Jim wrote: << The former is not just a subset of the latter. For one thing, I think high school teachers tend to be punctilious about some matters that university mathematicians are sloppy about. A case in point is the meaning of "imaginary part". Is the imaginary part of a+bi equal to b, or to bi? I don't think that I myself am consistent about this. I got the impression that Barry Mazur, in his book "Imagining Numbers", used a convention at odds with what's being taught in high schools these days, because one irate amazon reviewer of the book opined that Mazur was not qualified to write a book on advanced math because his misuse of the term "imaginary part" showed that he didn't even know basic high school math . . . . . .

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My own experience is quite the opposite. In high school I was never clear about what the imaginary part of a complex number a+bi is (ib or b ?). In college and subsequently I've learned that there is a universal, unequivocal convention: Im(a+bi) = b. Likewise, I recall in H.S. thinking the phrase "square root", or the square-root sign, meant *both* square roots of a complex number, and especially of a nonnegative real. In college & ever since, it's been made very clear that the square-root sign -- when applied to a nonneg. real -- is always the square root >= 0. Mazur glosses over this point and others, since there is no such thing as "the" square root of -15: it has two. (But it's no problem, since his intended audience is not at all the technically minded.) Based on the Amazon reviews, he seems to have succeeded well. In the same vein, all inverse functions that have multiple branches were described only vaguely in H.S. (Above all, in going from cartesian to polar coordinates, the inverse tangent was bandied about with great insouciance.) But in college I learned that, at least for the six inverse trig functions on the reals, there is a standard domain always assumed. (But it's entirely possible that, more recently, H.S. teachers have become more punctilious about such questions.) Finally, cones: the cone is one of many words in math that, unfortunately, are used differently in different contexts. In some contexts (e.g., real quadratic forms) modern researchers still discuss quadric surfaces and use cone to mean both nappes, as in H.S. In probably most math contexts a cone is generalized to mean any subset of a linear space L that is the union of all rays from the origin through any point in some subset S of L -- i.e., C(L) = {tv : v is in S and t >= 0}. In topology there is even a third, slightly different meaning of cone. --Dan