/sci/ - Science & Math

Sounds like your dream, OP.

There is an art to rigorous and elegant proofs.

If you don't like them, you're just not into that kind of math.

if you just want to calculate and crunch numbers, look into engineering

>>4420384
Though there is a large difference between learning to write proofs and just being handed proofs to study without explanation.

>>4420390
>>4420390
this is my situation, not all the time but most of the time anyway

im keeping a low b in the class at least

OP go read some Poincare. He was big on the intuitive approach. That said, you still need to be good at proof writing, as it is the main way of justifying mathematics. You just need to temper it with a creative and applied approach (but not so creative that you go off reservation).

:D

OP is a fucking idiot. All upper division math course are proof based. Who does not know that fact?

I went to UCLA, our mechanical engineering department offers a full year of upper division applied math courses. The classes are strictly engineering application of math.

>>4420780
>OP is a fucking idiot. All upper division math course are proof based. Who does not know that fact?

A lot of people don't.

At my school, a lot of math ed/math majors were surprised by the proofing classes. People entering with A's and B's in Calc I, II and III suddenly failed/bombed their introductory proof classes.

Proofs aren't easy. Especially when you get into Real Analysis, Topology, etc.

Conversely, I've hated maths my entire life. Then I entered university and got my face raped by rigorous proof after rigorous proof after rigorous proof.

Naturally, I became a maths major.

I lost interest in further math when I noticed that I didn't give a shit if this or that thing was a field or Abelian or what the fuck.

I know that this is coming for me, and I'm worried about how to prepare for the theory classes. Any tips /sci/ on how to prepare myself early for proofs and that new kind of difficulty? I need my As for my scholarships.

>>4420805
>that feel when no Galois Field

>>4420786
Person who's doing.. I think it's Calc 2 in America, but I'm in Europe. Basically just learned what integrals are and dipped toes in differentiation.

What would you have in a proofing class? Definitions, like (f(x+h)-f(x)) / h for derivatives, but obviously for more advanced shit?

>>4420849

When you obtain your book(s) for your theory classes, study the definitions intently and work on examples. Examples provide the best basis for comprehension/understanding.

Sadly, in higher-level math classes, there aren't a lot of examples. This is because those examples are specific and contain an enormous amount of information.

Get into the habit of drawing, making dots, small picture presentations. Imagine numbers converging to a point, or diverging. As I stated previously, learn the definitions intently and try to see how they make sense in the example.

Before reading over any proof, look at the theorem and try to prove it yourself using
(a) the definitions you learned
(b) intuition
(c) examples in which you learned even more information.

> the professor loves to teach seemingly only by rigorous proofs

that's pretty much every professor I've ever had

it's just what you get when you go to one of the best schools to study pure maths

>>4420854
Alright I can do that. That's what I'm doing right now in my lower level linear algebra class, going over what the definitions really mean, making connections to what is really going on, and I read intently over all the proofs in the book. Same with calc 2. I find it helps me to see what's actually going on. Also I'm currently reading The Road to Reality, and its helping me to create a bigger picture, so I hope that helps. I do find proofs to be pretty neat, especially when they are really clever, but I'm having trouble imagining myself making similar jumps without seeing it already done. That's what I'm mainly worried about.

>>4420852
Those definitions are babby tier for the most part. Actual definitions are like ...

The definition of an open set, in R:

We say a set O is open if every point p in O has a neighborhood containing p and is contained in O.

To understand that definition, you have to know what an open ball is. You also have to know what a neighborhood is.

In a topological space, a set U is open if it is contained in the topology, T.

A closed set would be:
Let X be a set. We say X is closed if it's complement, X`, is open.

Of course, I wouldn't expect you to know these, but I just want you to know that the definitions you see in Calculus I, II and III are relatively light when matched with those found in Real Analysis.

>>4420863

A good number theory book may be a nice introduction to proofs as well.

Theorems in number theory are notoriously easy to state and understand, but many require considerable mental gymnastics to prove.

>>4420868
That seems like a pretty good idea. I'll probably look into that.

>>4420866
>Of course, I wouldn't expect you to know these, but I just want you to know that the definitions you see in Calculus I, II and III are relatively light when matched with those found in Real Analysis.
Yeah I got just about nothing out of that, which is about what I expected.

Figured what we learn is very light, we hardly learn it properly anyhow. Guessing we just don't know enough maths to do so.

Well, thinking everything will get more interesting down the line. Right now there's an awful lot of just remembering the hows, not really about how we got the hows so to speak.

>>4420380
>YFW you realise that if you don't like rigorous proofs then by definition you don't like maths.

>>4420863
Certain proofs are hard and won't be asked on examinations. Others are relatively easy once you comprehend the basic mechanics in it.

A good portion of undergrad proofs just consist of abusing definitions and logic on a consistent basis. Know how to go from A to B to C and then D. If you can't prove it directly, do the contrapositive or contradict it. Suppose it's not true. If it's not, then you find such-and-such which contradicts the definition and/or hypothesis. Contrapositive form is unique in the sense that it is logical equivalent to the original conditional.

When you get into RA, Topology or Honors Linear Algebra(or LA 2), that's when it gets a bit tedious. Strong skills in logic aren't just necessary, but they're imperative. Intuition helps, but it's not really going to aid you too much in the long run. There are some proofs which you'll always find hard to regurgitate without needing some extra help.

Rarely do professors ask you to prove something you've never seen before. My RA class was a fucking nightmare, though, because the professor would ask questions/proofs we never discussed in class. Also, everything is trickier at the upper-level. Average was about a 50. Nothing is as straight forward. Everything is veiled in secrecy or there's some absurd trick to it.

>>4420877
>Rarely do professors ask you to prove something you've never seen before. My RA class was a fucking nightmare, though, because the professor would ask questions/proofs we never discussed in class.
Maybe in undergrad. That quickly changes.

>>4420868
I also recommend this.

Also, if you're interested in Math Competition exams, it's good to study some number theory. You learn interesting properties, such as modulo arithmetic, GCD, euclidean algorithm, et cetera. Wealth of knowledge in those books. It's a shame that it's underrated in undergrad study.

Probably also good to get a basic understanding of ordinals, cardinals, AoC/well order/Zorn's Lemma, etc for anyone actually going into math. Its stuff that's important to be aware of, and a lot of the ideas are pretty cool as well.

>>4420880
Right now I'm trying to get through The Road to Reality, which is honestly wear my heart lies, mathematical physics. However, once I have more free time over summer, I'll probably look into some number theory book. Also, is it bad that I'm pretty average at arithmetic? Every question I miss on a exam in linear algebra is usually do some dumb mistake such as a sign error. However I breeze through calc 2 exams because they involve variables. I'm assuming as I move up, it won't really make much of a difference if I'm not really awesome at it. My highschool let us use calculators in most math classes, so my skills went to utter shit sadly.

>>4420886
I've been doing grad level math and lost my TI83 years ago. Any class where not being able to use a calculator makes a difference doesn't reflect accurately on the majority of higher math classes.

>>4420886

You'll be fine. The amount of arithmetic goes down considerably the higher you go.

Actually, upper division math classes at my university basically consist of 5 to 8 students and a professor sitting around a table, presenting proofs to each other, and arguing over them. It must look like a philosophy seminar or something to someone walking by.

>>4420892
That's what I thought. It sucks when I have a B in a class simply because my professor won't give me much credit, even if all the real math is correct. Oh well.

>>4420878
Oh, yes. Of course.

In Grad, it's balls-to-walls insanity. It's a 'fuck you, I'm a PhD and if you want that shit, you best work for it.'

Remember, PhD is a pretty big fucking title. A lot of people seem to forget that, but it means that you're smarter than more than a good chunk of the fucking earth. At least I think so.

I'm just glad I didn't have that other RA professor in my University. His first exam had three questions.

First question: Construct R from Q.

This is actually contained in the Appendix of Rudin's Analysis text.

Second question: R is complete.
Third: Sets of definitions and small/light proofs discussed in class.

The third was worth only 10 pts. Needless to say, everyone bombed that exam.

>>4420886
Calculators don't matter in high end math classes, unless you're a stats major. It's what I despise the most about statistics. Constantly plugging in numbers, constantly.

>>4420905
Did they have to prove R was a field? Because otherwise that test sounds like it would be super easy for anyone who actually read the chapter.

>>4420915
Yep. The question, as it is stated in Rudin is:

>There exists an ordered field R which has the LUB property.
>Moreover, R contains Q as a subfield.

The proof of that theorem is about five pages long. It is not trivial at all.

Completeness for R, as I remember it, isn't a trivial proof either. It isn't necessarily pages long, but it's not childs play either.

>>4420936
>Completeness for R, as I remember it, isn't a trivial proof either.

I remember having to know how to prove that for my Calc I exam, back in undergrad. I don't think it was that hard.

There was a huge uproar about that exam, too. The withdrawal rate was the highest, ever, in the mathematics department. Not only was the exam excruciatingly tedious/difficult, but the professor had a penchant for verbally berating students. Sadly, he had tenure and they couldn't just get rid of him. Fucker was crazy.

Probably had to do with the fact that he didn't want to teach it in the first place and they just coerced him to do it because he was the only one who has a shitload of specialty in the field. Boy oh boy did the mathematics chair get riled up/pissed when he saw that exam.

>>4420938
I don't know analysis too well, but it seems like you'd have to do something like: given a cauchy sequence, take the dedekind cut of all rational numbers that are less than x_i for all but finitely many x_i, and prove that that represents the same number as the limit of the cauchy sequence. That at least definitely isn't something you would do in a calc class.

Am I the only one who hated analysis?

The material is fascinating and loved later classes like topology. But getting through real analysis fucking sucked. It was probably just because that was the "weed out" class at my university, but it seemed especially bad looking back at it.

>>4420947
I think most people dislike analysis in undergrad. You learn almost none of the cool parts of analysis.

>>4420945
>That at least definitely isn't something you would do in a calc class.

Maybe not in America. You people even think analysis and calculus are different things.

>>4420947
I did.

I like Topology a lot more than Analysis. I think it has something to do with the professors who teach it, too.

Every professor I've had that teaches it seems to be somewhat of a dick. They all have this, ''well if you don't understand it the first time you look at it, then you're obviously not cut out for math'' mentality. Francis Su on YouTube is really good, though.

>>4420953
Ah, yes.

The superiority of European Mathematics! It's why the US houses several wonderful mathematics departments, both in Undergrad and Grad. Mostly Grad, though.

>>4420953
They are. Calculus can be considered part of analysis, but the reverse inclusion certainly doesn't hold.

>>4420955
> ''well if you don't understand it the first time you look at it, then you're obviously not cut out for math''

Yeah, that was exactly the problem in the mathematics department at my university.

I can't speak for every curriculum, but at my university there was very little "build up" to a rigorous proofs class. Looking back, I think very few of us had any real understanding what mathematical research was like as an activity.

I can't remember the exact numbers, but of the 20 or so math majors that entered my Real Analysis I class sophomore year, I think 6 or 7 made it out. The rest dropped and switched majors.

> ''well if you don't understand it the first time you look at it, then you're obviously not cut out for math''

this is actually true, like it or not

except when they say "cut out for math" they mean you're not gonna go off and become the next Gauss

anyone can use mathematics to varying extents with a bit of practice, but to really delve into the depths of math takes a certain kind of person who just SEES it

>>4420961
Just because you people decided that the dumbed down version of the subject deserves a name of its down doesn't mean there is any actual difference.

ameribro here

do you britqueers really not call calculus calculus?

>>4420992
>not American
>must be British

Did you know that some people can speak languages other than their own native language? Amazing, I know.

>>4420905
this is super easy
everyone should know how to construct all the basic sets from the null set.
the second question seems weird since the most used method to construct R is by constructing the smallest complete set containing Q

>>4420400
I know it may make you feel like a dumbass but if your school has an upper-level math tutoring program go DO IT. Students can often explain things more intuitively than professors that seem to have their head in the clouds.

>im keeping a low b in the class at least

Based on this though, maybe you just need to apply yourself harder to get an A

>>4420999

you're still speaking english, faggot. most likely british english. i wouldn't be surprised if the french called calculus Le Caluclarrh.

>Caluclarrh
analyse

>>4420786
What? I knew that even when I was like eleven years old.
Why are people so dumb.

Speaking of which, I'm late for my number theory class. ttyl /sci/

>>4421076
Some people actually had childhoods and friends.

fuuuck

I actually love proofs; solving them, doing doing them writing them.

in fact, that's the only thing I love about hardcore mathematics. learning it is boring IMO, but being able to apply it even in such an abstract way is really exciting to me.

reading about linear algebra in high school actually got me excited about math

i hate calculus by contrast