Robert Hansen posted Dec 3, 2012 4:18 AM (GSC's response interspersed) > > On Dec 2, 2012, at 6:11 AM, GS Chandy > <email@example.com> 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? > Well, my problem is (was, see below) with the extra 'complexification of thought' caused by kids trying to understand (often unsuccessfully), the very concept of the square root of -1.
However, your explanation below of introducing this concept after quadratics takes away my objection. This is a very good reason to add the extra concept of "square root of -1". Indeed, when introduced like this, I believe the concept would help to clarify for most kids. By the way, at another level, this does indicate, I believe, the validity of my oft-expressed idea that we need to go to the formal and the abstract only via real and concrete.
(I'm not familiar with Dolciani, so shan't comment on any of that. I'm not able to see videos well on my system - but did appreciate the writeup about Max Beberman that you had linked to on the other thread).
Anyway, thanks for that explanation, which removes my doubts - and my objections.
GSC > > 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. > > http://glifos.cah.utexas.edu/index.php/MBeberman:E_mb_ > 0005 > > Bob Hansen