ℝolliℵg M∀th Thr∑a∂

you can just do basic arithmetic on the integers as an additive group, just teach your kids group theory ;-)

for division i guess you either need a euclidean ring or a fullblown division ring, in which case division is just multiplication by inverses

If you believe -1 x a is -a, then -1 x -1 is -(-1), and negative negative 1 is plainly 1.

But once you believe -1 x -1 = 1, I think you believe that a negative times a negative is a positive in general.

elegant

we had to prove all this bullshit in my first real analysis class, to give the impression of "rigour"--but we didn't even construct the real numbers (using dedekind cuts, etc), just stated the Completeness property as an axium--such a waste of time

Think it might be useful to think of multiplication as making a copy or n copies of something to replace the thing and multiplication by -1 as making an inverted copy. So say you have a white disk than multiplying by -1 you replace it with a black disk and vice versa, or better yet you have an Othello token and just flip it over.

is this thread a boys club? where the math ladeez at?

iirc harbl studied math but she said she has forgotten all of it and left it all behind and is a lawyer now

kid i was tutoring deferred his exam :-\

why negative times negative is positive

I feel like I did something like this in discrete math, you start with basic definitions of integers and parity or w/e and then do a formal proof or w/e?

lol n/m i'm drunk and listening to bill withers

how do i shot basic simplification of roots

http://farm8.staticflickr.com/7335/11318231686_aee01101ef_b.jpg

You're asking seriously?

Oh, I see you are making fun of the person who put the question marks.

no i'm asking seriously :((((

defeated by precalc ;_;

Multiply by conjugate?

rotation of axes??

last step looks like some bullshit, no? rationalize the denominator, b then u got sqrt(1 + 2/3sqrt(2)) not sure how much more u can smiplify tho?

it works on a calculator

google it!

sqrt((2+sqrt(2)) / (2 - sqrt(2)))-sqrt(2)

ok got it

Interesting post on zero indexing:

http://exple.tive.org/blarg/2013/10/22/citation-needed/

Author interviewed Martin Richards, author of BCPL and the supposed originator of zero indexing. Conclusion: it was a stylistic decision (i.e. it wasn’t commentary on zero’s inclusion in ℕ or whatever).

interesting to think about how stylistic concerns can be aligned naturally with mathematical principles (vs when they're not aligned). makes me think about what style really means and stuff.

0 is so not a natural number

mathematical principles are always about style

If you believe -1 x a is -a, then -1 x -1 is -(-1), and negative negative 1 is plainly 1.

I once gave this answer but in a much wordier way on this thread:

http://math.stackexchange.com/questions/9933/why-negative-times-negative-positive

considering prepping a talk for an undergrad conference in january, anyone got any topics to suggest?

https://www.simonsfoundation.org/quanta/20121002-getting-into-shapes-from-hyperbolic-geometry-to-cube-complexes-and-back/

great article on the classification of 3-manifolds, written at an extremely accessible level. basically this guy thurston conjectured 23 theorems that, once all proven, would result in classification. my topo prof proved a result that was used to prove the last three conjectures in one sweep, and article goes in some detail into his research. super interesting stuff, to me at least

thanks for the link -- that's very clear!

Yeah

it would be an interesting history of math to classify what programmes have led to the most research -- i suspect classification programmes themselves would probably lead the pack.

classification of surfaces seemed like it didn't take very long once they figured out what they were doing

oh yeah, finite simple groups, too

arguably, figuring out what you're doing is typically the hard part.

great little history of the classification of surfaces on appendix D of this book http://download.springer.com/static/pdf/997/bbm%253A978-3-642-34364-3%252F1.pdf?auth66=1387259121_9a9118105634f100257c6f624c9329f0&ext=.pdf (full pdf)

gah i should be studying for my analysis exam... blegh

which 2 should i take next semester out of these 4

real analysis 4 (measure theory, functional analysis)
differential geometry
topics in geometry & topology course on cube c0mplexes
discrete mathematics of paul erdos (taught by the great vasec chv4tal http://users.encs.concordia.ca/~chvatal/6621/)

real and discrete OR differential and cube complexes

i sort of hated analysis 3 but while studying for it and memorizing all those theorems i became really impressed with it and now have the urge to take the 4th. also i've heard measure theory is one of those things you've just *got* to learn and this guy would teach it properly

interesting, why those 2 diff pairings?

(xp)

i've always had better luck in school when i take courses with some connection to each other rather than courses which have different approaches

although ... is real analysis useful in differential geometry?

yes

pairing of most similar would be cubes + discrete, diff geo + ana

oh okay. that's what i'd do then.

(higher math n00b)

cube complexes was developed by algebraic topologists & geometric group theorists, people who exploit an analogy (functor or whatever) between topological spaces, infinite groups, and cayley graphs of infinite groups, to prove results in group theory & 3-manifold theory. so graph theory would come up

the simplest vers of diff geo is, like, in multivariable calculus taking a surface integral

yeah that's about as far as i got in geometry

that article i posted upthread goes into the CC stuff, with some quotes by dude who is teaching the course (and is like world champion of cube complexes)