ℝolliℵg M∀th Thr∑a∂

why do they schedule exams at 9am? who can even think that early?

http://math.berkeley.edu/~wu/AMS_COE_2011.pdf

Professional development (PD) for in-service math teachers is
generally taken to be \feel-good sessions". Some believe that its
main goal is to give teachers encouragement and sharpen their
pedagogical skills.
Others believe that teachers should be exposed to fun mathematics
(such as the Konigsberg bridge problem or taxicab geometry),
even in the face of their inability to deal with bread-and-butter issues
such as how to teach fractions, why negative times negative
is positive, what similarity means, or why the parallel postulate
is important.

anyone want to take a stab at 'why negative times negative is positive'? seems like a good one.

because negative divided by positive is negative.

i think algebraically it follows from that

"what is division?" is a good problem that i think i've raised on this board before. does 20/4 = 5 mean that if we divide 20 into 4 parts each part is 5 units large, or if we divide 20 into parts that are 4 units large we get 5 parts?

or how about this: if multiplication is defined as repeated addition, then multiplication by a negative is repeated subtraction, and subtracting a negative is obviously positive

"If any single quantity is marked either with the sign + or the sign - without affecting some other quantity, the mark will have no meaning or significance, thus if it be said that the square of -5, or the product of -5 into -5, is equal to +25, such an assertion must either signify no more than 5 times 5 is equal to 25 without any regard for the signs, or it must be mere nonsense or unintelligible jargon."

Baron Maseres otm

Use complex numbers. Rotation twice by 180 degrees is the identity

LOL imo if you're using complex numbers to justify arithmetic you've won the battle but lost the war

Well, take out the complex numbers but keep the argument.

or how about this: if multiplication is defined as repeated addition, then multiplication by a negative is repeated subtraction, and subtracting a negative is obviously positive

― the late great, 11. december 2013 00:48 (1 hour ago) Bookmark Flag Post Permalink

This isn't logical, right? Surely -5 - -5 5 times is +20?

that's because what you just described is -5 - (-5) - (-5) - (-5) - (-5) - (-5), no?

should i say subtracting a negative is the same as adding?

tbh i don't completely understand the objection frederick

Seems there are plenty of ways a mathematician could convince himself of why it has to be but not clear what is the most obvious common sense explanation for the layperson.

Actually I might have an idea. But there is not enough room to write it in the margin of this thread.

Oh, I get it. You're right. My fault.

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