i (the square root of minus one) and -i are not equal to each other, they are mathematically indistinguishable:

...  1 has two square roots that are not interchangeable in this way. Suppose someone tells you that a and b are different square roots of 1, and you have to figure out which is which. You can do that, because of the two equations a2 = a, b2 = b, only one will be true. If it's the former, then a=1 and b=-1; if the latter, then it's the other way around. The point about the square roots of -1 is that there is no corresponding criterion for distinguishing the two roots. This is a theorem.  ...

What struck me about Mark's account was that failed to mention that this a symmetry and that symmetries such as this one are important in abstract algebra.

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