@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)
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.
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)
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.
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?
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?
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*
@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.
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 ...
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
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.
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.
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).
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.
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.
@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.
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.
@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.
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.