There's an easy way to make the mistake. To define something, it must exist and be unique. Therefore "define the square root of a real number to be the real number so that when you square it you get the original number." This is wrong because "the" is broken in two ways: it implicitly asserts something exists (it doesn't for negative numbers), and that it is unique (it doesn't for positive numbers).

See also assuming unique antiderivatives, assuming unique coset representations of groups, etc. In general confusing a representation with a definition does the trick. See whether .999... equals 1. The anti-intuition comes from the fact that most people associate numbers with their decimal representation very strongly (and, from a mathematical theory standpoint, incorrectly) so that if two numbers "look" different in their representation on paper, they must also be different.