Monday, July 29, 2013

How do you multiply points?

Complex Numbers has got to be the topic with the largest gap between how cool I think it is and how I teach it. I've struggled to teach this to kids in a way that doesn't make me nauseous, and so far I've pretty much failed. At the same time, I think that I'm making progress on sharing what I love about the topic with kids.

(I've written before about this. I ran a Global Math Department session about this.)

Here's my most recent attempt at telling a story that I like about complex numbers. I aim for practically no prerequisites, and I try to suggest problems that I think would be interesting -- for kids? -- at the bottom of each page.

I am looking for feedback. Be constructive, but tell me where I've gone wrong here. Drop a comment if you're excited. I'll be hanging out in the comments section, looking forward to some good back and forth on this.

(You might be wondering where Page 1 is. It's after Page 14, duh.)



(Also, I'm sorry that you're looking at an ad from the Scribd script. Does anyone have a pdf sharing thingy that works better than Scribd?)

(Also also, yeah, you're going to want to biggify that by clicking the bottom right corner of the Scribd widget.)

Update 7/29/13: You know, on second thought it's probably too much to assume that people will read a 17 page booklet on the internet. Maybe I'll try this post again.

Update 8/21/13: I think that a good bit of math to precede this discussion would be one that showed how certain models fail to extend to other operations.

30 comments:

  1. Interesting development Michael,

    I follow up until you start looking for an analogy for what it means to 'multiply' two points in the plane with each other. That presumes that there is such an operation. You could certainly define one, which is what you do, but it goes quite a ways in one direction to ultimately make it to the application of a complex number property. I think part of the issue is that you are looking to define an operation on vectors with certain properties that match what is happening in the 1D case.

    It looks like you are defining multiplying points to be equivalent to multiplying by a scalar (page 9), as an arbitrary rule (ac,bd), and then another arbitrary rule (ac,bc). These don't match what really happens with more 'typical' vector operations like scalar multiplication, dot product, or cross product. You could certainly define operations as any of the three things you came up with, but I think expecting them to also have the other properties you seek makes an inconsistent system of rules. This seems like a stretch if the goal is to justify why complex numbers are cool.

    When I introduce the idea of complex numbers, I do so by merely stating that they are mathematical objects consisting of a sum of a real number and a pure imaginary number. I then ask them to order numbers from least to greatest: 3i, 6, 3+2i, 9 - 5i, 2+3i for example. They get really uncomfortable and will argue beautifully. That discussion establishes pretty quickly that order doesn't make sense in the same way as it did with plain real numbers. We have to then talk about operations in that context.

    Just a thought - interested to see what others say.

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    1. OK, now we're cooking. Thanks for the thoughts, let's dig in.

      1. Yeah, everything is pretty much prep work for multiplying two points. That's where the action is.

      2. I'm not sure I'm getting your comment in that second paragraph, but what I'm trying to do is build a definition of multiplication for complex numbers that preserves 1D multiplication. Those operations you mention aren't where I'm pretending to end up -- they're placing where I'm stopping along the way. If all goes well, each of those attempts are seen to be insufficient for our purposes.

      So the arbitrary rules are attempts to match a small set of facts that we're clinging tightly to: that multiplying by (2,0) is equivalent to scalar multiplication, that (1,0) is an identity, that the commutative property still works.

      As far as your approach to complex numbers goes, yeah, it's fine, but why would anyone want to think about 3 + 2i as a thing? The answer that most teachers give is along the lines of "Mathematicians imagine things all the time!"

      And I'm not sure about that. Mathematicians imagine whatever they want, but they imagine them when motivated by something: curiosity, a drive to generalize, a pretty picture, a something. And what's that something with complex numbers?

      So I'm trying to start with the cool stuff, and have the invention of the complex numbers be motivated by a drive to generalize to a higher dimension and interesting problems.

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    2. Gotcha. I see that you're trying to get the complex behavior without making the supposition that imaginary numbers exist first. I like having students substitute sqrt(-1) into x^2+1 and see that it works, particularly given that this is how the historical 'discovery' of the concept of imaginary solutions to equations happened. Even then it's a bit of hand-waving or trickery, even though it works.

      I guess part of my issue with the approach is that you keep calling this process _multiplication_, which has a very specific meaning. Points are vectors, so when you are talking about multiplying points, you are really talking about multiplying vectors. This isn't really something that is defined, except as part of your analogy.

      I like that you are approaching this so abstractly, but the fact that this works with complex numbers seems more like a 'you-me-math-teachers cool' than a 'students-en-masse cool'. My students tend to let me know that things are the former when I'm a bit too excited about what we are doing.

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    3. The concern about whether students will find this interesting is the only one that I'm really concerned with.

      I'd like to think that kids will find this interesting -- I can state the problem really quickly, and I think the constraints become clear quickly too, and the problems seem fun to me -- but this is all untested.

      I appreciate your skepticism, and feel free to go into more detail about what makes this (potentially) uncool for school.

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  2. We did complex multiplication like this in my summer capstone course. The seed was the idea that multiplying a real number by i rotated it 90deg. They immediately jumped to multiplying imaginary numbers and seeing that it worked out right (the same as if i was root(-1)) and then multiplying complex times complex and trying to make sense of it. It worked as a linear combination, but was unsatisfying geometrically. So I introduced Euler's notation, re^{i theta} and there was much reconciliation.

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    1. That approach sounds awesome.

      Did you give them i before making that connection? Because I'm proposing developing complex multiplication without the use of i^2 = -1.

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  3. First off, I like this A LOT. Going to steal.

    When talking about adding points, consider drawing the appropriate vector addition parallelogram to convince students of commutativity of addition.

    There seems to be a jump in difficulty when you talk about powers of (0,2). Perhaps instead of asking about (0,2)^2, ask about (1,0)x(0,2)x(0,2) to re-emphasize the action of (0,2) being repeated on the identity element. Just a thought.

    Considering how heavy you want to hit function notation in your course, perhaps NAME the point (1,0) something like P and ask "What does (0,2) do to P? What does (0,2) do to (0,2)xP?

    I'm assuming you left out on purpose some stuff between pg 16 ("close, but not quite") and pg 17 ("we've got it!"). Getting from your inconsistent rules to (a,b)x(c,d) = (ab-cd, ad+bc) seems difficult. For what it's worth, a cute fact is that (4,3)x(4,3) = (7, 24), harking back to Pythagorean triples (and you can build others similarly).

    To gealgerobophysiculus: Michael mentioned this, but it seems the goal here is to introduce the complex numbers without having introduced imaginary numbers. I really like your ranking task, especially since it naturally leads back into the idea of the magnitude of a complex number, but if I'm a kid, I'm not sure I would buy the idea that complex numbers can be represented geometrically or that, just because they can't be ordered, they should be represented in two dimensions. In fact, I might say that the fact that they can't be ordered should be evidence that they are fake, since I (the student) have never seen numbers that can't be ordered before. I would say that your task would be an awesome follow-up to Michael's exposition.


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    1. Thanks David!

      1. Parallelograms are a good idea. So are naming the points, though I'm not exactly getting how that's going to help with function notation.

      2. You're right to pick on the gap between (1,0)x(0,2)x(0,2) and (0,2)^2. It's the difference between seeing multiplication as operating on a point, and seeing it as combining two operations. It's subtle and tricky.

      3. Yeah, there is a gap between pg 16 and pg 17. I wanted to emphasize that I thought that this gap could be bridged by students, though per your comment, maybe I'm underestimating the difficult of this problem. Certainly, more examples would help. I think my next move would be to bring in points related to the 30/60/90 triangle.

      Actually, I'm having trouble seeing how you'd help kids see that (4,3)x(4,3) is (7,24). Or would you just give that to them?

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    2. 1. I just mean that giving the point (1,0) a special name might help to get at the tricky relationship mentioned in your #2, the correspondence between (multiplying any point by (0,2)^2 rotates that point 180 degrees and dilates by 4) AND (when you multiply (0,2)x(0,2), you get the point (-4,0)). Probably unnecessarily confusing....

      Hmm, you're right about the Pythagorean stuff, so perhaps it's best saved for enhancing practice problems later down the line. I wasn't thinking about the sequencing; you would need to either already know some trig or how to multiply numbers in C.

      I like the idea of using points from 30-60-90 triangles better. You could also consider the distributive property (e.g. multiplying by (1,2) should be the same as multiplying by (1,0), separately multiplying by (0,2), and adding the result), but that might be property overload. In other words, you have a rule for multiplying by (a,0), you next need a rule for multiplying by (0,b). You might start with the idea that (a,b) x (0,1) = (-b, a).

      Anyway, I already said this, but I really like this approach, since it's geometric from the start and builds the algebra from that, it gives the whole sequence a real group theory feel (which in my opinion, is totes awesome).

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  4. Michael

    Have you seen this post yet?

    http://researchinpractice.wordpress.com/2013/07/24/i-dont-get-it-vs-i-dont-buy-it/

    I'm running out the door right now but I'll browse through your doc and share comments soon.

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    1. Read the post, and liked it a ton. Thanks for sharing. Looking forward to your thoughts, sir.

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    2. I, too, enjoyed the other post but I could not get past thinking that many of my students would be troubled because they lack the inherent curiosity that seems to be required by that approach. While there is certainly some of that in your geometric approach you scaffold it well enough that I think everyone can come along for the ride. I jotted down a few notes/observations/questions as I was reading the document.
      Do you pause and see if there is a discovery between the 2 units long and the sort(52) units long? The magnitudes are certainly related here.

      Do you think of discussing (2,3) as (2,0) + (0, 3)?

      When you mention that (2,0) isn't just stretched it also changes direction - this is also true when multiplying by negatives so the directions change idea is there.

      The introduction of multiplying with three numbers is fascinating, but also a bit troubling to me. We are SO used to thinking of multiplication as a binary operation. Might the introduction of silly old associative notation help here?

      Thanks so much for a thought provoking approach. I have always relied upon appealing to my students' sense of completeness and faith in the power of the quadratic formula. I have tried to simply play the card of "Why should we be satisfied with so many quadratics simply being unsolvable?" Some kids play along, many do not.

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    3. Oh, so that you can write (2,1)x(2,3) as (2,1) x [(2,0) + (0,3)]? Brilliant. This must be what David Price was trying to get me to understand. (Sorry for being so dense, David.)

      OK, got it. That's freaking awesome.

      In order to finish off the multiplication, you to have an idea how to multiply (2,1) by (0,3), which is OK once we've gone through the whole thing but not before. Still, this would let you skip a lot of the stuff at the end of my notes, so that's good.

      The trade-off is that you need to bring back the Distributive Property in a fairly abstract form. I need to think more about this, but it's certainly an important part of this conversation. Thanks for bringing it in.

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  5. The best overview of complex numbers I have seen is from Kalid at BetterExplained. He starts with the assumption that you have seen the number 'i' before, so that's a little different from your approach, but it is definitely worth a read.

    Also worth a look is Steven Wittens's How to Fold a Julia Fractal. It's a high-level look at the complex plane, with AMAZING animations to help build intuition. When I picture adding or multiplying with complex numbers, this is exactly what my mental models look like.

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    1. I'm going to go on the attack here against Kalid's explanation. So, you've been warned...

      Kalid writes: "Proofs have two goals: show that a result is true, and show why a result is true." He then goes on to say that proof has no place in building insight, and that's why he just gave a bunch of examples in his explanation.

      To which I say bah!

      Proofs seem unnecessary to his approach because he's entirely failed to show that there's anything interesting or puzzling around here. We're dancing with the same math, but his approach makes it a blind date, and I'm working with slow-burning flirtation.

      (It's probably best not to worry about what represents "proof" in this metaphor.)

      Point is: I love the geometric interpretation of complex numbers, but who needs it? What problems does it solve? Why would anyone care to discover it?

      Or: How do you take something interesting and make it perplexing?

      Kalid writes: "Our goal is to see why we can multiply two complex numbers and add the angles."

      My goal is to figure out what on Earth it might mean to multiply two points. We'll discover -- not without a lot of help from me, EdRealist, if you're reading -- that with a few assumptions, it's just gotta be adding the angles (and the dilation factors). And then we'll see why inventing i creates a really nifty algebraic shorthand for all of our work.

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    2. Hi Michael,

      So I had a really nice reply and then Wordpress ate it because apparently I don't own the account I thought I did.

      Your arguments against Kalid are valid, but for the love of god read Witten (who you must have missed when I tweeted the link to you earlier today). His point that complex numbers are inherently polar is, IMHO, the cornerstone of a lesson plan. The end products of your scribd notes are either better done in R^2 or not done at all. Of course, your larger point was to show experimentation, the process and not the result. Still, I think you may be barking up the wrong tree with xy.

      My other comments I actually wrote up independently two days ago on a blog post I will shamelessly plug (http://wp.me/p1vixq-ov). Number 2 is of interest.

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    3. Thanks for the Witten link. I'd first seen it shared on Dan Anderson's blog, I think, and I shared the visualizations in class last year. That stuff is pretty.

      You write in your post: "Once we’ve defined complex multiplication - angles add, lengths multiply ..."

      But where did that definition come from? Why choose to define things this way? You can talk about the number line and stretching and flipping and how this is just an extension of that, but this is all very quick and hand-wavey. So you try to take it a bit more seriously: why do we define complex multiplication this way? How would someone come up with it if they didn't know about it already?

      And then, I think, you're getting close to what I'm doing here.

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    4. Ah. I had indeed handwaved over that myself, but after reviewing Witten the answer is that it fits 1 angle 0 * 1 angle 180 = 1 angle 180, i.e. 1*-1=-1. Of course, we don't care about the answer so much as the process that produced it.

      I still maintain that it's easier to reason about complex numbers in polar, and that this definition of complex multiplication is much more natural and intuitive then anything in a+bi form. I also maintain that a+bi is duplicative of xy components of vectors. So the students should definitely discover complex multiplication for themselves, but we should guide them as to where to look.

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  6. First, I like this general approach—it shows a level of playfulness and mathematical curiosity I was rarely exposed to in my math education.

    I particularly like the moves you are making on page 9 to go back and see that 2-d point multiplication agrees with the rules we've set up for 1-d, and then how you show the usefulness of the commutative property on page 11 that (2,3) x (2,0) must equal (2,0) x (2,3).

    I'm trying to use my previous understanding of vectors to figure out what this "point multiplication is"—it isn't a cross or dot product, so what is it? Can we treat points in the complex plane as vectors? (I think we can). If we can multiply 2 points in the complex plane using algorithms we've learned by rote before, generating a new point (vector), then this would seem that we are doing some sort of vector multiplication I haven't ever seen before.

    I think you have generated a 17 page long argument to discover complex numbers, which I find fascinating, but I'm not sure every high school student would agree. Luckily, I think most of the students you will be teaching at your new school would find just such an approach enjoyable and satisfying. And you've certainly come up with the some of the best arguments for really using and understanding the commutative property that I've ever seen.

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    1. "But I'm not sure every high school student would agree."

      Yeah, I don't want to get lost off in my own little world here.

      But I'm still having a hard time seeing how this isn't an improvement on the current motivation we give kids for complex numbers. This is a topic that -- currently -- kids actively think is stupid.

      Maybe it would help if I tried to put together some worksheets or something that would show how I think this could be used with kids.

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    2. John's comment solidified for me what makes me uncomfortable about this.

      This is awesomely playful and shows what you acting on your curiosities and questions of what is possible. The problem is that these are YOUR curiosities and questions. Getting students through this development will require worksheets and a very teacher-centered lesson to get to the discovery of the activity, which is a cute way that complex numbers straighten up this geometric problem. You may have students come along for the ride and admire your enthusiasm, but this is a long way into the woods to see an application of complex numbers that shows why they are beautiful. Without a definition for complex numbers, or the imaginary unit yet, this is a jump.

      I wonder if you'll leave as many students behind wondering 'why' with this approach as you or I do with an exploration starting with definitions. The development reinforces much more about vectors and algebraic properties than it does about complex numbers. Again, this development is SUPER AWESOME, but I don't know that it will do what you want it to do for students.

      To make this all about me - I wasn't super pumped about imaginary numbers when I learned about them in Algebra 2. I really think this wasn't because of the delivery or lesson, but because I wasn't intellectually ready. I had a solid understanding of how to manipulate them through operations, graph them, and find them as solutions to polynomial equations, but the 'why' factor wasn't there, and at the time it didn't matter to me. It wasn't until an abstract algebra course for my teaching masters degree that I learned to appreciate abstraction and exploring mathematical concepts for their own sake rather than application. It took ME that long. I'm not typical in a lot of ways, but I don't think I'm alone there.

      That said, I got really excited about complex numbers in college. Why? Applications in a whole bunch of different courses in different departments.

      1. Resistors and capacitors in AC circuits being modeled using complex numbers (called phasors).

      2. Light moving toward the boundary different materials in just the right way causes something called total internal reflection. Modeling the light using waves requires that a boundary condition for continuity be satisfied at the interface. That boundary condition requires that a portion of the wave have an imaginary component. That imaginary component can actually be measured using measuring devices. What?! Perplexed, to this day.

      3. Through Euler's formula, solutions to 2nd order differential equations modeling the movement of things from car shocks to circuits to chemical reactors can be solved using one form, not two. Beautiful.

      If anyone had tried to force me to see why these were beautiful in high school, it would have been forced and it wouldn't have happened either because the development was off, or because I just wasn't ready.

      A colleague that taught Calculus BC did a full extra unit on complex power series, Euler's formula and identity, and the idea that infinite series are like the DNA of a function. He did an amazing job piecing his development together and brought the students along for the ride. That said, his development took two or three times through it with students before he felt they actually got out of it what he wanted them to get out of it.

      Tricky thing, this business.

      I'm not saying don't do it. I've just been burned when I try to push students to discover something that I think is amazing. They are appreciative and respectful and will do what I ask, but the result usually isn't what I want it to be. Part of that is that I need to keep getting better at this. The other part is that it's really hard to force your own taste for perplexity on a class that might not share it.

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    3. Getting students through this development will require worksheets and a very teacher-centered lesson.

      Given your comments, I think my next step is to try and show how I see this working with kids.

      But I want to take this one up for a second...

      The problem is that these are YOUR curiosities and questions.

      Is there another way to do this thing?

      I mean, we always want to give kids things that they'll find exciting and interesting. How do we figure out what those things are? Experience for sure, but we try things that we're excited about, right?

      Taking your interests as a starting point and seeing whether it's possible to share them with 10th graders seems like a decent way to plan lessons, as long as you've got a pretty good BS-meter to make sure that you're not trying stuff that's going over kids' heads. And that BS-meter is usually experience, a rough idea of what's possible and what sort of math kids like. In this case, your alarm is going off, but mine isn't. Wish I understood why, and I hope that I'm not simply enamored by my own work.

      Here's what I'll do: I'll come up with some lessons/tasks, and I'll find a friend (or a spouse) and try this out on her before showing it to kids.

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  8. There's a lot of great insight there, especially in your number-line interpretation of real arithmetic. And the geometry of complex multiplication is quite beautiful - I see why you want to use it as a starting point.

    So if I'm reading right, you start from a 2D coordinate plane, and argue that, for things to work out nicely, it must have the multiplicative structure of the complex plane.

    But from what I understand, that isn't true. For example, we often treat the 2D plane as a vector space. And while we have two ways of multiplying vectors (dot products and cross products), neither one corresponds to complex multiplication.

    The lovely structure you're building around is unique to the complex plane. It seems the best way to reach it is the following path:

    1. Yeah, so negative numbers don't have square roots.
    2. But hey, what if they did?
    3. Let's take root(-1) and call it, I don't know, "i." It's a brand new number.
    4. But where does it go on the number line?
    5. I've got a crazy idea. How about we put it above 0 - that is, above the number line?

    From there, the complex plane emerges (semi-)naturally, and you can discover that multiplicative structure as a cool consequence.

    Not sure if this solves your original problem of making the complex plane feel meaningful. But I think it stays a little truer to the logical development of the concepts.

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    1. But from what I understand, that isn't true.

      We're in two-dimensions here, and it's not obvious what the 2D version of cross-product is if you're using the "mutually orthogonal" intuition. (I've seen a 2D analogy, but I'm just saying it's far from obvious that there is one.)

      As far as the dot product goes, yeah, that's possible. But only if you're willing to give up the idea that multiplication by (2,0) has the same effect as multiplication by 2. If you want "new" multiplication to contain and conserve the "old" multiplication, then (I was discovering in those notes) you end up at complex multiplication fairly quickly.

      Here's the path: conserving "old" multiplication, assuming that we still have commutativity, and then noticing that this means that "new" multiplication must involve stretching as well as rotating. Then we have test cases for a rule, and then we can bring in imaginary numbers as a cool way to keep track of all this stuff.

      As far as your story goes, I'm not such a fan of it. It seems to me that this kind of wondering is even farther away from our students' intuitions than this sort of wondering by analogy that I'm doing here. It rings false to my experience of math to suggest to kids that mathematicians regularly imagine conceptually impossible things and then run with them.

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    2. Fair point about cross products and 2D analogs.

      But I'm not sure why multiplication by 2 should need to be similar to multiplication by (2,0). Why not (0,2)? Why not (2,2)? Why associate (2,0) with the number 2, unless we're talking about the specific structure of the complex plane?

      I can't necessarily defend my story. I've only taught it that way to 11th graders who had already met the complex plane. I usually begin by reframing prior math experiences as a gradual expansion of the idea of "number." ("You can't take 7 away from 5. But what if you could?" "You can't divide 5 into 6 pieces. But what if you could?" "You can't take sqrt(-1). But what if you could?") They don't throw fits. But I can't blame you if it rings false to your own mathematical experiences.

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    3. (2,0) and 2 are at the same place on the original number line. (0,2) isn't at the same place. (Unless you imagine that the number line has been placed vertically, which I'm fine with.)

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  9. I like Gérard Michon's approach to complex arithmetic at http://www.numericana.com/answer/algebra.htm#complex.

    The key sentence is "Multiplying any number y by x is like using x as a new 'unit' step." To multiply 2+i by 3-i, we define our new unit step as 2+i (2 forward, 1 left), then we take 3 steps forward and 1 step right using the new unit step.

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    1. Which means we have to come up with a good definition of what it means to step forward or right using a 2-Dimensional unit step. Interesting!

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  10. I couldn't fit my thoughts into a comment, so I made a blog post: http://mathforum.org/blogs/max/complexity-perplexity/

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