WitCH 115: Not So Complex

Last year we took a multiwhack at VICmaths, Nelson’s Year 12 Specialist Mathematics textbook, specifically at Nelsons chapter on logic and proof: see here, here, here, here, here and here. This post is the beginning of a second multiwhack, this time at Nelsons chapter on complex numbers.

Why do this? Do we particularly have it in for Nelson? Not really, although the temptation is there. It is a very bad text, every chapter deserving of a multiwhack. But Nelson is hardly alone in this regard. Cambridge, for example, is also very bad and, being the respected textbook of record, is a much more important target. Indeed, we have already taken a doublewhack at the first part of Cambridge‘s complex numbers chapter, and we could have productively extended that whack to the entire chapter. But Nelson‘s complex chapter, while not as fantastically bad as their chapter on proof, is plenty bad, and is bad in specific and seemingly important ways.

Regular readers of this blog know well the back story. In 2021, VCAA stuffed up a complex numbers question on the Specialist Mathematics Exam 2, a stuff up which, still, VCAA refuses to acknowledge or to correct in their exam report. VCAA’s inexcusable failures resulted in their stuff up being repeated verbatim in Nelson (and Jacaranda). Then, in 2022, VCAA stuffed up again, in an almost identical manner, and once again, and still, VCAA refuses to acknowledge the stuff up or to correct their exam report (Word, idiots). VCAA’s subsequent defence of their 2022 question was completely mad.

This post, however, is not about VCAA’s lack of professionalism. Nor is it about VCAA’s incompetence. Nor is it about VCAA’s arrogance. Nor is it about VCAA’s cowardice. Nor is it about VCAA’s dishonesty. Nor is it about VCAA’s disconnect from reality. These are all givens. This post is about Nelson.

The madness of VCAA’s complex question defence, and the madness of the complex exam errors having occurred at all, seems to us very much in synch with the madness of Nelson‘s chapter on complex numbers. We have no idea why the two are in synch but that it is the case seems unquestionable to us. Thus, an understanding of all that is wrong with Nelson‘s complex chapter may provide an understanding of some that is wrong with VCAA. In doing this, we could have focussed more directly on specific wrongnesses in Nelson, and left alone the general wrongness and badness. But to do so would have arguably missed, or at least minimised, a broader point. In any case, these things are work and we get tired; WitCHes are easier.

Here, then and finally, is the first WitCH, on the first section of Nelson‘s chapter on complex numbers. Since Nelson launches the chapter with some presumed understanding, the first two snapshots are from Nelson’s companion Year 11 text, which introduces complex numbers from scratch, and which covers (very badly) the basic algebra and geometry. And just a quick note in self defence: we have tried very hard to be fair and to not cherrypick. Yes, we have selected the worst aspects and we will not reproduced some routine and less flawed material. But we will not intentionally exclude any meaningful explanations that somehow justify or mitigate the selected material. On the flip side, we will also not reproduce any of Nelson‘s nauseating CAS instruction, which is badly perverting and butt-ugly, but which is off the point of the WitCH(es).

19 Replies to “WitCH 115: Not So Complex”

  1. Looks like they’re saying there are real numbers neither rational or irrational. Actually on second thought, maybe that green rectangle is just labelling that 2nd outer ring.

  2. An immediate error is defining i to be the square root of -1, and claiming that to be equal to the fact that i squared = 1. What about -i, then? So to write that the square root of -4 is 2i, as they have done, is execrable. They’ll have to dig themselves out of a hole when dealing with the roots of quadratics, given that the quadratic formula includes a plus/minus symbol. Aside from that outright and inexcusable error, the presentation is simply hideous.

    1. Unfortunately that statement is pretty much universal in VCE textbooks at present.

      Assuming that index laws (specifically surd laws) apply for negative radicands is (in my opinion) where a large part of the rubbish originates, but there is a lot more rubbish to be found here, mostly in the form of assumptions or generalisations made without much thought (or so it seems)

    2. Thanks, Alasdair. Note that Cambridge, with four maths PhDs on the cover, makes the same outright and inexcusable error.* Go figure.

      *) At least it did. I’m pretty sure the error is still there.

  3. Let me be honest: I don’t know how to teach complex numbers, though I know it can be done well. However, like sex and other wonderful things, I note that it is tempting to engage too early and to create a mess. So please forgive me for being boring. (What follows is a turgid attempt to expand on what Alasdair alluded to.)

    The move to complex numbers should be a magical experience – a time when scales fall from one’s eyes, and the whole world is seen afresh (even if it is not at first understood). The downside occurs when this magic is replaced by *pretend wisdom* – like “little kids trying to talk big”, getting things wrong, and not realising their mistakes.

    The problems begin at the very beginning. They have their roots in the standard misrepresentation of *the square root sign* in early secondary school. There we find routine confusion (in textbooks as well as classrooms) between
    (i) solutions to the equation x^2 = 4, and
    (ii) the two solutions x = (+/-)SQRT(4).

    There are plenty of authors and teachers who think, and occasionally write, horrid things like,
    “SQRT(4) = (+/-)2”.
    This is wrong – and a recipe for future trouble. (So I suggest that preparation for complex numbers goes back to how the SQRT symbol is introduced and used at ages 11-14.)

    At first “SQRT” is simply an operator which is applied to a specific number to give another number (SQRT(4) = 2). As such, it is a “proto-function”, but not yet a function. However, when we come to collect all these values together, and to think in terms of “variables” and functions, we know that there will be problems unless “SQRT” is *single-valued*.
    So first we exclude “negative x” (restrict the “domain” to the non-negative reals).
    Then we notice that, for k>0, solutions to x^2 = k come in pairs, with one solution being positive, and one negative.
    We can then make the *systematic decision*, and commit henceforth to writing “SQRT(k)” to denote only the *positive* solution.

    This then makes “SQRT” into a *function*. When we write “SQRT(k)”, we know exactly what it stands for.

    To summarise: when k is a positive real, the solutions to x^2 = k are like *identical twins*: they are “two of a kind”.
    However, one of the twins (the positive one) can be systematically identified – like the twin who emerges from the womb first. So we can write “SQRT(k)” without any ambiguity.

    But this should warn us of trouble ahead when we come to writing such things as “SQRT(-1)”.

    If we allow ourselves to imagine a solution “i” to the equation “x^2 = -1”, then its negative “- i” will be an equally good solution: x^2 + 1 factorises as (x + i)(x – i), so solutions come in +/- pairs.

    But it is completely wrong to pretend that “+i” is “the proper square root of -1”, and “-i” is its negative, lesser twin.
    These two solutions are more like Tweedledum and Tweedledee: they are identical twins; and we can see that there are two of them; *but there is no way to tell them apart*.

    This creates didactical challenges which are rarely discussed.
    The boring pure mathematician in me may feel inhibited; but the creative teacher may see this as too good an opportunity to be missed, and recognise that one can simply write “SQRT(-1)” and start to calculate for fun.

    But the notation “SQRT(-1)” (that appears everywhere), and the way it is then manipulated is in fact a nonsense! So the didactical error (if I am not mistaken) is *not* that we dare to write such nonsense (once upon a time we dared to write the at-the-time-nonsensical “3 – 5”), but that we *persist in writing it* when it becomes clear that it is genuinely nonsensical, and fail to clarify as soon as possible (but not too soon!) why it has to be replaced.

    “SQRT(-1)” is inescapably *two-valued*, and (like the twins Tweedledum and Tweedledee) there is no way to tell the two values apart; so none of the calculations and equations one writes using this symbol respect the principle one has spent the previous 11+ years trying to inculcate (namely that names should have clear meanings, and that ambiguity is to be avoided).
    So having introduced the notation as a creative wind-up, one should perhaps explain the problem clearly and move on sooner rather than later to using “+/- i”, with the clear understanding that “+ i” is in no way “positive”, or special (nor is it really “on the positive imaginary axis”: as complex conjugation will one day clarify, the complex plane is intrinsically ambiguous).

    This raises questions about “end of course assessments” that have still not made this transition, and that expect students to “Simplify SQRT(-12)”. (The standard solution generally presumes that students will use “SQRT(ab) = SQRT(a).SQRT(b)” as though it were a valid identity, ignoring the uncomfortable fact that, if SQRT(-2) and SQRT(-3) have the conventional meanings, then “SQRT(-2).SQRT(-3) = – SQRT(6)”.)

    1. Thanks very much, Tony.

      As you suggest, one might get away with introducing complex numbers with \boldsymbol{\sqrt{-1}}, and there’s an argument that it is natural to do so, but this must be done with great care, and with an eye to making the necessary correction as soon as it is natural, and definitely before it is required.

      The thing about the \boldsymbol{\sqrt{-1}} error that *really* gets to me, in both Nelson and Cambridge, is the immediate leap to the \boldsymbol{\sqrt{-4}} error. They’ve introduced complex numbers 30 seconds ago, and already they’re embarking upon unjustified and unjustifiable complex algebra. It demonstrates that the authors are not even trying.

      The entire chapter is like this. The chapter is riddled with unjustified statements, which are often so poorly phrased that they are incoherent or plain wrong. For complex numbers. It’s like pissing on the Mona Lisa.

      More generally, all of Specialist is like this. Specialist. The subject is just a pile of semi-coherent, semidemi-justified mathsy stuff. And pretty much no one cares or has any clue that anything is amiss. The teachers don’t. The students don’t. The textbook writers don’t. And the Godawful VCAA drones definitely don’t.

      Methods is a much worse subject, but Specialist is much more hateful. It is an absolute crime. And pretty much no one cares.

    2. I agree with all of your arguments regarding the didactical points.

      When I (rarely) teach complex numbers (which is outside the curriculum in Germany), I never use the square root symbol, or even the square root as a function on C. Everything relevant can by tied to solutions to quadratic equations, and for actual numerical computations it is sufficient to use positive real roots. It also has the advantage that this easily generalises to the interesting stuff, solutions to polynomial equations.

      But there is one point where I disagree. Let me phrase it as a question. If there is no (reasonable) way to define the “sqrt” function on C, because there is no way to distinguish between i and –i, how is it possible to define the “Im” function then, which does distinguish between i and –i?

      Clearly, there is no way to define “sqrt” as a continuous function, or such that the usual rules that apply to positive reals also hold for complex numbers. But if the “Im” function can be defined such that Im(i)=1, why not the “sqrt” function such that sqrt(-1)=i?

      In other words: I totally agree that it is a mortal sin to define i:=sqrt(-1), but it is okay to define sqrt(-1):=i.

      1. Tony has offered you an apology, below, but I’m not sure I understand the Im problem.

        There are two ways of setting up the complex numbers. The first way is to simply declare/imagine/rabbit-out-of-hat a number whose square is -1. We call that number i, write all the complex numbers as a + bi, and off we go. Then Im(a + bi) = b.

        The second way is to define complex numbers as ordered pairs (a,b), with the algebraic definitions. We then define i = (0,1) and then, again, Im(a + bi) = b.

        What then is the problem? True, however we do it, there is another number, let’s call it I, whose square is -1. Then, however we do it, we can write all complex numbers as A + BI, and we can define IM(A + BI) = B, and we could deal with these capitalised complex numbers in the same manner. But I don’t see how this negates or undermines a clear meaning for i and Im.

  4. Thanks for this and particularly to Tony as I have learnt something. I don’t think we did complex numbers in my four years of Pure Maths at uni and can’t remember this treatment at uni (in spite of an excellent maths education there). Is there a textbook written properly that I could see?

    1. I’m very surprised that a course in pure maths wouldn’t include complex numbers. How was the course structured, and what subjects were taught?

    2. JJ, if you did four years of pure maths without having done a subject in complex analysis then there is something distinctly not excellent about your excellent education.

      The fundamentals of the complex number field (including the basics of complex algebra) sometimes gets forgotten at uni, considered to be one those things that has been done in high school, although it is seldom done that well at school (even if not as badly as in Victoria). You might check out Spivak’s Calculus or Stillwell’s Elements of Algebra.

      1. Thanks Marty – I may have forgotten. Though I would have thought I would remember something like that.

        After all Stillwell was one of my lecturers!

        1. There was a time, not long ago, when “pure maths” at Monash was the thinnest of gruel. (No fault of Stillwell’s). It’s still not exactly a hearty soup.

  5. I must have been very lucky to have had the competent mathematician lecturing many years ago. A shrug and a brief comment “there is no such thing as an imaginary number(s). Complex numbers are simply pairs of numbers”.
    Addition and multiplication are then defined, former in a very natural way, mult in a pecular way.
    Historically though…
    Then I apologise for refering to 1st in a pair as “real” and the second as “imaginary” ( rolling eyes). We have to. The textbook, the syllabus terms. And “handy” if you want to impress students who do not do Specialis. Perhaps.
    Then, “can you find a complex number which, when squared, gives (-1,0) ?”
    “Aye, Sir” . Or “i, Sir”.

  6. In response to hjm.

    1. I don’t actually see it as a ” mortal sin” to write SQRT(-1), One always has to be prepared to “lie a bit” when breaking new ground (there is a lovely discussion on this at the beginning of one of Ian Stewart’s books defending the need to “simplify” when explaining mathematical ideas to a wider audience).
    Indeed, in some ways this “abuse of SQRT notation” may be a more natural classroom approach than the modern algebraist’s artificial question: “What about solutions to x^2 + 1 = 0?”. In the “real” world, this question doesn’t arise, since squares are never negative. (Like many weird things in mathematics, complex numbers don’t seem to have arisen in this way. Instead they seem to have forced themselves on our attention, when the formula (1543) for the *real* solution of a real cubic equation – a solution which has to exist – turned out to involve square roots of negative quantities that cancelled out.)

    2. All I meant to say was that,
    (i) when one first writes SQRT(2) one should stress the advantages of defining its meaning *uniquely* (as the *positive* solution of x^2 = 2), and
    (ii) when one later comes to write SQRT(-1), one should be aware that one is breaking with this important principle, and so be ready to explain why we have to move away from this to write complex numbers using “+/- i” as the two (indistingushable) roots of “x^2 + 1 = 0”.

    3. This move is simple but subtle.
    One is specifically *not* saying that “i = SQRT(-1)”.
    One is saying: “We can’t give a single meaning to “SQRT(-1)”. Any attempt to extend the SQRT notation in this way is inescapably ambiguous, so one cannot give “SQRT(-1)” a clear meaning. Our old idea of “SQRT(k)” as a specific solution (the positive one) of “x^2 = k” is no longer available: the quadratic x^2 + 1 = 0 has two solutions which are *truly indistinguishable*, so they come as a +/- pair, with *neither* of them being special. If one of them is called “i”, then the other is “- i”; but neither of them is “positive” or special. They are entirely interchangeable.”
    This will only be half understood when it is said first; but it will be slightly better understood when they discover that complex conjugation respects everything – addition, multiplication etc., so that “i” and “- i” are genuinely interchangeable: one can rename “- i” as “i” and nothing changes. (So we discover that “x^2 + 1 = 0” is in no way special: the “non-real, complex” roots of *any* polynomial equation with real coefficients come in indistinguishable conjugate pairs.)

    4. As so often, this is the first instance of something even more bewildering. SQRT corresponds to the exponent “1/2”, and is two-valued. nth roots become n-valued. But everyone was stumped when (as with asking about “SQRT(-1)” and finding it was inescapably two-valued) they started puzzling about how one might interpret “log(-1)”.

  7. Random thought…

    Students learn about squares and square-roots pretty early in school. Hopefully primary school, but in many cases, likely in Year 7.

    When we say 3^{2}=9 and then say \sqrt{9}=3 they are treated as two sides of the same coin.

    But when we say (-3)^{2}=9, it is hopefully also mentioned to students that this does not mean \sqrt{9}=-3.

    So there is a lesson learned early to be careful with the square-root symbol.

    When the statement i^{2}=-1 is written, with some care, writing \sqrt{-1}=i is not too problematic. Showing that (-i)^{2}=-1 also would be a nice way to bridge the gap partially, although it does risk over-complicating (no pun intended) the situation.

    The underlying issue, to me, is that the qualifiers that make this OK are not mentioned, not considered or both.

  8. I owe hjm an acknowledgment – and maybe an apology.

    When I said that I did not know how to teach complex numbers, I meant that I had not done it regularly enough to have sorted out in my own mind what works, and what doesn’t; what is helpful and what isn’t.
    It was (and is) clear(?) to me:
    – that one probably has to start by writing things like “SQRT(-1)” as the only way one can initially refer to a conjectured “entity whose square is -1”; and
    – that at some point (maybe sooner rather than later) one has to explain that, whereas “SQRT” is deliberately set up as a (single-valued) function on the positive reals, this new usage is an abuse of the SQRT notation: if we allow the equation “x^2 + 1 = 0” to have a root, then it has two roots; but (unlike with SQRT(2)), there is no way to tell these two roots apart (like Tweedledum and Tweedledee).
    At the first stage, we use “SQRT(-1)” as a tentative notation for something new.
    At the second stage, “SQRT(-1)” is recognised as being inescapably two-valued – and it is then natural to drop “SQRT(-1)” and to introduce our two identical twins “+/- i”.

    There is subtlety here – so there is scope for teachers to devise their own strategies.

    (i) Between the two above stages, we are probably all going to fumble along, writing things like “SQRT(-4)” and “SQRT(-12)” a though they are “well-defined” (within this magical New World) – even though they are not.
    For all I know, it may even be natural to interpret the quadratic formula for “b^2 – 4ac < 0" in the form of roots "u + vSQRT(-1)". It is unclear to me at what stage such notation may cause trouble – if only because the "+/-" in the quadratic formula provides partial (confused) protection from the fact that the "SQRT" is no longer a (single-valued) function.

    (ii) The move from writing "SQRT(-1)" to writing complex numbers in Cartesian form "u + iv" using "+/- i" does need some explanatory comment to set up the change. I suspect hjm was alluding to this in suggesting that one had to say something like "Henceforth write SQRT(-1) = i".
    I winced at this, but can see myself saying something similar – and being completely ignored by the class! ("We have been writing "SQRT(-1)" as though this had a clear meaning – like SQRT(2). In fact it has twin-meanings, neither of which can be identified separately. But they form a +/- pair, so it is best to give them fresh names "+/- i" and to avoid abusing the SQRT notation – writing complex numbers in the form "u + iv". This looks a bit like just replacing "SQRT(-1)" by "i" [or by "- i"] – which is fine provided we understand in retrospect that we were using "SQRT(-1)" all along to denote "one of the two indistinguishable roots of the equation x^2 + 1 = 0".")

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