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Ok, I can't take it anymore. I'm a newcomer to the delights of math, but I'm completely lost as to what's complex and what is more elementary and then what is foundational.

I wanted something foundational, so I asked my calc. prof about number theory. When he asked why I was curious, I told him and he said that's not a good place to look. He said to go through calculus and on to abstract algebra then number theory. He said the foundation of mathematics is in calculus. Well, I don't plan on going further than two semesters of calculus in college so I'm trying to figure out how I'm going to tackle this when I gradate.

So, what's hierarchy? Where's analysis come into the mix? topology? linear algebra? number theory?

The dover series on mathematics gets great reviews across the board on amazon, but I don't know where they all fit in. Does anyone have experience with these books?

I know there probably isn't a very rigid hierarchy, but I'm sure there's a general layout as to what is lower level and what is higher level; what should be learned before learning something else, etc.

Number theory is about finding primes and modular arithmetic and such. Foundational in one sense but impractical in the real world. Calculus is the math that you would use IRL. And/or in physics or other classes outside social studies (which is more about statistics).

Number theory is fun if you're into computer programming or are a serious math geek, otherwise not so much. IMO. There's a subset of number theory that applies to cryptography. If that was to be your life's work, you'd probably know it by now.

tl;dr Calculus first, number theory later.

-------------------- You make it all possible. Doesn't it feel good?

What do you know/understand about math at the moment? I find it very hard to suggest anything without knowing a bit more about your current math level. What do you mean by newcomer?

Take Real Analysis. Let's just say that some classes start out by proving that zero is unique... It goes over what calculus is (measures, Lebesgue Integration etc...), and a lot of the foundations of math. It's a really important class, then from there you can move onto complex analysis, topology, what have you. Make no mistake though, it's a serious class

Quote: I know there probably isn't a very rigid hierarch

That was my thought as I read your post.

I would say that the easiest is arithmetic, then algebra and trig, then basic calculus. After that you can study the rest of maths at either a high or low level.

Quote: Annom said: What do you know/understand about math at the moment? I find it very hard to suggest anything without knowing a bit more about your current math level. What do you mean by newcomer?

Well, I'm in single variable calculus with analytic geometry right now, the first of two semesters of calculus at my university. Obviously I have had algebra and trig, although I would like a more comprehensive understanding of them.

When I said I'm a newcomer to the delights of math, I was just meaning that I've recently taken more of an interest than, "I need this class for my major."

Quote: Chespirito said: Take Real Analysis. Let's just say that some classes start out by proving that zero is unique... It goes over what calculus is (measures, Lebesgue Integration etc...), and a lot of the foundations of math. It's a really important class, then from there you can move onto complex analysis, topology, what have you. Make no mistake though, it's a serious class

Thank you. This sounds like almost exactly what I'm looking for.

In my university, the calc sequence comes first. Then they require Elementary Linear Algebra and Ordinary Differential Equations. Then Statistics. After which, you are required to take Advanced Calculus (Real Analysis), Modern/Abstract Algebra, and a proof based Linear Algebra (Real Analysis & Modern/Abstract Algebra are all proof based). Then you are free to take classes based on your interest and specialization.

Topology, Partial Differential Equations, Combinatoric courses, etc are all upper division, grad courses.

I myself took elementary Linear Algebra, proof based LA, and a grad level LA, plus grad level Numerical Analysis I & II, Stochastic Diff Eq (I'm going for numerical analyst). So the higher level courses would cater to your specialization.

In summary, there is a core requirement after which you're free to study what you want.

Quote: Chespirito said: Take Real Analysis. Let's just say that some classes start out by proving that zero is unique... It goes over what calculus is (measures, Lebesgue Integration etc...), and a lot of the foundations of math. It's a really important class, then from there you can move onto complex analysis, topology, what have you. Make no mistake though, it's a serious class

True that. That class doesn't fuck around. Abstract Algebra is also full of number theory. If that helps.

Start with calculus (with at least some exposure to multivariable calculus) and linear algebra as a foundation. The next step would be to pick up an elementary text on proofs, because after foundational mathematics pretty much everything is proof based. But this is a great thing, because it's in the proofs that mathematics becomes really interesting.

Beyond this there really is little hierarchy. However the more you know, the more you will get out of a certain subject. For example, I have taken complex analysis, and it would have definitely been easier if I had taken topology before hand. Likewise, number theory makes use of results from across the whole spectrum of mathematics like geometry and abstract algebra.

I only have one dover book that I've read, and it serves the purpose I needed it to. The book is titled Introduction to Linear Algebra and Differential Equations. I bought it because I wanted a review of those subjects. For that purpose it did the trick, but I already knew the material. If you want a standalone book, I recommend something a little more pedantic, at least for these foundations classes. Something with copious examples, and examples that are walked through.

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