What is Reddit's opinion of Principles of Mathematical Analysis (International Series in Pure and Applied Mathematics)?

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I don't know how much the AIME has changed in the last 20 years (sorry), but I would recommend being able to get at least a 7-8 on it. I got a 12 my senior year of high school, but I was one of the top 2 students in Math 25 (they don't actually keep track, but we were all friends in the class so we compared how we did), so there's plenty of room.

What I would most encourage is that you get really good at proofs. I don't know what 25 will be like next year (in my time, it was taught by a different professor every year so it was always different), but it will be proof-based and it will be challenging. Your knowledge of multivariable calculus will be helpful in your other classes, because you probably won't learn anything directly applicable in 25! But you'll be doing a ton of topology proofs, so make sure you're good at that. Maybe find a (cheap Indian) copy of the Rudin. I'm serious, get it shipped from India if you have to, but do not pay the amount they're selling it for on Amazon, holy shit.

Rudin has published many books on analysis each quite different. It is important to know which you are talking about.

Check out the Amazon review “like drinking math from a fire hose” https://www.amazon.com/Principles-Mathematical-Analysis-International-Mathematics/dp/007054235X#aw-udpv3-customer-reviews_feature_div

It is a book for maths people. Maybe you have done enough math to get something from it, but I suspect that it is a poor choice.

I looked for a textbook that I had to use in college in the 80s:

Principles of Mathematical Analysis (International Series in Pure and Applied Mathematics) 3rd Edition by Walter Rudin. I found a beautiful pdf of this that one can download and for sale brand new for $142.49.

Yes, go forth and pirate!

IMO the absolute best book for intro to real analysis is Principles of Mathematical Analysis by Rudin

I think real analysis kicks most people in the butt. Its really abstract and jarring in relation to any lower level math. Its kind of your introduction to higher level math and the type of thinking that analysis forces you to use.

The way I got better? Practice writing more proofs and then do it again. Rudin has some awesome problems in it, and you can teach yourself some incredible things about the inner workings of calculus.

You need some math but not all math. Statistics uses only probability theory, and that uses only real analysis. You don't need algebra except linear algebra, combinatorics, complex, or many other areas of mathematics. If you do a PhD in statistics, you will be expected to take a course in measure-theoretic probability. A prerequisite for that is advanced undergraduate real analysis, for which an appropriate textbook is "baby Rudin". So it would be helpful if you had that course (whatever it is called where you are) and all of its prerequisites, prerequisites of prerequisites, and so forth. You need a lot less for a masters in statistics. Multivariable calculus should be enough for that.

OTOH, even having a PhD in math isn't a good preparation for statistics. Some familiarity with the ideas of statistics is helpful.

This is super helpful, thank you!

And nothing against simulation, I know it's a powerful tool. I just don't want my foundations built on sand (I'm familiar with intro stats already).

Would Rubin's book on Real Analysis suffice: http://www.amazon.com/Principles-Mathematical-Analysis-International-Mathematics/dp/007054235X

Or are there even more advanced texts to pursue for Real Analysis?

For the whys and hows, you're gonna need a full-blown analysis textbook like baby Rudin. Calc I and II at most universities don't even scratch the surface when it comes to understanding the whys of anything. Anyways, yeah. Engineering is cool.

Sure. Here are some basic articles:

https://en.wikibooks.org/wiki/Real_Analysis/Limits

https://mathcs.org/analysis/reals/infinity/countble.html

Here's a decent book for the introduction to real analysis: http://ramanujan.math.trinity.edu/wtrench/texts/TRENCH_REAL_ANALYSIS.PDF

But if you really want to get into the meat, you need to get some of the grad level books: https://www.amazon.com/Principles-Mathematical-Analysis-International-Mathematics/dp/007054235X

https://www.amazon.com/Principles-Mathematical-Analysis-International-Mathematics/dp/007054235X

Very expensive hope it’s the one

Baby Rudin

real adj: of or relating to practical or everyday concerns or activities

As such, every example I gave is a real thing—integers, imaginary numbers (which allow tunneling, antimatter, and computer rotation to occur, and thus you use daily on Reddit), gender, race, and even personhood. They are all just abstract real things.

And, as complex numbers exist separate of your experience, they are also quite real from a philosophical standpoint. They are not platonic ideals that I have defined, but rather a result of the nature of space-time. They would work the same in all bases, if we redefined all the operations on them, and so on. In fact, Rudin (which most people use to study analysis) has you prove these facts outright. The linear analysis book I teach from also shows many of them.

In conclusion, they are real philosophically. They are real by denotation (definition if you must). I have shown these points with sources, while you have only your ableism and your stubbornness, no evidence nor argument.

In short, get lost until you learn how to execute persuasive language instead of crying when you don’t get your way.

I

I read down below that the highest "math class" you took is calculus 3. Well, looks like you haven't started learning math yet.

Generally, calculus is applied real analysis. Here, "applied" is too big a word since calculus is made up of very simple problems tailored for high school math students and/or college students who aren't math majors(engineers, med students, biology majors etc).

If you are in calc 3, I suggest you continue taking math classes because at around this point actual math starts kicking in, complete with theorems/lemmas/corollaries/proofs. You'll realize math proofs are held up to much higher standards of rigor and quality as compared to any other science.

Baby Rudin is intro real analysis textbook meant to introduce math majors to calculus as it's done by mathematicians. Check out the table of contents. Notice how the whole book has a definition -> theorem -> proof structure. No other science does such rigor. Remember, though, this book is just the beginning of your journey into analysis.

When mathematicians say "basic algebra", they mean topics like those in Basic Algebra 1 by Jacobson. Again, check out the TOC and the structure of the book. It's all about theorems and proofs. Major topics include groups, rings, modules etc.

But these only scratch the surface of what's out there in math: graph theory, combinatorics, measure theory, category theory, algebraic geometry, differential geometry, complex analysis etc. Each one of these subjects are humongous and ever expanding.

I suggest you take a look at an actual math topic. I promise it's real cool and interesting.

If you want to do more math in the same flavor as Apostol, you could move up to analysis with Tao's book or Rudin. Topology's slightly similar and you could use Munkres, the classic book for the subject. There's also abstract algebra, which is not at all like analysis. For that, Dummit and Foote is the standard. Pinter's book is a more gentle alternative. I can't really recommend more books since I'm not that far into math myself, but the Chicago math bibliography is a good resource for finding math books.

Edit: I should also mention Evan Chen's Infinite Napkin. It's a very condensed, free book that includes a lot of the topics I've mentioned above.

Is this the book you are recommending?

Imho Zorich rocks. It gives you a modern take on analysis, esp in book 2. Kinda heavy on the theory but has way more general stuff later on, such as integration on manifolds... Now our educational system is built on the soviet model so experience may vary. Rudin also has cool stuff.

http://www.springer.com/us/book/9783540403869

http://www.amazon.com/Principles-Mathematical-Analysis-International-Mathematics/dp/007054235X

Rudin.

Try Baby Rudin. I think the first chapter covers what you are looking for very thoroughly.

You might also find Analysis: With an Introduction to Proof to be rather helpful.

Rudin

Since redditmeupbaby has covered some of your alternative choices, I'll talk about undergrad math versus olympiad math. Firstly, undergrad math requires less ingenuity than olympiad math. The number of techniques you can employ is limited, and most problems can be solved by applying these techniques directly. If you can do the IMO, you have sufficient problem solving skills.

Undergrad math poses challenges which are not emphasized in olympiad math, however. One challenge is setting up a rigorous foundation for your thinking. redditmeupbaby mentioned real analysis. Check out the first few chapters of Rudin. Given any simple statement from those chapters, you have to know whether it is right or wrong and why it is right or wrong. Even though certain things may seem obvious, you still have to prove it at least once. Only after that are you (usually) free to forget the proof and use that fact freely in your reasoning. You will have to destroy your current intuition and replace it with a new one grounded in propositions and theorems. This should be an abstraction of thought: once you have proved a result, you need not re-prove it whenever you want to use it. If you do not achieve this, you will be helpless once you leave the comfort zone of objects with many concrete examples.

Another challenge in undergrad math is understanding the concepts involved, how they interact, and why they are relevant. Algebra is a good example. Take for example the determinant. It is insufficient to know just the formula for calculating it. You have to understand what the determinant really means and why it is useful. This could involve generalizations to different contexts e.g. higher dimensions, or introduction of more new concepts e.g. multilinear forms. Once again, this is a form of abstraction.

In general, I would say that you should not have any problem. Just keep an open mind and you'll do great.

P.S. Are you Singaporean by any chance?

If you don't mind I'll only reply to this post.

> For over for thousand years we have known that the area of a circle can be defined via pi. yet if I ask you today to accurately define the circumfrance of a circle with area = 1 it is imppossible without rounding or boring me forever refining your answer

For a circle, A = πr^2, where A is the area and r is the radius. Furthermore, C = 2πr where C is the circumference and r is the radius again. To go from an area to a circumference we can solve for r in the formula for A and substitute it into the formula for C. This yields C = 2π√(A/π) = 2√(π)√(A). Hence, a circle of area 1 has a circumference of 2√(π).

> Can you you believe that it is impossible to know the area of a circle if I know it's circumfrance?

No, because that isn't true. By the same method as above, we find A = C^2 / 4π.

The formulas I gave are exact, true statements about the mathematical object that is a circle. They involve the irrational number pi. Irrational numbers are strange, but well defined mathematically. I don't understand what you're trying to say about them.

I think it would help you if you learned the relevant mathematics (algebra, geometry) and physics (mechanics, special relativity) before you try to talk about them. They're very interesting subjects.

Edit: Here's something you may find interesting. An excerpt from Walter Rudin's Principles of Mathematical Analysis where infinity is defined. Statements about infinity such as those about the limits of sequences can be made rigorously using the definition. Hence, there is really no "unresolved issue" with infinity in the sense that you seem to be talking about, at least in mathematics.

Once you feel you have a decent grasp on your algebra, i'd recommend Rudin its hardcore but by far the best way to begin your foray's into higher math.

Why shouldn't everyone just begin with Baby Rudin?

https://www.amazon.com/Principles-Mathematical-Analysis-International-Mathematics/dp/007054235X

Here you go OP.

I prefer the third review of this classic math book!

http://www.amazon.com/Principles-Mathematical-Analysis-International-Mathematics/dp/007054235X

Rudin is good for analysis, the lecturer wrote most of the course straight out of Kreyszigs book for the functional analysis class I just took.

I'm not really sure what the best references are for representations and Lie theory, I just had a class where the book used was Bakers, but I don't think it's meant to be very good, I never usually end up looking at texts for my classes anyway..