HN Books @HNBooksMonth

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.

My favorite LA books are Linear Algebra by Friedberg/Insel[0] which is a combination of Axler style book with more computation oriented one (Terry Tao has a set of lectures based off this book). Another one I like is Modern Intro To LA by Henry Ricardo[1] which implicitly introduced me to Replacement theorem which is really overlooked in a ton of LA books. Again, this book's a rigorous mixture of both theory and computation done very well. If you've never seen higher level math before, there's Linear Algebra: Gateway to Mathematics by Robert Messer[2]. It has tons of commentary about elementary set theory and proof techniques along the way. Whenever someone mentions Axler's book, someone else brings up Treil's book. But there's a third one in the same league/group which is Linear Algebra: An Introduction to Abstract Mathematics by Robert Valenza[3]. Other favorites are Coding the Matrix by Philip Klein[4] for Python aficionados and Linear Algebra Through Geometry by Banchoff/Wermer[5] for those who like geometry.

If you are way beyond all this, you can still pick up new things from Advanced Linear Algebra by Steven Roman[6].

[0] https://www.amazon.com/Linear-Algebra-Stephen-H-Friedberg-eb...

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

[2] https://www.amazon.com/Linear-Algebra-Mathematics-Robert-Mes...

[3] https://www.amazon.com/Linear-Algebra-Introduction-Mathemati...

[4] https://www.amazon.com/Coding-Matrix-Algebra-Applications-Co...

[5] https://www.amazon.com/Algebra-Through-Geometry-Undergraduat...

[6] https://www.amazon.com/Advanced-Linear-Algebra-Graduate-Math...

⬐ dman

+1 for counting from zero

⬐ sn9

Seconding Klein's Coding the Matrix. It's a great alternative introduction and actually implementing linear algebra ideas in Python is both useful and a great aid for learning.

The auto-grader is helpful for making sure you got the right solutions.