Real Not Complex: Free Math Resources

by phonebucket • 6 years ago

168 25 6 years ago

I am almost always disappointed in such lists as they almost never contain the books I have never seen before. I got lucky today.

The list has [0] Multivariable Calculus by Don Shimamoto. I saw it on Amazon first(so I have seen it before, but not on lists such as the linked one), but didn't want to pay $40 for this strikingly beautiful book. Turns out it's free.

The real analysis section has [1] by Lafferriere, Lafferriere and Mau Nam. It's short, sweet and very pretty. No bloat. Logically organized. For example, the section "Applications Of The Completeness Axiom" contains everything you need to know as pertains to the topic (within elementary analysis) in one place. Such things usually get lost in the weeds in regular analysis books. Another book I like there is [2] which is on Lebesgue Integral and Measure Theory by Cheng.

[0] https://drive.google.com/file/d/1IB-fSe-KR6mbo89iFTJVSA0CrFf...

[1] https://pdxscholar.library.pdx.edu/cgi/viewcontent.cgi?artic...

[2] https://www.gold-saucer.org/math/lebesgue/lebesgue.pdf

thaumasiotes6• 6 years ago

The real analysis section has [1] by Lafferriere, Lafferriere and Mau Nam. It's short, sweet and very pretty. No bloat. Logically organized. For example, the section "Applications Of The Completeness Axiom" contains everything you need to know as pertains to the topic

I guess this is a matter of taste. I went to look at this and it really bothers me that, true to the name they give, the "completeness axiom" is presented as an axiom. My analysis class used Strichartz' The Way of Analysis, which is certainly not "short and sweet" -- the professor described it to us as "chatty" -- but it goes ahead and actually proves the "completeness axiom".

Compare the completeness axiom

Every nonempty subset A of ℝ that is bounded above has a least upper bound. That is, sup A exists and is a real number.

with Strichartz' theorem 3.1.1

For every non-empty set E of real numbers that is bounded above, there exists a unique real number sup E such that
1. sup E is an upper bound for E
2. if y is any upper bound for E, then y ≥ sup E

Maybe Laferriere, Lafferiere and Mau Nam could use a little more "bloat". :/

nointer• 6 years ago

In the first one, they already defined upper bound and least upper bound beforehand, so the statement is more concise than the Strichartz's one. On the other hand, the proof of the completeness axiom, or rather a detailed construction of the reals, is a part I always enjoy in analysis textbooks.

JadeNB3• 6 years ago

How does Strichartz prove it? You certainly need some axiom beyond, say, the ordered-field axioms; for example, the analogue of Theorem 3.1.1 isn't true for ℚ in place of ℝ.

mixture• 6 years ago

Disclaimer: please, excuse the typos, grammatical and stylistical mistakes below as I am pretty freaking drunk at the moment.

Friendly reminder:

while the material in the linked books might appear trivial to many, the style might be discouraging to those who've never seen "actual math" before. If you're in the latter group, try out the books below (some are not free, but they're still helpful [libge*n might be helpful]):

1. BOOK OF PROOF by Richard Hammack. This FREE book is your passport into mathematics proper. It teaches you how to read math books. After this book you'll stop fearing the word "proof" and start demanding it wherever you go.

Link: https://www.people.vcu.edu/~rhammack/BookOfProof/

2. The book by Don Shimamoto is super trivial if you know linear algebra. The best books in the genre (imo) are (a) Linear Algebra: Step by Step by Kuldeep Singh whose secret is that he uses all the exercises from other LA books as his 100% worked out examples (which seems to be a tradition for Indian authors; I have tons of examples from other branches of math). Indian authors teach you how to drive a car by putting you at the steering wheel instead of lecturing you about it. BTW, if you know math at the level of, say, 7th grader, then you're perfectly suited to study from Singh's book. If that isn't clear, do you know how to add and multiply fractions? If yes, then you're perfectly ready.(b) A modern Intro to Linal by Henry Ricardo. This book is great in its ability to explain certain introductory concepts. For example, the linked Shimamoto book starts out with (going by memory) "the elements of R^n are finite sequences of length n... bla bla bla". Well, Ricardo explains what that means. It also helps if you know what a "sequence" is. But we're getting there. Another author who does a good job of it is Sheldon Axler, I believe. He does have his own idiosyncrasies, but I digress. The only difficulty here is, probably, the idea of diferentiability of multivariable functions.

Links:

https://www.amazon.com/Linear-Algebra-Step-Kuldeep-Singh-ebo...

https://www.amazon.com/Modern-Introduction-Linear-Algebra/dp...

http://sites.msudenver.edu/wp-content/uploads/sites/385/2017...

3. Now to analysis of real kind as opposed to the complex one. If you've never heard anything about this, then you should hook up with Lara Alcock's "How To Think About Analysis". To the best of my recollection, half the book is about how to think about the term "sequences". She goes above and beyond in trying to explain you what that means. The rest of the book is so-so. Once you learn about sequences, you can move on to Jay Cumming's book. It's a bit more involved with proofs and shyt, but still extremely accessible. Laughably so. But note, he only stays grounded on the real line and does not venture into the Euclidean Spaces proper. However, there's another easygoing dude named Raffi Grinberg who'll take you through R^n by hand, no problem. If you master this book, then you're at the level of what's referred to as "Baby Rudin". Next, if you're feeling adventurous, there's a new book by Sheldon Axler that introduces you into the modern analysis which is FREE FOR NOW. GRAB IT WHILE YOU CAN CAUSE I SAW IT ON AMAZON.

Now if you're a business type who ain't got no time for all this foolishness, try "Intro to Analysis: From Number To Integral" by Mikusinski's. It starts with axioms for reals and takes you straight through to Lebesgue Integral in no time. Every section has very few exercises. Say, about 4. It also skips through the traditionalist Riemann integral. Because why not.

Links:

https://www.amazon.com/Think-About-Analysis-Lara-Alcock-eboo...

https://www.amazon.com/Real-Analysis-Long-Form-Mathematics-T...

https://www.amazon.com/Real-Analysis-Lifesaver-Understand-Pr...

https://link.springer.com/content/pdf/10.1007%2F978-3-030-33...

https://www.amazon.com/Introduction-Analysis-Number-Integral...

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Now to a few examples of Indian authors' propensity to give you everything you need to know about the subject unabashedly. Check out:

-- "Sequences and Infinite Series" by N.P. Bali. It's 100% worked out proofs after proofs after proofs. If you read through this book, I imagine you could easily crush any "sequences and series" portion of any elementary real analysis exam in existence. The author also has a book on real analysis which is structured very similarly to the aforementioned book. No idea what N.P stands for, though.

Now to a book you've never heard of. "Data Structures and Algos" by Gav Pai. It easily blows Cormens, Sedgewicks and Skienas out the park by giving you, as you guessed it, all the 100% worked out examples you need to understand the topic.

"Fundamentals of Complex Analysis by Bhat" is another one. All its problems are immediately followed by solutions. Look, it understands you're not at 40K a year school with 10 profs to help you out with any of your questions. Books like this are written for students, not to impress the profs.

I just remembered some other Indian textbooks on abstract algebra, discrete math etc, but I cannot remember their names. They are also all about giving you all the examples you need to grok the concept at hand.

Koshkin• 6 years ago

On the other hand, your time is never free - to the point that no matter how much you pay for a math book, you will end up paying incomparably more with your time spent studying it.

phonebucket2• 6 years ago

On the other hand, your time is never free - to the point that no matter how much you pay for a math book, you will end up paying incomparably more with your time spent studying it.

This is true.

However, I find that I tend to buy much more than I read. Not just because I tend to hoarde these things, but also because I try to seek out sources that click with me before I work through them.

throwaway_pdp092• 6 years ago

I utterly agree with @Koshkin. the monetary cost of a book is virtually irrelevant compared to the days or weeks time cost in reading it.

I take your 2nd para in a way, the cost of a book being so cheap when I do read up on something I tend to buy 2, sometimes 3, books on the subject so where I get stuck on one the other will see me through.

Still, I am discomforted by your book-buying habits; a book unread is a valuable resource wasted :) But each to their own way.

thaumasiotes6• 6 years ago

Counterpoint: https://www.youtube.com/watch?v=qIOIHywqwTY

jimhefferon• 6 years ago

Very true. A good deal on something bad is not really a good deal.

But it is also true that "freely available" is nto the same as "worth nothing" and some of the works here, at least, are well worth the study (to their intended audience), I believe.

adele11• 6 years ago

Possibly - If you are optimizing for time that is.

But what if you are optimizing for enjoyment?

manthideaal• 6 years ago

A more recent version of the last link in the Statistics and machine learning section is (1)

(1) https://github.com/percyliang/cs229t/blob/master/lectures/no...

photon_lines• 6 years ago

If you guys like visual / intuitive guides, I'm trying to release my own and you can find them using the links below:

- Linear Algebra: https://github.com/photonlines/Intuitive-Overview-of-Linear-...

- Maxwell's Equations: https://github.com/photonlines/Intuitive-Guide-to-Maxwells-E...