Michael Smith
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 Nov15 answered Complete course of self-study Oct9 awarded Supporter Oct3 comment Permutating dance partners with least distance moved Thanks that is useful link and solve all the parts except minimizing how far the people have to move to swap partners and what this problem is called in graph theory Oct3 revised Permutating dance partners with least distance moved improved title and added some combinatorics tags Oct3 comment Permutating dance partners with least distance moved I looked up Maximal independent set here en.wikipedia.org/wiki/Maximal_independent_set but it doesn't seem to capture the same idea Oct3 comment Permutating dance partners with least distance moved @Ross - that does seem to solve the first part and the rotating lines part. It does not address what this is called in graph theory (Qiaochu Yuan commented: I think he means counting the number of maximal sets of disjoint edges in a complete graph on n vertices but did not give a reference at the time). And it also does not address how we minimize the amount of movement required by the people. Oct3 asked Permutating dance partners with least distance moved Sep12 answered Historical textbook on group theory/algebra Sep12 answered What are some good math specific study habits? Aug28 comment On the Math Mindset I got "A survey of modern algebra" by Birkhoff and Mclane while I was in grade 12 and while I enjoy it now I did find it a bit abstract at the time. When I am getting into a subject I like to have more motivation and history in a math book. If you are into Algebra then "Rings, Fields and Groups" by RBJT Allenby is a much easier read that still covers most of the same ground. And the author's enthusiasm for math really shines through. Given that you are not finding your current teaching method motivating I think finding books that explain why the material is interesting are useful. Aug11 comment How to tell $i$ from $-i$? That is an interesting question - is there an algebraic definition of orientation in the (complex) plane. For the question that was asked for an elementary explanation of difference between i and -i and I think it is reasonable to use the complex plane and the natural notions of geometry that students usually already know. Here is my attempt at describing clockwise in words - if you multiply a complex number by another complex number of length 1 that is close to 1 then if the y coordinate increases it is an anti-clockwise rotation and if it decreases it is clockwise. Aug11 comment How to tell $i$ from $-i$? Yes there is an automorphism (a mirror reflection of the complex plane) that swaps i to -i and changes clockwise to anti-clockwise. I don't think that affects the above way to explain the difference between i and -i Aug11 answered How to tell $i$ from $-i$? Aug1 awarded Teacher Jul31 answered How to make notes when learning a new topic Jun3 comment How can we find and categorize the subgroups of R? Yes that is true. And the point I am trying to make is that I think there is a infinite hierarchy of dense subgroups between Q or these other non-Q subgroups and the full group R. And I am wondering how to classify those... My intuition says that this is somehow related to the diagonal proof of the uncountablity of the reals and the point that @rschwieb made on the related question math.stackexchange.com/questions/152263/… about stripping away leading digits doesn't matter. Jun3 comment How can we find and categorize the subgroups of R? @benmachine - thanks for correction of definition of dyadic rationals - I updated the question text to reflect this. Jun3 comment How can we find and categorize the subgroups of R? I am thinking that the structure of the subgroups of R might be clearer if we look at the generators of them. For example Z = [1], nZ = [n], aZ = [a], Dyadic =[1/2^n n in N], Q = [1/p p prime] etc. Then it is clearer which subgroups are subgroups or supergroups of others. Perhaps this idea can be extended to infinite (uncountable?) sets of irrational generating elements... Jun3 revised How can we find and categorize the subgroups of R? added 28 characters in body Jun3 comment How can we find and categorize the subgroups of R? @benmachine Oh, thanks for that example! That is interesting - we have a set that spans the subgroup but no subset is linearly independent! In fact we can remove any number of elements (finite or infinite) from this spanning set and so long as an infinite number of elements remain the subset still spans the subgroup... However I am not clear how to prove that no basis exists for dyadic subgroups of Q. I tried to get a contradiction from expressing 1/2^n as Z linear combination of basis elements but don't see that yet.