HN Books @HNBooksMonth

Every undergrad level math subject (real analysis, linear algebra, topology etc) has a few core theorems around which everything revolves. Unfortunately, math textbook writers bow to the tradtition of writing bloated books where the core ideas get lost in the sea of chaff. The exercises sections usually further muddy the waters with ridiculous number of irrelevant garbage. This makes learning math not only inefficient, but also ineffective. Thus, in the present there's probbly no such book or method.

However, you can make learning LA massively more enjoyable if you choose the right books (read "books with tons of examples" that clarify theorems, definitions and the like). For example, Linear Algebra: Step by Step by Kuldeep Singh [0]is an awesome book.

[0] https://www.amazon.com/Linear-Algebra-Step-Kuldeep-Singh/dp/...

⬐ rxsel

What do you think about a MOOC approach?

⬐ Mihalis

Can you recommend other books in the same vein?

While "all the high school math" is somewhat clear, the "all the college math" should be clarified a bit.

Are we talking about math for social studies majors, engineers, physicists, economists or mathematicians?

For example, there are books like [0] for engineers which serve as a boot camp of sorts. There are no theorems, proofs or deep math in it. There are many different kinds of engineers, so books like this don't include everything an engineer needs to know. For instance, there's no automata or group theory in [0]. It has a part at the beginning called Foundation Topics which could serve as "all the high school math one needs". In fact, this whole book could probably serve as "all the math an advanced high school student needs".

There are books like [1] for physicists. They are meant to introduce physics majors to a wide array of math topics in a relatively pain-free way. This book is much more rigorous and contains a LOT more material than [0]. Basically, a theoretical minimum for a physics major.

Economics majors also get somewhat rigorous math load where measure theory features a lot more prominently than it does in other majors (except mathematics proper).

List of subjects for math majors vary from place to place, but you'll have much easier time down the road if you master the (rigorous) rudiments of linear algebra (vector spaces), group theory, number theory and real analysis. Of course, knowing more math (say topology, complex analysis, category theory, combinatorics etc) is always good.

Before you get started with math for mathematicians, you'll want to learn their jive. A good intro is [2]. It's free and a really nice book. Another really nice book with non-existent pre-reqs is [3].

A pitfall that awaits a lot of people new to math is a concept of "multivariable calculus". This concept is a mess and means everything to everyone. Oftentimes it means surface level discussion of concepts in scalar fields (functions from R^n to R) and a little bit of talk about differential geometry of curves and surfaces (functions from reals and planes to R^n). The treatment is often not rigorous and n is limited to 2 and 3. After this laughable bullshit, one is thought ready to jump straight into the rigorous analysis of manifolds, granted they know a bit of real analysis. This is like jumping from 3rd grade straight to 9th grade. Along the way the most important thing missing is the rigorous treatment of vectors fields (say, at the level of Rudin). Some nice books here include [4]. Since diff geo (in particular, that of curves and surfaces) is its own thing entirely and there are a lot of really nice books for that like [5]. Also, note there's a nice new intro to "manifolds and stuff" [6] which is like what's calculus is to analysis.

Before I forget, most intro to stats books are written for science majors and are entirely inadequate for math majors, but there are elementary intro to stats books for math majors like [7].

Originally, I wanted to write a more fleshed out and huge comment, but I am running out of time.

Good Luck.

[0] Engineering Mathematics by Stroud/Booth

https://www.amazon.com/Engineering-Mathematics-K-Stroud/dp/0...

https://www.amazon.com/Advanced-Engineering-Mathematics-Kenn...

[1] A Course in Modern Mathematical Physics: Groups, Hilbert Space and Differential Geometry by Peter Szekers

https://www.amazon.com/Course-Modern-Mathematical-Physics-Di...

[2] Book of Proof by Richard Hammack

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

[3] Linear Algebra: Step by Step by Kuldeep Singh

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

[4] An Introduction to Analysis by Wade

https://www.amazon.com/Introduction-Analysis-4th-William-Wad...

[5] Differential Geometry of Curves and Surfaces by Tapp

https://www.amazon.com/Differential-Geometry-Surfaces-Underg...

[6] A Visual Introduction to Differential Forms and Calculus on Manifolds by Fortney

https://www.amazon.com/Visual-Introduction-Differential-Calc...

[7] Statistics for Mathematicians: A Rigorous First Course by Panaretos

https://www.amazon.com/Statistics-Mathematicians-Rigorous-Te...

⬐ qwerty456127

Thanks. So many great recommendations. Not a single link to a flash cards collection that I could use with Anki however.

⬐ yesenadam

It's better making cards yourself - not just better, using someone else's pack seems utterly pointless to me. Why would you want to miss this step?

It reminds me of the situation with books of transcribed jazz solos, which at first seem like gold. But they're actually worse than useless, as you miss out on everything you learn from transcribing solos yourself.

⬐ qwerty456127

I get the point, yet..

> Why would you want to miss this step?

Because I don't really have much time for every step. I don't need it to be fun, I need it to be efficient (requiring as little time and effort as possible). In fact I had always wanted an app of a sort that would feed me math in concise pre-digested twitterish pieces.

Of course I can't expect somebody to do this for me but if somebody has done and shared I'd like to make use of that. I am certainly going to share if I find time to make the cards.

My answer to your question is math. Learn to read and write proofs. Any intro to proofs will do: those employed in discrete math, the ones in analysis, the diagram chasing ones, whatever...Working with math proofs will definitely straighten out your thinking and whip your mind into shape.

Some suggestions to get you started:

Book of Proof by Richard Hammack: https://www.people.vcu.edu/~rhammack/BookOfProof/

Discrete Math by Susanna Epp: https://www.amazon.com/Discrete-Mathematics-Applications-Sus...

Mathematical Proofs: A Transition to Advanced Mathematics by Chartrand et al: https://www.amazon.com/Mathematical-Proofs-Transition-Advanc...

How to Think About Analysis by Lara Alcock: https://www.amazon.com/Think-About-Analysis-Lara-Alcock/dp/0...

Learning to Reason: An Introduction to Logic, Sets, and Relations by Nancy Rodgers: https://www.amazon.com/Learning-Reason-Introduction-Logic-Re...

Mathematics: A Discrete Introduction by Edward Scheinerman: https://www.amazon.com/Mathematics-Discrete-Introduction-Edw...

The Real Analysis Lifesaver: All the Tools You Need to Understand Proofs by Rafi Grinberg: https://www.amazon.com/Real-Analysis-Lifesaver-Understand-Pr...

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

Abstract Algebra: A Student-Friendly Approach by the Dos Reis: https://www.amazon.com/Abstract-Algebra-Student-Friendly-Lau...

That's probably plenty for a start.

⬐ mezod

great, thank you!

⬐ agentultra

A more graduate-level book but one I found pleasing:

A Logical Approach to Discrete Mathematics: https://www.amazon.com/Logical-Approach-Discrete-Monographs-...

And a more pragmatic approach to the same material (with a lot of cross-over in terms of proof-style, etc):

Programming in the 1990s: http://www.springer.com/gp/book/9780387973821

But one I particularly enjoyed early on was written for liberal-arts level students of maths (who might've been traumatized by maths in the past):

Introduction to Graph Theory: https://www.amazon.com/Introduction-Graph-Theory-Dover-Mathe...

It will actually get you into writing proofs in set theory within the first couple of chapters.

⬐ bordercases

Oh gosh the equational logic rabbit hole.

To add to the fire: http://mathmeth.com/