## Re: Complex numbers & fractals

1

Thanks, that was a lot of fun to go through. Great tool for building intuition.

Posted by: dalriata | Link to this comment | 12-17-17 7:16 PM
2

Q: What does the B. in Benoit B. Mandelbrot stand for?

A: Benoit B. Mandelbrot.

Posted by: Doug | Link to this comment | 12-18-17 6:41 AM
3

Posted by: bill | Link to this comment | 12-18-17 8:13 AM
4

Not Benoit Balls Mandelbrot?

Posted by: Barry Freed | Link to this comment | 12-18-17 8:43 AM
5

That was a great article. I like the kickoff--that math is all about a series of consistent choices, and we just name some of them Algebra, Geometry, etc.

Posted by: Mooseking | Link to this comment | 12-18-17 12:00 PM
6

That explanation of complex numbers and square roots of complex numbers is just lovely.

Posted by: Unfoggetarian: "Pause endlessly, then go in" (9) | Link to this comment | 12-18-17 12:21 PM
7

They didn't mention that we get the word absurd from negative numbers or surds. Negative numbers were considered ab surd.

The big problem isn't that math isn't interesting. It's that math is like music or speaking a foreign language. You can appreciate it well enough, but to actually do it involves memorization, familiarization, conceptualization and practice, practice, practice. Still, an amusing enough page.

Posted by: Kaleberg | Link to this comment | 12-18-17 5:03 PM
8

Maybe they didn't mention that because it isn't true:

Etymology: < Middle French, French absurde (adjective) unreasonable, contrary to common sense (late 14th cent.; beginning of the 13th cent. in Old French as absorde ), (noun) (with indefinite article) something absurd, an absurdity (16th cent.), (with definite article) the absurd, absurdity (17th cent.) and its etymon classical Latin absurdus out-of-tune, discordant, awkward, uncouth, uncivilized, preposterous, ridiculous, inappropriate < ab- ab- prefix + surdus surd adj. With use as noun compare classical Latin absurda , neuter plural (Quintilian). Compare earlier absurdity n.

Ok, what's up with "absurdity"?

Etymology: Originally (in sense 1) < post-classical Latin absurditat- , absurditas dissonance (4th cent.), perversity (5th cent.) < classical Latin absurdus absurd adj. + -tās (see -ty suffix1; compare -ity suffix). Subsequently (in sense 2) < Middle French absurdité (French absurdité ) absurd act or statement (1371-5) < post-classical Latin absurditat- , absurditas . Compare Spanish absurdidad (c1440 or earlier), Portuguese absurdidade (c1727), Italian assurdità (a1573).

The first example usage for "absurdity" is indeed musical, from 1429: " There ware difformitee the beutee of Absolon..There Dauid harpe and the musik of Jubal ware absurditee."

There is a usage of "absurd" applied to negative numbers from 1557 ("8−12 is an Absurde nomber. For it betokeneth lesse then nought by 4."), but the first usage given comes from a piece of writing called "The determinations of the moste famous and mooste excellent vniuersities of Italy and Fraunce, that it is so vnlefull [sic] for a man to marie his brothers wyfe" (1531), and there it just means "contrary to reason".

Posted by: nosflow | Link to this comment | 12-18-17 5:26 PM
9

thanks nosflow, that was awesome.

Posted by: alameida | Link to this comment | 12-18-17 8:38 PM
10

Still, both "surd" and "absurd" derive from Latin "surd," which means "deaf" via Arabic. Language Log was on the case (at great and hilarious length), as expected: Ab Surd

Includes bonus Lewis Carroll poem:

And what are all such gaieties to me,
Whose thoughts are full of indices and surds?
x2 + 7x + 53
= 11/3

AND bonus bonus Cordwainer Smith quatrain:

She wasn't the woman I went to seek;
I met her by the merest chance.
She did not speak the French of France,
But the surded French of Martinique.

As they say, "read the whole thing!"

Posted by: DaveLMA | Link to this comment | 12-19-17 6:38 AM
11

At each step, we feel hoodwinked: we were only shown a part of the puzzle! As it turned out, there was a 'better' set of numbers waiting to be discovered, more comprehensive than the last.

This pretty much exactly describes how I reacted to my chemistry education. Every year they'd explain that actually, last year's model wasn't right, and this is how things really work. Only to do the same thing again the following year. Though, strangely enough, I didn't rebel against maths.

Posted by: Ginger Yellow | Link to this comment | 12-19-17 7:02 AM
12

You think it's an accident they give the Nobels in Stockholm?

Posted by: Mossy Character | Link to this comment | 12-19-17 7:07 AM
13

11: If they skipped right to the final model, would it have been comprehensible?

Posted by: Walt Someguy | Link to this comment | 12-19-17 7:40 AM
14

Good Post)

Posted by: griffin963 | Link to this comment | 12-19-17 7:44 AM
15

11: If they skipped right to the final model, would it have been comprehensible?

Probably not. I'm not saying 14-year old me was being reasonable. Just that that sense drove me away from science subjects at that age.

Posted by: Ginger Yellow | Link to this comment | 12-19-17 7:47 AM
16

Statistics does that a lot. You learn the simple model that isn't actually properly specified before you learn the, usually more complex, model that works. It annoys me sometimes, but I do find that I usually run the simple model anyway as an error check because the coding is more simple.

Posted by: Moby Hick | Link to this comment | 12-19-17 7:50 AM
17

The worst offender is philosophy. They start out with how they are going to answer questions about epistemology or what have you, drag you on for a whole semester, and then reveal the answer is a nicely worded version of "I don't know."

Posted by: Moby Hick | Link to this comment | 12-19-17 7:52 AM
18

The problem with 11, AIMHMHB, is that it trains people to believe that the simple and intuitive explanation for anything cannot possibly be the correct one, and thus trains them to be Slate writers.

Posted by: ajay | Link to this comment | 12-19-17 8:16 AM
19

re: 17

To be fair, really advanced practitioners move on from, "I don't know" to "It depends".

Posted by: nattarGcM ttaM | Link to this comment | 12-19-17 8:28 AM
20

I did a teaching evaluation of a chemistry teacher who I have a very high opinion of, and he used point 11 very explicitly as his teaching framework. (It was pitched as Inquiry Based Learning but it was very, very different from what that means in math.)

He took a historical approach and a knowledge-acquisition approach - with each model, they discussed what seemed to work, and then what new technology of the time caused failures in that model, and how the next model integrated the new information more accurately. I thought it worked nicely.

Posted by: heebie-geebie | Link to this comment | 12-19-17 8:28 AM
21

20 is actually a pretty good idea.

Posted by: ajay | Link to this comment | 12-19-17 8:35 AM
22

To be fair, really advanced practitioners move on from, "I don't know" to "It depends".

Some people also favor "it isn't really a problem at all".

Posted by: nosflow | Link to this comment | 12-19-17 8:39 AM
23

This is a really nice link, neb! I shared it with other people.

Posted by: heebie-geebie | Link to this comment | 12-19-17 8:53 AM
24

20 is kind of how we were taught maths in the English equivalent of high school. The curriculum as a whole wasn't historical in a chronological sense, but we'd have regular "circus hours" where teachers would riff on, say, Euler and Gauss, or Euclidean geometry from first principles, or Russell and set theory.

Posted by: Ginger Yellow | Link to this comment | 12-19-17 9:17 AM
25

20: A bit of Lakatos in that.

Posted by: JP Stormcrow | Link to this comment | 12-19-17 1:23 PM
26

Three cheers for robust, methodological, falsification.

Posted by: Moby Hick | Link to this comment | 12-19-17 1:35 PM
27

we'd have regular "circus hours" where teachers would riff on, say, Euler and Gauss, or Euclidean geometry from first principles, or Russell and set theory.

what kind of terrible circus is this?

Posted by: Robert Halford | Link to this comment | 12-19-17 1:41 PM
28

I blame PETA.

Posted by: Moby Hick | Link to this comment | 12-19-17 1:48 PM
29

22. I thought "it needs more study" was the go-to phrase.

Posted by: DaveLMA | Link to this comment | 12-19-17 1:56 PM
30

26: Proofs and Refutations Lakatos.

Posted by: JP Stormcrow | Link to this comment | 12-19-17 1:56 PM
31

27: You haven't been to a circus until you've seen Tarski pull a rabbit into a hat.

Posted by: dalriata | Link to this comment | 12-19-17 2:52 PM
32

30: What's funny about that book is that the process he described has ended. There are rigorous proofs of Euler's original theorem that no one questions. The way the topic has progressed has rather been in forms of greater and greater generality -- the modern version is in terms of the Euler characteristic in algebraic topology.

Posted by: Walt Someguy | Link to this comment | 12-19-17 2:53 PM
33

Euler? Euler? Euler?

Posted by: Moby Hick | Link to this comment | 12-19-17 2:54 PM
34

Education recapitulates discovery.

Or at least it should if you want to train people in how science actually works. If you want people to memorize facts and how to use equations and not actually believe that they really describe reality (a la Eric Mazur's student's illuminating response: "How should I answer these questions? According to what you taught me? Or according to the way I usually think about these things?") then by all means present the current theory as received wisdom.

Posted by: F | Link to this comment | 12-19-17 6:37 PM
35

Ooh, now do quaternions.

Posted by: Eggplant | Link to this comment | 12-19-17 11:13 PM
36

I worked through the examples on that page, and, honestly, all I could think was "pretty pictures". I'm sure the key to it is thinking of all numbers as imaginary, but I can't. I'm an instinctive platonist. I'd sooner believe that real numbers are real than that mutable humans are. So, with imaginary numbers in those animations, a huge part of me thinks, "but why do the pushmepullyou? while another part can't do the vectors and work out why pushing this and pulling that should leave the green dot where it does.

Posted by: NW | Link to this comment | 12-20-17 2:07 AM
37

I'm sure the key to it is thinking of all numbers as imaginary

No, no, just think of all numbers as real - it's just that some of them don't correspond to things you can see and perceive in the physical world. And this isn't just true for things like i; it's also true for things like "a trillion". I can't imagine a trillion anything. Not even otters. But I have no problem with the concept.

Posted by: ajay | Link to this comment | 12-20-17 2:41 AM
38

I bet I could picture at least 100,000 otters.

Posted by: Moby Hick | Link to this comment | 12-20-17 6:05 AM
39

I had an argument with a Platonist on whether the quaternions "really existed".

Quantum mechanics seems to require complex numbers, which makes them seem pretty real to me.

Posted by: Walt Someguy | Link to this comment | 12-20-17 6:18 AM
40

The fact that the ratio of a circle's diameter to its circumference isn't a rational number despite circles being real suggests to me that '1' isn't real.

Posted by: Moby Hick | Link to this comment | 12-20-17 7:11 AM
41

Perfect circles aren't real.

Posted by: Cryptic ned | Link to this comment | 12-20-17 7:27 AM
42

It's easy to get a perfect circle is you start with a perfect sphere.

Posted by: Moby Hick | Link to this comment | 12-20-17 7:29 AM
43

One trillion is 10^12 (nowadays, anyway). 10^12 = (10^4)^3. 10^4 is ten thousand. Can you imagine a thousand otters in a row? Do ten of those. Now make an ottercube with that as a side.

Posted by: dalriata | Link to this comment | 12-20-17 7:29 AM
44

Can we give them typewriters to test another theory?

Posted by: Moby Hick | Link to this comment | 12-20-17 7:31 AM
45

Perfect circles aren't real but we can approximate them as closely as we'd like to. If you're willing to make it quite large I'm not even sure to what degree Planck units get in the way of creating a locus of points whose perimeter/diameter radius differs from pi to an arbitrarily small degree.

Posted by: dalriata | Link to this comment | 12-20-17 7:34 AM
46

Can you imagine a thousand otters in a row? Do ten of those.

That's a lot of otter poop.

Posted by: Spike | Link to this comment | 12-20-17 7:50 AM
47

Perfect circles are all in the shoulder swing.

Posted by: heebie-geebie | Link to this comment | 12-20-17 7:55 AM
48

Also, always draw your circle first and then put your axes through the center, on the chalkboard. If you draw the axes first, it's much harder to make your circle on center.

Posted by: heebie-geebie | Link to this comment | 12-20-17 7:56 AM
49

With great shame I never learned the secret of how to draw circles well. The rare times I have to whiteboard one it's just embarrassing.

Posted by: dalriata | Link to this comment | 12-20-17 8:09 AM
50

Now make an ottercube with that as a side.

You can describe it and you can imagine a cube with, effectively, a little label saying "it's ten thousand otters long, deep, and high", but that's different from being able to imagine a trillion otters.

I mean, the famous example of imagining a regular chiliagon can't be addressed (well) just by saying "well, it's a regular, thousand-sided polygon, done".

Posted by: nosflow | Link to this comment | 12-20-17 9:05 AM
51

You're just bitter because you don't have a good imagination.

Posted by: Moby Hick | Link to this comment | 12-20-17 9:07 AM
52

The halacha follows R. Nosflow. https://cabin-pressure.livejournal.com/60389.html

Posted by: ajay | Link to this comment | 12-20-17 9:10 AM
53

I can imagine it, it's just that the resolution of my imagination is fuzzy.

Posted by: dalriata | Link to this comment | 12-20-17 9:31 AM
54

Picture a regular n-gon. Then take the case that n=1000.

Posted by: heebie-geebie | Link to this comment | 12-20-17 9:53 AM
55

Picture Germany being supreme over all for a year.

Posted by: JP Stormcrow | Link to this comment | 12-20-17 10:51 AM