/sci/ - Science & Math

>>12441413
that's a weird essay from my point of view:
the claim that "R is the biggest jump in sophistication" is also very subjective, throughout my education i've seen people baffled by fractions or complex numbers rather than R

to me, "IVT and real numbers" is an even more basic concept than "roots of numbers"
when you first encountered [math]\sqrt{2}[/math] how did you think about it? well i'm sure most people do one of the following:
1) they have intuition about "continuity and IVT" and they understand here that there's gonna be a number that gives 2 when squared
2) they think "well whatever this [math]\sqrt{2}[/math] is, it's somewhere between 1.4 and 1.5, it's also between 1.41 and 1.42, and so on and so on, so it's somewhere there on the fucking line", now you could argue that this is intuition about "infinite decimal expansion" or about "taking a limit of an increasing/decreasing sequence" which is very close conceptually "supremum/infimum of a set"

i'm confident you could give an axiomatic definition of "real numbers" to a high schooler based on the description "real numbers contain the rational numbers, there's the order < on reals, and bounded sets have supremums"
this is how you attain rigor with regards to real numbers

Why does N exists? is it because the axiom of set theory tells you the set exists? no, the axiom of infinity was invented, because we wanted set theory to conform to our intuition
why does Q exists? because you constructed it out of N? no, it's the other way around: when you are an undergrad studying set theory, the fact that you can build "stuff that looks like rational numbers" gives us confidence that the language of set theory is expressive enough to include the math we knew so far
does Z exist because you constructed it out of N? no, same argument as above
and just like Q and Z, the construction of R in terms of dedekind cuts or cauchy sequences serves as an argument that set theory makes sense; NOT THE OTHER WAY AROUND

>>12224371
yes i do it and i think everyone should do it
I see a new theorem -> i think about the proof
I have some ideas -> fuck around and try to prove it for a while
I don't even have a clue where to start -> read the beginning of the proof, then try again
In the case i can't prove the theorem myself, i "read" the proof as in "verify that every next sentence follows from the previous sentences" and then i contemplate until i understand better. By this i mean: even the tricky proofs contain one or two smart ideas and the rest is details. you need to extract the key idea, once i understand the proof well, i'll be able to say something like "oh so i just define ABC and then observe that XYZ holds and the rest is obvious"

obviously often im too lazy but i think that's how a proper reading of a mathematical text looks like

Do you own any springer yellow books?
I ordered one for myself a few days ago - on the order page, it said it's "print on-demand". I've heard books made this way are often of poor quality (smudged ink, bad binding etc). Do you have any bad experiences?

>>11708417
How did you get "there are [math](2^{k-1})^2[/math] ways to pick A and B"? Do you remember they're supposed to be disjoint? I don't think it's correct.
The actual answer is [math]4^n[/math].

>>11542592
For example, start with any differentiable function f:R->R, then define g:R^2 -> R^2 by g(x, y) = (x, y + f(x)), then |det Dg|=1.

>>11466780
you're doing fine, don't worry
having said that: you're in high school, there is no need to hurry into university-level topics and research
a good mathematician is someone who can solve math problems. learning more advanced topics and fancy theorems is only a tool to that end. At my uni I see a lot of people who can regurgitate definitions but will fail to solve any problem that isn't an obvious application of a theorem they learned
i have been doing math competitions as a high school student and I feel they are worth more than people say. I don't mean "you need to score high to be a good mathematician". But time spent on them is not wasted.

>>10378187
"gluing every interval to its successor" doesn't really describe a path connected topology.
For example, the intervals 1..2..3.. are connected to each other, but then comes [math]\omega[/math] which doesn't have a predecessor, so it isn't connected in any way to the previous ones.
Well-ordered doesn't imply "every item has a previous one".