WitCH 117: The Roots of Disunity

There is still a big issue in the previous WitCH that I believe no one has noted, but let’s continue. Here are excerpts from the next subchapter of Nelson‘s chapter on complex numbers.

 

12 Replies to “WitCH 117: The Roots of Disunity”

  1. This only applies when n is an integer. I don’t like the notation with the nth root of 1. Typo in step 4 of the worked example, principal arguments for k = -2,-1,0,1,2. It would be useful to mention that the equal spacing between the solutions is about an angle.

    1. Thanks, Deejay. The use of \boldsymbol{\sqrt[n]1} here is hilarious. One can consider general n, but of course the text is only considering n a positive integer, and so should declare this.

  2. How many mathematics teachers have taken a subject on complex analysis at university? How many universities offer such a subject?

    1. 1. Few, I would guess, but it doesn’t matter. With a coherent text, the teacher would be fine. The problem is whether the textbook authors have taken an appropriate subject (and, if so, what grades they received).

      2. Almost all, I would guess, although the subject may suck. Unimelb’s subject used to suck, and maybe still does. Monash’s subject used to suck, and I believe still does.

        1. Hi, U S. It’s a good ten or fifteen years ago, and a lot has changed in unimelb maths. Mostly for the worse, but I have no idea what the the complex subject(s) are like now. But I’ll explain.

          When I was at or semi-at unimelb there were two 2nd year complex subjects, regular and advanced. It was (at least) the advanced subject that sucked, and it sucked in a way that pure maths subjects too often do: the subject was too pure for its own good.

          Complex analysis is an insanely beautiful and rich subject. There’s simply no way you can fit in everything you’d want to teach in a first subject. So, time is incredibly precious and you want to choose what to cover, and the depth in which to cover it, with great care. You want to leave as much time as you can for the really good stuff. Moreover, a lot of the analysis technicalities are not so important for getting a working sense of complex. In a 2nd year real analysis subject, the details are the subject: you either get your hands dirty with hard ε-δ proofs and the like or you’re just piss-farting around. But a complex subject, while still needing some care with definitions and the analysis-topology basics, is not so dependent upon delving into the foundations.

          I tutored the advanced unimelb subject, and what I remember is weeks of fussing with the ground level stuff. I remember kids having to work out the principal values of inverse trig functions and their domains and whatnot, and thinking who the hell cares?

          I’m sure there was also, eventually, good stuff in the subject, but I can’t remember it. What was memorable was how much of the subject was boring and ugly. For a subject like complex, it was a hell of an achievement.

      1. If the fundamental theorem of algebra is taught, as per WitCH 118, I’d argue that it does matter that teachers are taught Complex Analysis.

        1. Hi, Joe.

          In practice, FTA is not taught at all and is not in the syllabus at all except for: all (non-constant) polynomials have roots. I don’t see how a course in CA helps with that.

          Of course, any solid background is a plus. And the systemic weakness of the vast majority of new teachers is a disaster. But there is plenty of PD I’d be advocating before a course on CA.

  3. Cheesus. It would actually be a challenge to make this incoherent garbage less comprehensible. This is not even wrong. Or, to put it in another way: If mathematics is the art of identifying different things, then you must be very advanced down under.

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