Divisibility Rules.

January 31, 2011

Sometimes when I’m at a restaurant with friends and the check comes, we need to evenly split the bill.  If you’re a math major, then you’re already assumed to be able to calculate this in your head; nonetheless, no one wants to hear that they owe  $3.53333333333… because very few people have fractions of pennies lying around.  It suffices, sometimes, to round up or down a penny or two to make things come out nice.  But this means you should have a good understanding of when numbers divide into other numbers.

Here’s an example (which motivated me to post this): every month, the bill for the internet is around $65.  I have two other roommates (and a freeloading daschund) so we split it up into three even parts.  Now, I know that 3 does not divide into $65, so I just "round up" to $66 and then use the excess as credit, since that’s easy to divide by 3.  But what happens if you lived with 10 other people and you needed to divide up a $2033 bill.  Does this work?  What about $2035?  Did you have to use a calculator?

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On my website (linked from somewhere here) I asked the question, "I want a more topological proof of the infinitude of primes."  A peer of mine (I don’t know if he wants his name on this blog) took one look at my site in passing and as soon as he got to that part, he looked over to me without missing a beat and noted not only that he knew of such a proof — but he knew exactly where the book which had it was in the library.  More impressive: this student is not even studying things related to topology!

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Before this post, I outlined another post which attempted to find the average price of being a mathematician — or, in other words, what does it cost to be able to get a BA in math, including tuition, tools, books, and so forth.  A lot of my estimates were arbitrary, so I’ll post about a slightly different topic: what tools do I use on a regular basis to do mathematics, and how much do they cost?

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End of Jan 2011 Update.

January 24, 2011

It’s almost the end of the month, and I haven’t been posting as much as I’d like.  I’ve just started the spring semester, and I’m getting used to juggling my schedule. 

In addition, as it stands, my homology primer has been bothering me; it’s not rigorous enough to be lecture notes and it’s not simplistic enough for it do be of any more use than, say, Hatcher.  This is a significant problem: I set out to compute the cellular homology for certain structures, but without the notion of singular homology (which I don’t even mention!) it’s difficult to justify certain notions and to define new ways of computing and it feels sort of cheap to just say, "it works because it does — trust me." 

I’m not sure when I’ll have a chance to revamp the homology primer, but what I may do is just handwrite a bunch of examples (it’s much faster) and upload them.  The upshot to this is that it’s really quick for me.  The downshot is that any typos will be much harder to fix.  C’est la vie. 

 

Either way, I will be posting again soon. 

In this post, we’re going to do something that I thought helped me through homology, but that others no doubt think is a huge waste of time: we’re going to explicitly compute the homology groups of trivial or nearly-trivial things explicitly, without appealing to any theorems regarding homeomorphisms or the like.

Recall the things that we’ve done so far.  The steps to find the homology groups in this post (and, more generally, in life) will be as follows:

  1. Find the chain groups and their generators.
  2. Compute explicitly the image and kernel of the boundary maps.
  3. Find a nice way to express the kernel and image of these boundary maps by potentially using different, but equivalent, generators (we’ll get to this step soon; it’ll make more sense then).
  4. Find the quotient \displaystyle \frac{Ker(\delta_{i-1})}{Im(\delta_{i})}, which is equal to the (i-1)-th cellular homology group.
  5. Reflect on this solution: does it make sense?  Is it nice?  Do we love it?

So without further delay, let’s dig right in.

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