Mathematics - 2021-04-23 (page 3 of 3)

@zacts Thanks for the book recommendation, it's on my pile (and seams to be relevant seeing as all this started with me trying to work out magnets anyway)

zacts

cool

Andrew Micallef

@TedShifrin, You've taught this stuff for a while, is diff geo the kinda thing someone making technical drawings of mechanisms would benefit from?

Looking at the examples I feel like the answer is a definitive yes

zacts

I kind of like how this #-theory book interprets 5 x 4 versus 5(4) visually. The former is a 5 by 4 rectangular arrangement. The latter is visually five groups of four squares.

Ted Shifrin

I doubt that's necessary. But perhaps you have some specific examples for me.

That's the elementary school definition of multiplication, @zacts.

Andrew Micallef

I don't have the diagram here, and it feels like it's taking us too off topic, but then I look at leslie and coppers conversations and should propbably stop worrying. I dabble in CAD, and a while ago was trying to work out the path a particular element would move, given it was pushed in one direction, and puled in another (I would need to draw it, but it is in essence similar to the dog on a leash example you gave)

zacts

10:25 PM

imgur.com/a/zgUXzy6

^ @TedShifrin

Andrew Micallef

tiny text

zacts

It's very subtle, but I like thinking of it in this way.

Ted Shifrin

Subtle?

We talked about the meanings of division in here a month or so ago.

Andrew Micallef

I'm being told I need to stop harassing people on the internet

ciao internet friends

Ted Shifrin

Are you and Leslie becoming one now?

zacts

10:29 PM

The difference in meaning between 5 x 4 versus 5(4) seems to be a subtle syntactic difference, yet the representation between the two, in this text, is quite different.

Ted Shifrin

I think it's crap. They're thinking of one as $4\times 5$.

The commutative property is transposing the picture.

zacts

The thing that was interesting to me was that the goal wasn't that the final quantity was equal, but that you could represent that final quantity in two different ways visually. The goal wasn't the actual final quantity itself, rather it's how it's visually represented in two different ways.

@TedShifrin what do you think? What is it that you think is crap about that?

Ted Shifrin

One is $5/times 4$, and the other is $4\times 5$. Those are the two pictures. 5 groups of 4 versus 4 groups of 5.

zacts

Oh I see.

So 5(4) is the same as 4 x 5.

Ted Shifrin

Yeah, that's the crap.

zacts

10:40 PM

Now that you put it that way... lol. Ok.

Ted Shifrin

The two questions for division are these. $6/2$ asks EITHER if we divide 6 into 2 equal groups, how many in each group? OR if we want to divide 6 into groups of 2, how many groups will there be?

These seem quite different.

Andrew Micallef

I wouldn't object to having a lawyers income...
I just returned to throw in that visual representations can easily be misleading. It's a trap that scientists fall into *all the damn time*

zacts

@TedShifrin So you would have a different number of groups depending on which perspective you are looking at it. In the first case you have 2 groups of 3. In the latterr case you would have 3 groups of 2.

Andrew Micallef

@robjohn had a great example of a visual proof that was a visual illusion the other day...

Ted Shifrin

So this reverts to the commutativity of multiplication that we already observed.

zacts

10:45 PM

but the final answer to 6/2 is still 3.

ok cool

Ted Shifrin

@AndrewMicallef I saw that one decades ago, yes.

Andrew Micallef

what was it called, I'm trying to find it, i feel it would be fitting

Ted Shifrin

I don't know a name.

Martin Gardner had it, so you can search here for that.

zacts

@TedShifrin So is it useless to consider groupings like this due to the commutative property of multiplication?

Ted Shifrin

Not useless, but that book's notation is totally misleading.

zacts

10:49 PM

@TedShifrin thanks, good to know.

Ted Shifrin

Division algorithm $12 = 2\times 5 + 2$ is definitely 2 groups of 5 when we divide by 5 with remainder.

Andrew Micallef

found it

Apr 18 at 5:17, by robjohn

Missing square puzzle

The missing square puzzle is an optical illusion used in mathematics classes to help students reason about geometrical figures; or rather to teach them not to reason using figures, but to use only textual descriptions and the axioms of geometry. It depicts two arrangements made of similar shapes in slightly different configurations. Each apparently forms a 13×5 right-angled triangle, but one has a 1×1 hole in it. == Solution == The key to the puzzle is the fact that neither of the 13×5 "triangles" is truly a triangle, nor would either truly be 13x5 if it were, because what appears to be ...

leslie townes

the one with the triangle that isn't a triangle?

yes.

somewhat connected to fibonacci numbers.

Ted Shifrin

Leslie will go into geometry shock soon.

Andrew Micallef

There is a analogue for this in science, where you have a graph of two curves that look more or less identical, then you stick a star above one, and all of a sudden they are totally different curves

leslie townes

10:53 PM

cliff stoll had a good little talk about a related thing. a gimmicked ruler and piece of paper. youtube.com/watch?v=9yUZTTLpDtk worth the five minutes.

Ted Shifrin

A rising star or a falling star?

@leslietownes I watched enough. He's good!

leslie townes

i would have loved something like that in middle school.

Ted Shifrin

My science teacher senior year of h,s. Was amazingly creative. I loved him. Half chemistry, half physics. Only good science teacher I had in 4 years.

No good math teachers, really.

Geometry teacher punished me for my enthusiasm asking questions.

leslie townes

i had two decent math teachers, a great physics teacher, and an insane chemistry teacher.

he let me do fairly dangerous things in the laboratory. under a fume hood and in small quantity, but dangerous things.

Ted Shifrin

And I was in the Boston suburbs, supposedly top school systems.

leslie townes

11:02 PM

all of my best teachers had previous careers. retired from industry of varying kinds and then became teachers.

Andrew Micallef

@leslietownes Remonds me of my yr 8 science teacher. I could not stand working for someone like that

gtg

zacts

I'd love to get into number theory, but I don't know where to begin from where I'm at, and which resources would be relevant towards this goal.

leslie townes

first we made soap. i found that boring, i wanted to make something i could set on fire that would do sparks or something. we made something with sodium azide in it.

zacts there are a fair number of decent 'elementary number theory' books that approach the subject from first principles and use methods that would have been recognized by the originators of the subject. there are also books that use more advanced methods. it depends on where you want to go.

robjohn

@AndrewMicallef when was that? I'd like to see it :-) Oh, that one... that was in reply to a movie that was posted by PM2Ring

leslie townes

niven, zuckerman, and montgomery is a college level text that i do not think presumes much in the way of prerequisites (beyond facility with arithmetic and mathematical induction).

zacts

11:07 PM

@leslietownes I don't yet know Algebra or Analysis.

leslie townes

almost all of that stuff is clever descent arguments.

i'm trying to remember the author of the one number theory book my hometown library had in it.

it was right at my level. no theory needed.

ha, it seems to be gone from the collection. it was really trashed when i read it in the 1990s.

one weird feature about elementary number theory books is they tend to be preoccupied with stuff that preoccupied the founders of the subject. i've never been able to care about quadratic reciprocity, but there it is in every book.

the only number theory book i own is serre's cours d'arithmetique. i don't recommend it for beginners but it is a wonderful book.

i may also have a copy of neukirch that i inherited from a former officemate. never opened it, never planning to.

zacts

"books that use more advanced methods" what would these methods be?

leslie townes

well if a book presumes a knowledge of group theory then you can get a lot of mileage out of that. you don't have to prove special cases of group concepts, you can just use them.

robjohn

@TedShifrin I had a summer math and science camp during one summer where the teacher gave the question about how many regions can be formed from connecting lines between $n$ points on a circle. The one that starts $1,2,4,8,16,...$ then the next is $31$. I came up with the correct polynomial for the answer, but he was not prepared and said that I should beware of patterns.

leslie townes

and if you know analysis that opens up other ideas. a lot of very intuitive results can be proved with infinite sums and products but you need at least some analysis to justify them.

robjohn, that is quite funny.

my analysis instructor in college did something similar. she set a trap involving infinite series. i walked into it, only i didn't, i proved the series, whatever it was, was divergent and grew like log(n). i think my explanation was simpler than whatever she had in mind. i was told to pay attention and that series weren't limits.

about which, well, OK.

i love that problem about regions on a circle. i used it when i taught 'intro to proof' at berkeley.

they canceled that class after i left, i think because they realized it made more sense to just require people to take discrete math, which for some reason they didn't do for a very long time.

robjohn

11:23 PM

@TedShifrin The only good teacher I had in high school was during the summer before high school when I took geometry. I kept going back to him and he gave me calculus problems because I had taught myself calculus in junior high. He and I kept in touch and had occasional lunches until he passed away some years ago.

The only good MATH teacher

leslie townes

my calculus teacher lent me all of the books he used in college. i was particularly taken with a numerical analysis book. i'd completely forgotten about that until you mentioned that.

that might have made a difference in my life. i should write him.

Dear Sean, All of this is your fault. - Leslie

he lent me baby rudin and i didn't even open it. i wanted to compute digits of pi on my calculator. which i ended up doing. press enter, get another digit after waiting 20 seconds. i wish i still had that program.

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$Mathematics$ Mathematics