WitCH 116: Polar Bare

This is our second WitCH on Nelson‘s chapter on complex numbers. As with our first WitCH, we have not excluded any definitions or arguments or explanations from the text that would fill apparent (and actual) gaps in the selected material; the rest of the subchapter consists of routine examples and less problematic (but far from unproblematic) exposition.

UPDATE (14/01/23)

With hindsight, it makes sense to include an excerpt from the next, very short, subchapter as well. (The excerpt below is followed by less annoying proofs of De Moivre for n = 0 and n < 0.)

65 Replies to “WitCH 116: Polar Bare”

  1. Every now and then, this book hints that it is going to answer the question WHY? but then never seems to get around to it. Which is a pity.

    Because without an attempt to answer the WHY? question, the rest of it all just seems a jumbled mess.

    And that is before we get to the section on straight lines.

    I will make one specific nit-pick though. Writing \tan{(\theta)}=\frac{x}{y} is fine as a starting point, but there needs to be some attempt to explain what happens outside the first quadrant…

    1. Thanks, RF. Note that I’ve added an excerpt from the next subchapter, on De Moivre, although it’s not related to your comments.

      I think there is much worse in the above, but you are correct, that the text somehow presents itself as if it’s going to offer solidly mathematical explanations, but then either provides nothing, or provides something completely bizarre. If the text were more straight-forwardly pedestrian, it would still be deficient but a lot less annoying.

  2. “The ray is drawn with an open circle at its origin because Arg(0) is undefined.”

    This is the first mention of Arg(0) being undefined. That should clearly be stated at the beginning of the chapter.

    De Moivre’s theorem is only true for integer exponents. The text implies that it’s true for non-integer exponents.

    I wonder, what are people’s thoughts on teaching e^{i\theta} = cis(\theta) during specialist maths?

    1. There was a 2014 MAV SAC 3 Application Task (back in the days when there were five SACs and the first two were tests) that was part of the MAV’s Fresh Starts program (now called MAV SACs Suggested Starting Points) that introduced and used the Euler formula:

      “Elementary Functions of a Complex Variable”.

      Students calculated things like ln(-4), ln(i) etc and solved trigonometric equations like sin(x) = 2 etc.

  3. I wonder if the author after writing this section tried to teach from it

    The introductory section has bolded the terms argument and principal argument, but it never explicitly distinguishes between the two terms. If you aren’t going to bother to explain the difference, why bother mentioning “argument” in the first place? A comment on why the the interval (-\pi,\pi] was chosen or why it has length 2\pi would be nice.

    Why is there no proof of the product and division formulas? It seems odd given they have no issues using compound angle formulas in the De Moivre proof. It would be nice if they explained the geometric significance of multiplication and included a diagram to illustrate ‘multiply the mods and add the args’.

    The straight line section is a mess. The first line promises some connections to the cartesian plane but it never seems to make these connections. It introduces the point (a, b) in \mathbb{R}^2 and never explicitly associates this with the complex number z. Why can’t it be consistent with the introduction where it defined z=x+iy? It then talks about the two identities where it treats z and its conjugate like numbers. In the next line it states that z+\bar{z} is horizontal and parallel to the real axis. But z+\bar{z}=2a is a real number? It doesn’t make sense to say the real number 2a is horizontal and parallel to the real axis. Do they mean the vector from 0 to z+\bar{z}=2a+0i is horizontal and parallel to the real axis (which is a bit redundant)? I’m not sure I see the point of this whole example.

    In the final section they never seem to explain that \theta is the angle measured anticlockwise from the positive x-axis. Are we to assume from the diagram? I wonder how the textbook expects students to sketch a semi circle where u and v have the same imaginary component using their method.

    1. Very few things are defined in this chapter and those that are defined are using the word “defined” quite loosely.

      I was tempted to check the Year 11 book to see if things were defined there, but that should not be a consideration.

      Modulus is defined poorly. Argument is not really defined at all, except maybe by stealth with the use of a diagram and reference to Principal Argument.

      As for the whole Arg(0) part…

      …since the book refers to complex numbers being vector-like, I wonder if this idea was covered with the introduction of the zero vector.

      1. Thanks, RF. I think you are correct, that *some* of the sloppiness with formalities is because the authors are assuming that students are familiar with the Year 11 coverage. One of the things that really irritated me in the first WitCH was Nelson starting the year 12 material with “Remember …”. It was an idiotic way to start, and it’s the wrong damn word: they meant “recall”.

        In any case, and as you suggest, any use of the Year 11 coverage as an excuse for a “that’ll do” approach in Year 12 would be absurd.

    2. Thanks, tango. A couple comments in reply, below, and i think there’s a very big screw-up that you haven’t mentioned, but you’ve covered it all well.

      I think the definition of principal argument is in the ballpark of adequate, although of course it’s very far from good. As I replied to joe, I couldn’t see that the text ever defines the Arg notation.

      I agree on the lines section. Just a mess.

      With the last section, the semicircle diagram is misplaced, and should appear after the rays discussion. And, the choice of θ for the triangle angle is hilariously bad.

  4. Question: does the text ask students to “remember” or “recall” the definition of a complex conjugate?

    If not, there are more than a small number of problems with this section.

    1. That’s not a reasonable question. If the text has announced from the beginning that one should remember/recall material on on a topic then they don’t have to have the official Remember Flag at the beginning of each sub-topic time. But it also doesn’t matter because the Remember Flag is not a magic wand that you can wave to avoid presenting material in a coherent manner.

      1. Sure. I ask mainly because of how the conjugate can be written in polar form and whether there was any attempt to link these ideas.

        I’ll focus more on what is written and less on what is missing, perhaps.

        1. I think it’s fair enough to focus upon what’s missing. I’m just suggesting chanting “remember” doesn’t fix things.

  5. I’ll continue with the Nelson complex WitCHes soon, but there’s one biggie (and a couple smallies) here that no one has yet mentioned.

  6. Second to last line in the Straight Lines section. Should u and v in the cis brackets be arguments, not the complex numbers themselves?

    Unless I have completely missed something here, taking the cosine and sine of a complex number is rather pointless, the argument of said complex numbers is surely required instead..?

    1. Hah. I hadn’t even noticed that. You’re right, that is plain nuts. Of course, the entire passage is nuts, but cis(u-v) is the cherry on top of the nuts.

    1. Thanks, Guardian. It seems nice (although it’s not like it’s hard).

      By the way, I’m still waiting for someone to identify another big screw-up in the above …

      1. Well, you probably never want to put the ray before the semicircle – basically it’s just like “poking the bear…”, a very dangerous move.

        In fact the language choices and all explanations in this whole section “poke the bears”. I sort of imagine the feelings (and maybe possible puns intended by your title “polar bare”).

        The provided induction proof is problematic and flawed. I couldn’t buy the logic jump between Zs and Ns… if they really want to establish the truth for all integer n they would still need to further elaborate by letting n=0, n=-1…etc for the completeness…

        Another thing I noticed (but might not be Marty’s intention) was the incomplete extension on semicircle – depending on the size of angle on the RHS there are possibilities of being a minor arc or a major arc.
        I also wonder why they didn’t make it clear: u and v are distinct/different complex numbers.

        1. Thanks, Guardian.

          The “bare” was to indicate being bare of mathematics. The language and presentation is appalling throughout, but I don’t understand your first point. The proof of De Moivre does include the arguments for n = 0 and n < 0, as I indicated in the parentheses.

  7. The argument of a complex number is not properly defined in this piece of text. Is it obvious that it is an angle between the positive real axis and the line joining the origin and z?

    And the definition tan(theta) = y/x is only valid for -π/2 < theta < π/2. That looks like quite a serious issue to me…

    1. Thanks, Deejay. These have already been pointed out. (I don’t think you’ve worded the θ issue quite correctly.)

    2. The equation tan(\theta)=\frac{x}{y} being true does not imply arctan(\frac{x}{y})=\theta and this really needed to be made clear.

      But I suspect there is something else here, hiding in plain sight that Marty is hinting at.

      Still pondering.

          1. That’s not the way WitCHes work. I could be mistaken, but I’m pretty sure there’s a very big that no one has mentioned.

      1. Diagram: (pi/2 + theta) instead of (pi/2 – theta)

        Also: The ray is drawn with an open circle at its origin, its origin should be u.

          1. Arg(z) = theta specifying a ray etc. is not an EQUATION.

            One solves an equation but not a specification!
            Sketch the ray according to its specification.

            1. I’m ok with calling anything with an equals sign an equation (even if another label might be more appropriate). In any case, that’s not what I’m thinking of.

              1. The correctness of the statement with ” right hand side/left hand side of the diameter from u to v” depends on the positions of u and v in the Argand plane.

                No diagrams in that box.

                  1. For +pi/2
                    Im(u) less than Im(v) right of diameter
                    Im(u) equals Im(v) above
                    Im(u) greater than Im(v) left

                    For -pi/2
                    Im(u) less than Im(v) left of diameter
                    Im(u) equals Im(v) below
                    Im(u) greater than Im(v) right

                  2. For +pi/2
                    If Im(u) is less than Im(v), the semi-circle is on the right side of the diameter
                    If Im(u) equals Im(v), the semi-circle is above the diameter
                    If Im(u) is greater than Im(v), the semi-circle is on the left side of the diameter

                    For -pi/2
                    Im(u) less than Im(v) left
                    Im(u) equals Im(v) below
                    Im(u) greater than Im(v) right

                    Please let me know if further clarification is required.

                    1. Yu, knock it off. Write respectfully, with careful English. I’m patient, but it’s not my job to decipher nonsense.

                    2. Well, it’s an English sentence, but it’s not clear: what does “the above” mean?

                      Start again. Say exactly what fails to be correct, and provide a specific example to illustrate the failure.

                    3. Thank you to Anothermouse

                      A typo in your counter example, the example on the left.
                      Arg(z-[3+i]) – Arg(z-[1+2i]) = pi/2
                      [1+2i] instead of [1+i]

                    4. ThanksYu. In fact, both the 1+i’s should be 1+2i.

                      The locus defined by each Nelson equation is obvious if it’s constructed from rays and an unprincipled argument is carefully used. But the whole section is in disarray. Always the risk when bumper car stickers masquerade as mathematics. When the chapter was written, the authors may well have declared “Here’s mud in your i!”

  8. One of these days I’m going to get one of the WitCH things wrong and make a complete idiot of myself. Maybe I’ve done it here, but I don’t think so. I’ve checked the existence of the error that I’m claiming numerous times, and I’ve checked that no one has noted the error numerous times.

    To annoy readers a little further, I spotted the issue pretty much immediately upno reading the section, but confirming the error took a little puzzling. Which reminds me a bit of this WitCH. So, I think I understand why readers cannot see the error, but I don’t think it’s great that they cannot. Not awful, but not great.

    Anyway, here’s a further hint: why is arg better to work with than Arg?

    1. There was a paper in the MAV 2005 Conference Proceedings:

      Subsets of the Complex Defined by \displaystyle \text{arg}(z-z_1) - \text{arg}(z-z_2) = \theta \displaystyle \left( z_1, z_2 \in C, 0 \leq \theta \leq \pi \right)

      that discuss all this in detail (and indirectly answers your question). It presents a geometric approach (using two well known angle theorems of the circle) and algebraic approaches (using vector, trigonometric and parametric methods) for finding these subsets. It examines specific examples and gives a general taxonomy.

      (Back in the old days when the MAV Conference had presentations about mathematics).

  9. Some problems I see, but I don’t know how important they are.

    Saying that straight lines in the complex plane are “similar” to those in the cartesian plane seems odd.

    z + \bar{z} and z - \bar{z} are numbers, not lines, so they can’t be vertical, horizontal, or parallel.

    I don’t see how points being equidistant immediately implies a perpendicular bisector.

    The text jumping to vectors is very confusing. They should make that clear. It’s very odd, like they’re just ditching C as a field and just treating it as a vector space over R, with how they use the term “multiple.”

  10. OK, final question from me on this WiTCH.

    Is the issue related to functions which are not one-to-one but the text assumes they are without justification?

    I believe that Arg(z) refers to the principal argument whereas arg(z) refers to a more general angle/argument.

  11. The assertion that Arg(z-u) – Arg(z-v) = pi/2 is a semicircle connecting u and v isn’t quite true: the correct LHS would be Arg((z-u)/(z-v)), since Arg(x/y) = Arg(x) – Arg(y) is only true up to mod 2pi. e.g. if we suppose that Arg(z-2i) – Arg(z-2) = pi/2 is the semicircle through (0,2) and (2,0) passing through the origin, then this is nonsense since in fact Arg(-2i) – Arg(-2) = -3pi/2.

      1. That’s probably overly polite. To add on: I can’t see anything majorly wrong with the de Moivre proof, but (1) the inductive step could be about four lines shorter and (2) it’s slightly deceptive to say “we only need to prove … for n in Z” when the statement, as written, already fails for non-integer n.

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