New types of math are invented all the time, but they are usually sub areas of the very broad areas you know about. No one just works with algebra or founds a new field of that magnitude anymore, that sort of low hanging fruit has for the most part been picked. Many of these fields are so...
If this is in response to my post, then please let me clear up that what I meant was that it is not optional if you want to obtain a PhD subsequently. There is nothing stopping you from just getting a Bachelor degree if you don't intend to pursue a PhD.
I have no experience with engineering...
The usual European model is (there is some variation between countries, but the following is the most common):
Bachelor: 3 years
Masters (not optional): 2 years
PhD: 3 years
While the American model is:
Bachelor: 4 years
PhD: 4-5 years
So it works out to about the same. From what I hear...
Everyone is different, but the following are some approaches I have used when self-studying.
Look at the exercises. If any look particularily scary or you have no idea how to begin on some, then do that exercise.
Another approach is simply to do the last 5 exercises or so in each section...
No there is not a general consensus.
Linear algebra is for many people their first introduction to proof based and abstract mathematics, and if you are not prepared for it then that can be hard, and for some people very hard. If you are able to understand proof-based and abstract mathematics...
If you are mathematically mature enough and have no problem with proofs and abstract arguments, then there should be no problem in starting with abstract algebra and topology. Most students are not however.
At some point you will need calculus, but it is entirely possible to pick it up later...
Some reasons of the top of my head.
1) Obviously you would rather not pay, especially with how much grad school costs unsupported.
2) If some grad school accepts you, but without support, then it is likely that some other grad school at about the same level would accept you with support. At the...
I would like to echo the comment by eri and ask what you mean by valuable? Clearly knowledge is knowledge, so in that respect it will be just as valuable. It may be harder to demonstrate competence without a letter grade, but if you will take more advanced classes building on it does it really...
It's uncommon to be able to solve olympiad level problems with no preparation. Personally I prepared for 5 months for our national qualifiers and at that point I might have had a shot at problems 1 and 4 on IMO (the easiest problems), but usually couldn't even solve these. After a total of about...
I only have experience with the IMO so I won't speak for physics, but I guess the situation is mostly similar. It's 2 years since I participated so some new instructional material that I'm unaware of may have come out in the meanwhile.
Unfortunately 3-4 months is not a lot of time to prepare...
Depends heavily on what country you are in, and whether you have already trained somewhat.
If you are in the US and have not strayed much of the regular curriculum, then it is probably not feasable, but why not give it a try? The competitions leading up to it are great fun, and who knows, you...
If you win the lottery the first 10 times you buy a ticket, you will be surprised when you lose on the 11th ticket.
"Smart" people with a shot at going to grad school at Harvard have probably gotten into a lot of "top 5%s" throughout their academic career (top 5% GPA, top 5% undergrad, top 5%...
How do you define dumb? The impression I get from my Chinese friend is that average Chinese students certainly are more hard working and probably know more facts. If this is how you define intelligence I suspect Chinese students are more intelligent, but personally I would be vary of trying to...
It's a bit disheartening to hear that people (even grad students who presumably want to do research in the area) cheat on take home exams.
Take-home exams are pretty common at my university and I have never heard of anyone cheating. I suspect and hope only a very small minority of the...
I just finished my first year of math at university. I've only had math courses (calc, linear algebra, abstract algebra, analysis I+II, probability&statistics, differential geometry, discrete math) and all my written exams have been open-book 3.5-4.5 hours (this semester I will also have some...
The guide has some good advice on the errors beginners often make, however the style of proof it uses is NOT standard. I suspect it uses this format for there to be no ambiguity, however in real math proofs we use English, but we use it carefully. "For all real x there exists a unique integer n...
My advice would be to do one or more of the following 3:
1) Computer science relies on mathematics and you will very likely need to take some math courses and will definitely need math in your CS courses. Thus one way to prepare is to brush up on your high school math, and possibly start...
If you do the same things at the two colleges, then graduate schools probably won't care. Your undergraduate institution to a large extent only shows the graduate school a bit about what you were like in high school, and they don't really care. What they care about is what you're like when...
Everything is coming from the perspective of a math student and applies to math and other degrees close to it. I have no idea how much applies to other degrees.
According to http://www.usnews.com/articles/education/best-colleges/2009/08/19/how-we-calculate-the-college-rankings.html" the...
I would like to second the Basic Mathematics book recommendation. Fundamental, yet very challenging and its focus is more on getting you in the right mindset than teaching you a list of facts. I believe this will prepare you more for college-level mathematics than any accelerated advanced honors...
No one can really give you a straight answer. Natural ability has some effect on how you do in higher level maths, but no one really knows how much. I have never seen a person who really felt math was interesting who wasn't able to do it. This may of course be because they have dropped out...
How many problems do you feel this way about? If it's only an occasional problem then it's perfectly normal. If it's every other problem, then you may not have a good enough grasp on the subject matter.
I would suggest that he does not do this. Not being able to solve a problem and thinking...
It will of course depend on your university, but from the experiences I have with graduate students (mostly from European universities, don't know if it matters) it seems that at most places you're allowed to audit courses if you like (i.e. take them, but without credit) and take whatever...
Depends on the course. An undergraduate course may teach you the necessary commutative algebra as you go along and skip some generality to keep the prerequisites down. For a full general course in algebraic geometry using something like Hartshorne's algebraic geometry you will need a strong...
Probably no more than the standard requirements are mandatory, but there are lots of interesting CS and EE problems in biology so for some specializations and applications of CS and EE it may be useful (for instance a well-known biology/CS problem is finding good algorithms for protein folding).
f(n) = \sum_{i=1}^n i^3
It's pretty clear* that an expression of this form is a polynomial so we just need to show it has degree 4.
We have,
f(2n) = \sum_{i=1}^{2n} i^3 > \sum_{i=n+1}^{2n} i^3 \geq \sum_{i=n+1}^{2n} n^3 = n^4
so f grows at least as fast as a fourth degree polynomial and...
If you're advanced enough to have some exposure to independent formulation of theorem and proofs, then I don't think you'll learn much from this type of problem solving. For people unfamiliar with math at this level, these types of problems are approachable and may help them adjust. I remember...
I'm sorry, but while math is usually presented deductively most of it isn't deductive in the first place. Consider for instance how when you're reading a graduate textbook a good student may very well be able to do all or at least most proofs by himself, but many of the theorems were once hard...
I'm myself an undergraduate so the following is speculation. I suspect it won't penalize you directly, but you'll likely have less time for each degree than single-majors. If other people do some extra coursework they may be at an advantage, not because of their single-major, but because of...
I think that is a large part of it, but once you get beyond the initial paranoia there are still two scary things:
a) These are prescription drugs. Thus, at least legally, they are not on the level of alcohol, but rather the same as cocaine. You can get jail time in many countries for simple...