On Dec 2, 2012, at 6:11 AM, GS Chandy <firstname.lastname@example.org> wrote:
> I've always wondered why (and still do wonder why) we even have to call it the "square root of -1" at all [whether that's taken to be 'minus 1' or 'negative 1']. > > Why not just call it, say, 'i'? > [Which has the property i (x) i = -1].
I am not sure I understand the problem with calling it the square root of -1. When they are introduced it is obvious that it is the square root of -1, why not go with that?
Dolciani introduces them in trig using axes in an entirely arbitrary way with no indication of any use. In a much later section, one that probably no student ever reaches, the book uses them to establish the fundamental theorem of algebra.
I would introduce them after quadratics and simply study those quadratics that we were unable to solve (because the component inside the radical was less than zero). First the simplest cases, with no real part involved, that establishes i = sqrt(-1). Then the cases that call for a real part as well as an imaginary part. At this point I would state that we are simply extending our set of numbers in the same manner in which we did when we couldn't solve x = 4 - 8 until we introduced negative numbers. Likewise there will be application for these strange numbers just as there were for negative numbers. Although, those applications are generally difficult to do in an algebra 2 class. I have a few years before my son tackles algebra 2 so I guess I should start hunting for decent applications. Another aspect, also difficult, is how complex numbers simplify the math in some situations, just like logarithms do in other situations.
One of Beberman's videos deals with isomorphism. I am not sure why he was teaching isomorphism to high school math teachers. Unfortunately a key element in the death of "New Math" no doubt. Nonetheless, it is an interesting video, but I think only if shown after completing the development of an an actual case of isomorphism, like the use of logarithms to perform complex arithmetic, but without mentioning isomorphism. In other words as a deeper tickler of what you just did. Likewise, you could introduce the notion of "field" after completing a development of complex numbers, but both notions (field and isomorphism) would be very frail (virtually useless) until the students dive more into the ramifications later on. But these ticklers do serve as inspiration (to the mathy) to go further down the rabbit hole.