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40,000 HackerNews book recommendations identified using NLP and deep learning

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p_m_conApr 3, 2021

Calculus Made Easy - 1910. It is even entertaining.

"What One Fool Can Do, Another Can.
(Ancient Simian Proverb.)"


gisborneonJune 11, 2020

Spivak’s Calculus is the best-written textbook I’ve ever encountered and one of the more beautiful examples of book design also.

sn9onMar 6, 2017

Calculus is a prereq for taking the course in which this book is used. I'm pretty sure Differential Equations is also a required course for CS majors at MIT, too.

jonnybgoodonSep 3, 2017

I'd rather see a beautiful, minimalist Calculus textbook for that price. However, Spivak's Calculus is hard to beat.

KoshkinonJuly 3, 2020

Calculus by M.Kline would be not a bad start. For a broad (yet detailed) overview, Mathematics by Aleksandrov et al. is exceptional.

CurtMonashonDec 27, 2013

Spivak's Calculus contains a 3-page proof that e is transcendental. I didn't work through it, but it's unsurprisingly based on the famous series expansion of e.

craigchingonJan 10, 2016

I've been refreshing on Calculus and I found that Kline's book was good at application as well as a bit of history, at least I never got the history part at University and I found it very interesting.


karma_fountainonJune 11, 2020

I like Spivak's Calculus but I think it's a lot of effort to learn Calculus from (probably very rewarding though). I've currently been studying from Real Analysis 1 by Terrance Tao and I find the explanations to be great, https://www.springer.com/gp/book/9789811017896.

JimmyLonJune 15, 2009

If it's gonna be someone, he's not a bad candidate for it - his Calculus books are high-quality textbook that age nicely.

profosauronDec 16, 2018

Calculus by Spivak is good. Abbot's real analysis textbook is also quite popular.

disabledonJune 11, 2020

For formal engineering, Calculus, Sixth Edition, by Earl Swokowski, Michael Olinick, and Dennis D. Pence is the ultimate book to use. Also, my twin brother used it to teach himself Calculus 2.

Swokowski wrote phenomenal books, in math, just in general.

eigenvectoronJan 2, 2013

That may well be true in your discipline, but there are some excellent, concise texts in undergraduate mathematics. For example, Apostol's Calculus or Spivak's Calculus on Manifolds. Not the most accessible texts by a longshot, but not your average Pearson or Wiley drivel either.

wyld_oneonFeb 15, 2018

Book reuse is also key. I was able to use my Calculus book for three semesters. And like you say the content does not change that much.

tokenroveonNov 14, 2013

All of these are great. I'd like to add, maybe not for everyone, Calculus by Spivak. For me, it was the calculus book for which I had been looking for a long time.

rsanchez1onJune 8, 2012

Have to agree on Calculus Made Easy. Found it in high school and quickly learned intro calculus better than anyone in the AP Calculus class.

iamcreasyonJuly 17, 2020

Calculus Made Easy is exactly what you are looking for. http://calculusmadeeasy.org/

radicaldreameronApr 19, 2019

Sort of related... but is David Spivak related to the author of that famous Calculus textbook by Michael Spivak...?

sidekonJan 19, 2011

Also, 20 pages of, say, Spivak's Calculus is simply more than , say, 20 pages of Michel de Montaigne essays.

jcofflandonOct 18, 2018

The beginning calculus book for me is Gilbert Strang's Calculus.


Calculus Made Easy seems too dumbed down to me.

pinchyfingersonNov 2, 2010

Strang's Calculus book is available for free. Is the Spivak book a much better resource? I'm not familiar with either one, but I've seen both recommended before.


R3G1RonJuly 4, 2020

There are many books on that front, particularly the ones related to recreational math or intro higher math (see https://mathvault.ca/books for instance). Spivak's Calculus as an intro would be an interesting start, though Stewart's Calculus is dense but more accessible.

mdkessonJuly 22, 2012

Nothing hurts more than regret. If you are serious about this, go for it. to test yourself, grab a Calculus book - I recommend Stewart's Calculus - and challenge yourself to work though it at a reasonable, but predetermined pace (say 1 chapter per week).

krepsjonJuly 19, 2009

I quite liked Stewart's Calculus. http://www.stewartcalculus.com/
And it is really helpful to use some CAS (like Maple, Octave or Maxima) to visualize problems.

apricotonFeb 6, 2017

So it's like Calculus textbooks. 13th edition versus 12th edition: let's add 8 pages of new contents, mostly about the early life of Leibniz, but most importantly, let's scramble all the exercise numbers.

vector_spacesonDec 17, 2019

The listing price of the non-international edition isn't a thousand dollars -- it's about a hundred new, with several $50 listings for a used copy. Which is pricey, but not as egregious as e.g. Stewart's Calculus which runs close to $250 new for the latest edition

guru4consultingonMay 10, 2019

A while ago I saw someone posting a book about Calculus in HN forum that was written decades ago, probably 40s or 50s. I liked it and could not find it back. Does anyone have link to it?

thanatropismonDec 15, 2016

I've only seen naturals start with zero in Calculus 101 books. Any "introduction to real analysis for third-year undergrads" will do 1, at least in my experience.

patroclesonSep 8, 2008

Work through Spivak's Calculus, then Springer Verlag's Undergraduate Texts in Mathematics

SICP is as foundational as Spivak, however Comp Sci still doesn't have the equivalent of UTM, you have to target what you're interested in and go from there....

HiroshiSanonSep 18, 2018

maybe you could compile a list of problems and use anki to regularly train yourself?

There's also Schaums outline for Calculus

You can also try going through Art of Problem Solving Calculus (though it's much more difficult than the typical calculus text)

FranzFerdiNaNonDec 14, 2018

I think he uses Calculus by Soo T. Tan. Although you can probably match his video's to most calculus books as they mostly follow the same order.

acangianoonJuly 26, 2010

It looks like an understanding of Calculus and basic Physics is all that is required to read this book. Ballkpark estimate (how ironic, having to do a ballpark estimation before having read a book on how to do ballpark estimates): between 10 and 30 hours.

octonJune 6, 2010

Apostol's Calculus would probably fit your needs. I do not remember its discussing applications at all, unlike the Salas and Hille book according to online descriptions of it. From what you wrote it does not look like you are concerned with that, though.

InclinedPlaneonNov 2, 2010

Strang's Calculus book is fantastic. I bought it long ago merely to have it (having already learned Calculus).

dhumeonDec 6, 2011

As a disclaimer to all of this, I was the type that had fun proving stuff in seventh grade and had already read through Spivak's Calculus by Grade 10.

This makes your earlier claim of never having seen most of the concepts rather dubious.

b_emeryonNov 2, 2010

Haven't read his calculus book, but Strang's linear algebra book is the best math book I've ever read. It's actually readable! Certain chapters are available on line. Based on it I would definitely try his Calculus book if I needed one.

johndoenutonDec 26, 2016

- Learn some math (starting with Calculus by M. Spivak)

- Read more books

- Basics of music theory

tokenadultonSep 22, 2013

What would be a good book series (preferably a classic one that's stood the test of time) on math (Algebra, various Calculus topics, Statistics, etc)?

I'd like to edit this some more during the edit window for this comment. To start, the books by Israel M. Gelfand, originally written for correspondence study.



An acclaimed calculus book is Calculus by Michael Spivak.


Also very good is the two-volume set by Tom Apostol.



Those are all lovely, interesting books. A good bridge to mathematics beyond those is Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach by Hubbard and Hubbard.


A very good book series on more advanced mathematics is the Princeton University Press series by Elias Stein.


Is this the kind of thing you are looking for? Maybe I can think of some more titles, and especially series, while I am still able to edit this comment.

dominotwonDec 28, 2013

> Spivak's Calculus starts with a set of 13 axioms

I am reading third edition and it only has 12 not 13. Did that change in later editions?

JoeAltmaieronMay 28, 2015

Ok using a Calculus book as an example was not helpful. They are indeed a commodity. There are dozens of new ones printed every year; they contain the same centuries-old truths in different words. E.g. if you found a really good one, would you throw it away and buy a 'new' one every year? Of course not.

scandinaveganonApr 25, 2017

> So that would point to Feynman's calculus book being "Calculus for the Practical Man" by J.E.Thompson rather than Silvanus P Thompson's "Calculus Made Easy", whose second edition came out in 1914. I would not be surprised that Feynman read and used both.

It's just that the quote on fools is very prominent in the beginning of Calculus Made Easy, and the author continues to hilariously refer to both other people and himself as fools. I searched inside Calculus for the Practical Man [1] on archive.org for the word "fool" without a single hit.

I'm sure Feynman read both, but I was interested in the origin of the quote, since I learned it a few days ago and think it's very inspirational. Feynman was constantly arguing that everyone has the capacity to figure things out, it's just that they rarely practice it.

[1] https://archive.org/details/calulusforthepra000526mbp

acangianoonMar 15, 2010

Agreed, Calculus by Spivak is essentially a bridge to Real Analysis.

coreyonNov 14, 2010

I'm working through Calculus by Michael Spivak. If you want a thorough knowledge of calculus(who doesn't?) then I wholeheartedly recommend it. While it's more deep than broad, it will surely help you learn to start thinking more like a mathematician.


stavrianosonApr 20, 2009

Of the math textbooks I've used, two've been good enough to keep so far:

Linear Algebra:


on the more computer-sciency side, Introduction to Algorithms has a lot of good word-of-mouth. I'm not such a fan, myself, but here it is:

springcoilonAug 11, 2009

I second the OCW reference.
Some of the Calculus for Dummies, type books are good.
Its good to remember that Calculus and Linear Algebra don't have to be that complicated.
I also recommend scan the books before you buy them, I wasted far too much money at college on txtbooks that I ended up despising

mindcrimeonJune 18, 2017

Likewise. I started with pre-algebra level stuff and I just login when I have some free time and watch more videos and do more exercises. I figure every time I jump on there and do some of it, I'm improving that foundation.

I've also been running through the series of Youtube videos on Calculus I by Professor Leonard. The plan is to go through his entire sequence (Calc I, II and III) and then move on to Linear Algebra (I've already been dabbling in that as well, mostly with the 3blue1brown videos).

It's not easy, but I think it's worth the effort to build up that math base. It increases the scope of things you can read, study and understand, which is pretty valuable.

nimonianonMay 5, 2020

For my own part, it was all about finding the right book. I enjoyed my degree in pure maths, and after finishing I really missed studying so I tried out dozens of books. Eventually, I hit upon Calculus by Spivak. This book fundamentally changed me and I solved every problem in it fastidiously - it was just so well made, funny in parts, it had cliffhangers, and the problems were just amazing, linking together over chapters like characters in a novel. For the first time, I was solving problems just for fun and I was improving much faster than I did during my degree. I wrote all my solutions in LaTeX so I had an organised body of work to look back on and watch grow, adding to the satisfaction (I lose paper; pdf is better for me).

So my advice: find a great book, make notes and expand on what the author is saying with diagrams or additional lines of working, focus on solving problems, do it out of love.

disabledonNov 13, 2020

This is effectively the best reference notes on the internet for calculus 1, calculus 2, calculus 3, and differential equations. You can definitely become proficient in these subjects on those notes alone, with practice. Paul’s Online Math Notes were my main reference material back in 2008-2009 in university. I am glad to see that it is still very popular.

The absolute best Calculus book for undergraduates majored in the formal engineering disciplines is Calculus, 6th edition, by Swokowski, Olinick, and Pence.

ISBN-10: 0534936245
ISBN-13: 978-0534936242

My (fraternal) twin brother used this book to teach himself calculus 2 in preparation for another course he needed to take. There are lots of supplementary materials that you can get for the book, like solutions manuals, study guides, and even a linear algebra supplement (although the latter is hard to find—I should probably scan all of the materials)

krambsonFeb 10, 2011

To this day I still pull out Apostol's Calculus textbooks. If you're looking to learn calculus really well, or just brush up, these are the ones. http://www.amazon.com/Calculus-Vol-One-Variable-Introduction...

CurtMonashonDec 4, 2014

My favorite first-year calculus text by far is Michael Spivak's Calculus. When I was in school in the 1970s, he was by common consensus THE great and differentiated writer of math books -- but that was based on a small sample size, and Calculus is the only one that should be in this discussion.

There were a lot of other books back then that I think of as likely to have been the best in their time, but those were all more in the vein of texts for classes that I just happened to feel served me well.

eruonSep 29, 2015

Interesting enough, you can also build your whole geometry on the operation of reflection and a bunch of axioms. See Calculus of Reflections at eg http://rd.springer.com/article/10.1007%2Fs00022-012-0123-5 (surprisingly for Springer, not behind a paywall).

There's algebra and calculating, but with reflection operations here, not with coordinate pairs.

nudetayneonJuly 19, 2013

I simply have to disagree. I've only met two teachers I didn't think were dipshits and that's over the course of a B.S. in bioengineering, a masters in CS, and a trip through an MD/PhD program. These are all at top ranked schools, too. These people may have been smart, but they certainly weren't quality instructors. In my opinion, teaching should be something people do when they retire for fun. The two non-dipshits were doing exactly that.

Students should absolutely be able to pick it up on their own, but most classes use crappy textbooks and the teachers recite from the crappy textbooks. Stewart's Calculus is used by a ton of schools and that book is utter garbage. The guy is using it as a way to print money, which is fine, but you'd think after 10 versions, he'd eventually get things right.

eggyonMay 12, 2019

I loved working through Ken Iverson's, the creator of APL, books in J: "Calculus", "Algebra", and the "Concrete Math Companion" (a companion Knuth and Patashnik's book "Concrete Math" in J)[1].

I also like the Physics and Math in SICM (Structure and Interptretation of Classical Mechanics), and Scheme maps well to math equations, at least for me [2].

Julia does follow the way math is written, but I prefer Haskell. The book "The Haskell Road to Logic, Math and Programming" [3] is great, and this article on Geometric Algebra (GA) in Haskell is excellent [4].

[1] https://www.jsoftware.com/books/pdf/

[2] https://mitpress.mit.edu/sites/default/files/titles/content/...

[3] https://www.amazon.com/Haskell-Logic-Programming-Second-Comp...

[4] https://crypto.stanford.edu/~blynn/haskell/ga.html

jeffreyrogersonDec 21, 2020

As I'm sure you know, Spivak[0] (whose lecture notes the book is based on) was one of Milnor's advisees at Princeton. I was told by another Princeton PhD who was there shortly after Spivak (may have overlapped, I'm not quite sure) that whenever Spivak had a problem he thought was interesting he would discuss it with Milnor, who invariably had already thought about it and knew the answer. This experience dissuaded Spivak from an academic career because he felt he could never be as good a mathematician as Milnor. At least that's the story I was told.

[0]: Same Spivak who wrote the famous Calculus book and a series on differential geometry.

kjhughesonApr 21, 2017

Summer of 1980, going into my senior year in high school, I mentioned I'd be taking Calculus next year to a co-worker a couple years older than I. He said he had the best book in the world on Calculus, and he loaned me his copy of Silvanus P Thompson's Calculus made easy. I thoroughly enjoyed that book, benefited from its intuitive explanations, and forever appreciated his recommendation.

If I may similarly influence anyone here, for themselves or someone they know, to read Calculus made easy to supplement their calculus coursework, I will be happy to have paid the favor forward in some small way.

By the way, Kalid Azad may be our modern day Silvanus P Thompson. And he has better tools[1], which he wields masterfully, than just pen and paper. Recommended too.

[1] https://betterexplained.com/

j2kunonApr 12, 2014

Sheldon Axler's "Linear Algebra Done Right" has my highest recommendation if you want expertise in linear algebra.

As a followup, Paolo Aluffi's "Algebra: Chapter Zero" is the best synthesizing text for abstract algebra for a beginning graduate student. The thing that makes it so amazing is the writing style: it introduces and demystifies category theory, and then discusses groups, rings, modules, linear algebra, fields, r-modules, and advanced topics (toward the end) with the unifying theme of how they work and relate as categories. It is very much a book that focuses on the why over the what and how. And there are many many many juicy exercises.

I would recommend against Spivak's Calculus on manifolds. It's too dense and too focused on advanced topics (unless you're an ivy league undergrad, you don't learn cohomology). That being said I don't have an analysis reference that fits your request.

As a CS person I can continue with book recommendations related to computing-related topics in math (computational algebraic geometry comes to mind). Let me know if you're interested.

JachonMay 15, 2018

Schaum's Calculus was invaluable to refresh my memory of some of the details of "Calc 2" so I could be sure of passing a waiver exam (most schools would have waived it automatically on account of my AP credits but my school limited me to how many I could waive that way...) and get on with Calc 3. The book covered some Calc 3 too so continued being useful. I have a few others in the series, very handy.

"The reader who has read the book but cannot do the exercises has learned nothing." -- J.J. Sakurai

(Incidentally, I tried reading Sakurai's Modern Quantum Mechanics on my own once and was immediately curb stomped. Lots of prep work required for that one...)

artagnononOct 26, 2020

Motivation and direction are important when starting out. I decided pretty early on that I wanted to be an algebraist, but would have to build some mathematical maturity before I could get there, so I had a rather shallow goal in the beginning: to be able to solve the previous years' math GRE papers. Off the top of my head, these were some of the books that I worked through:

1. Spivak's Calculus.

2. Johnstone's Notes on Set Theory and Logic.

3. Gamelin's Complex Analysis.

4. Hoffman & Kunz' Linear Algebra.

5. Dummit & Foote's Abstract Algebra; just the group theory.

6. Munkres' Topology; just the general topology.

Once I was happy with my preparation, I strived for a deeper understanding of Group Theory. I bumbled through Herstein, but didn't understand it very well. Then, I stumbled upon Artin's book, and worked through it using the outline provided on the MIT OCW course page, and I could confidently solve most of the exercises.

For category theory, the top resources that I would recommend are:

1. Mileweski's Category Theory for Programmers, the video lecture series. As is always the case with video lectures, this one can help motivate a Haskell programmer uninitiated in category theory.

2. Goldblatt's Topoi. It's fairly dated, but teaches category theory well, via its application to topoi.

3. MacLane's CatWork. I'm not especially fond of this one, but it's necessary to work through it.

The most important thing to understand when learning category theory is that it cannot be learnt in a vacuum: the subject is entirely vacuous, and you need to use it in other mathematical disciplines to give it meaning.

Good luck.

ivan_ahonAug 31, 2016

Calculus books also fall in the same ball park: https://www.amazon.com/gp/bestsellers/books/491544/ref=pd_zg... some of them are almost $300, OMG...

ABeeSeaonApr 3, 2021

They are probably terrible for self-study, but I want to mention them anyway. I am a huge fan of David Bressoud’s Calculus (and analysis!) books. His first one doesn’t have exercises and his second one has a lot of physics.

They are all heavy on written narrative and interlaced with history. I find all four of them absolutely fascinating.






julesonJan 27, 2012

I have seen some of da Vinci's paintings in real life. Among them his famous Mona Lisa. I didn't find them particularly extraordinary among the other paintings in the Louvre, though I already said that he was a good painter.

What are his important discoveries in anatomy, optics and civil engineering? What did da Vinci discover about optics that Euclid hadn't already discovered 1700 years earlier? It seems to me that his contribution in those fields is mainly as an illustrator. Which of his "inventions" actually worked? How do you explain his nonsense engineering like flying machines?

Perhaps the case that da Vinci was an illustrator-wannabe-scientist/engineer is overstating it a bit. I'm not saying he made no contributions whatsoever, just that there are other people far more deserving of the deification that Leonardo da Vinci enjoys, like Newton, Galilei, Gauss, etc.

When people talk about da Vinci's contributions they always talk in vague terms. Now take Newton and we can easily find numerous contributions that are still extremely important even in our age:

* Calculus - the basis of physics, engineering, and a large part of practical mathematics

* Newton's laws of motion

* Newton's law of gravity

* Major contributions to optics

* Newton's method - one of the most important algorithms

The trouble with these is that they require far more effort and study to appreciate compared to a bunch of drawings, especially if you don't try to look critically at the drawings and try to determine whether the machines depicted actually work.

kjander79onJuly 15, 2021

I love introductory textbooks. Some of my favorites:

"Starting Forth" - Leo Brodie

"Computer Science, Logo-Style" series - Brian Harvey

"Algebra", "Functions and Graphs", "Methods of Coordinates", "Trigonometry" - I.M. Gelfand (does anyone know if he completed his Geometry textbook? I've never seen it. I've also never seen his "Calculus" or "Combinatorics" books, although I know they exist. I suspect his "Combinatorics" book is more advanced than he intended to produce.)

"Sage for Undergraduates" - Gregory Bard

khyrykonAug 4, 2012

I used Calculus (4th edition) by Larson, Hostetler, and Edwards for my calc 1 and 2 courses and I would not recommend it for independent study.

Consider Calculus by Briggs and Cochran instead ( http://www.amazon.com/Calculus-Early-Transcendentals-Briggs-... ) for good explanations with example problems and for the superior images, which are particularly helpful for multivariate calculus.

arvidonFeb 4, 2008

You really need to learn Math by doing it there really is no other way. Books are really just references and guides and can give you good problems to work from. Either find a friend who will study with you or get direction from a math professor.

I do not know Spivak's Calculus but his advanced books (by Publish or Perish) are excellent, So I assume his calculus book is also. Especially Calculus on Manifolds and A Comprehensive Introduction to Differential Geometry. Anyone who wants to understand calculus on higher dimensions should read Calculus on Manifolds.

If you want to learn from the masters and you have the confidence, audacity and intelligence. I would suggest Fundamentals of Abstract Analysis by Andrew Gleason and Geometry and the Imagination by David Hilbert.

Just a warning. These books are for people adept at mathematics and are willing to spend hours on a page or two. If you are not, then avoid these books.

septimus111onAug 28, 2017

Not knowing anything about you, I'll assume that

- you are starting with the equivalent of a high school level of maths

- you want to take a ML course or read an ML book without feeling totally lost

As some commenters have said, Calculus, Probability and Linear Algebra will be very helpful.

Some people like to recommend the "best" or "most important" books which you "should" read, but there is a strong chance these will end up sitting on a bookshelf, barely touched.
So I will recommend some books which are perhaps more accessible.

- Calculus by Gilbert Strang

- Linear Algebra by Gilbert Strang

For Probability: I don't have any favourites, sorry.

lkozmaonJune 15, 2009

You must be referring to the MIR series of textbooks. Demidovich's Calculus was another classic.

learc83onJuly 7, 2014

I remember my Calculus I and II text book always seemed to skip an important step.

1. Some Math

2. A therefore obviously B

3. Q.E.D.

It was always the "obviously" part were I was lost.

The proofs in my more math heavy CS classes (especially Automata) were always a lot clearer even though I thought the material was conceptually harder than calculus.

JimmyLonFeb 17, 2012

Sedra/Smith was a classic - I remember I picked mine up used from the campus bookstore, and it'd been cycled through the course so many times it was almost perfectly annotated and highlighted for the way one particular prof taught the material.

Of all the textbooks I used in my schooling, Sedra & Smith is on the short list of ones I remember explicitly - along with Silberschatz & Galvin on operating systems (with the ridiculous dinosaurs on the cover), Oppenheim and Willsky's Signals and Systems, EOPL, and Stewart's Calculus.

romwellonJuly 28, 2018

It is a demanding book -- I never got to really finish reading it cover-to-cover -- but it's very helpful nonetheless even if one reads the parts they find easy.

For example, the first dozen pages on Calculus have imparted a better understanding of the subject than all the indecipherable (if only due to sheer volume) tomes of Stewart. It was an invaluable boost.

johnsonjoonMar 24, 2018

I don’t have any resources for analysis directly to recommend that haven’t already been said, but there’s some good videos of Calculus by 3blue1brown on YouTube called The Essence of Calculus [1]. They are really well made and explain Calculus in a way that you get an intuitive feel for it. It may be helpful to learning analysis to understand Calculus really well, but I’ve never taken analysis so I can’t say for sure.

[1]: https://www.youtube.com/playlist?list=PLZHQObOWTQDMsr9K-rj53...

hackermailmanonApr 17, 2018

If you want to fill gaps, any typical undergrad calculus text will do such as Stewart's Early Transcendentals book. You do enough of those 8,000+ exercises and your highschool gaps will fill themselves. My favorite beginner math books are Thomas VanDrunen's Discrete Mathematics and Functional Programming because it's entirely done in SML, and Apostol's Calculus because you end up doing so many exercises you absolutely will never make a silly algebra mistake in a proof or forget a trig identity ever again. Often Apostol will just defer to endless calculating in the chapter exercises if he doesn't have anything he wants to add to the material, this is really good practice if your basic math education is shit like mine was.

ivan_ahonAug 15, 2012

I second everything said here. All very good advice.

Just one thing:

> introductory texts [...] University Physics by Young, Freedman, and Ford [...] to learn calculus, Calculus by Larson and Edwards [...] should take you a year or two to consume [...]

I disagree. Two years? I think six months should be enough time for the modern youth to learn all of first year physics, calc one and two included.

The only limiting factor is the time it takes to do the exercises, because, like you said, the main part of learning physics happens when you are figuring things out on your own. I bet that one or two semesters with a good book with exercises with answers in the back can be enough to learn how to use all of first year math and physics. If only there were such a book that teaches calculus, mechanics, E&M and linear algebra, all in one sitting. ;)

I will close on a personal recommendation from 1914. Calculus Made Easy by Silvanus P. Thompson

plinkplonkonMay 21, 2009

"Any particular math texts you'd care to recommend?"

I know you asked the question to Mahmud, but here is what I think.

To start with, Undergrad texts (the best I've found) to prepare you for Machine Learning.

Calculus (Strang) available free http://ocw.mit.edu/ans7870/resources/Strang/strangtext.htm

Linear Algebra(Strang) not free.

Probability( by Bertsekas a new edition just came out. expensive but VERY good).

Information Theory (also useful in ML) David McKay. freely available at http://www.inference.phy.cam.ac.uk/mackay/itprnn/book.html

Convex Optimizaton by Boyd. (free http://www.stanford.edu/~boyd/cvxbook/). Videos available on the Stanford site. http://see.stanford.edu/see/courses.aspx

Plenty of books at the graduate level - Spivak's Calculus and Rudin's Introduction to Functional Analysis should get you most of the way through mahmud's list of things to learn

After all these, you should be able to pick and choose for yourself. Enjoy!

robert-wallisonApr 28, 2016

The manga guide to Calculus is a great book. It helps me reason on a topic that is hard for me to grasp. Someone must think SQL is a difficult topic too.

TACOP is extremely dense and put me to sleep every time I tried to read it. I recently donated my copy of 0-4 to my work library, where nobody was interested in reading it.

The fact that we both have opposite views of your example books shows that this is a difficult process for librarians.

hackermailmanonMay 10, 2019

I have Thomas' Calculus from the 1960s, the same book that Knuth promotes in interviews as being responsible for him choosing a math degree. Very high signal to noise ratio without being as terse as Apostol https://www.amazon.com/Calculus-Analytic-Geometry-Supplement...

However I really enjoy algebraic calculus and wish somebody had taught me this in highschool https://www.youtube.com/playlist?list=PLIljB45xT85CSlgGh3681...

psykliconJuly 23, 2018

It sounds like you would most enjoy a Calculus book that introduces you to Real Analysis. So, take a look at Spivak's Calculus, which is rigorous and written for people passionate about math. Another rigorous alternative is Apostol's Calculus.

These books will prepare you for a more concise Real Analysis text such as Rudin's Principles of Mathematical Analysis.

allegoryonAug 10, 2014

Agree with this, as he says pumping another post into the internet.

Last proper break I had was in an 1997. Went to see a friend but didn't make it. Stopped off at an interesting looking charity shop (thrift store for my US cousins) and bought a copy of Calculus for the Practical Man. Got rather into it eating a petrol station sandwich and ended up staying in a Travelodge for a week with myself and a pencil and my old Nokia 5.1 turned off.

When I tell people this they assure me that I'm mentally ill but it was like you meditation focussing on something intently without distraction for for that period of time even though I've never used the knowledge gained.

Now I have a family so I'm good for an hour here and there but sometimes I get up in the night for distraction free time and sleep on the job ;-)

CurtMonashonDec 22, 2013

On a more elementary level, Michael Spivak's Calculus is an amazing book on first-year calculus.

He also wrote Calculus on Manifolds and a 5-volume series on Differential Geometry. Those, to put it mildly, were rougher going.

I actually once got a crush on a dorm-mate in large part because I saw Calculus on Manifolds on her bookshelf. It wasn't reciprocal, however. Later in life, she went on to run the Bureau of Labor Statistics.

ivan_ahonSep 29, 2012

> if you want widespread adoption, you need to target a course.

You may be right about that, but I think there is also value in a single book which covers all these subjects in an integrated manner. Especially Calculus and Physics, which are very co-dependent.

Imagine the use case of someone who wants to learn science on their own. Perhaps he/she is taking an online course which has calculus, physics and linear algebra as prerequisites. Traditionally this would mean he/she has to get three different textbooks (400+ pages each) and slog through all that material. There are excellent free books out there on all of these subjects so money will not be an issue, but going through 1200+ pages will be very time consuming.

This is the gap I want to fill: (1) textbook for self-learners, (2) add-on material for a university class, (3) reference book for adults who want to review the material they learned while at uni.

For audiences (1) and (3) having an 5-in-1 product is definitely a good thing. For audience (2) having all the extra material might make my book appear off-topic and decrease interest. In the next iteration, I am going to think about splitting the narrative to make books for individual subjects to cater more to (2).

stiffonDec 30, 2012

It's your study that gives you new skills, not the books, having said this, the following guided the most valuable learning experiences in my life:

"What is Mathematics?" by Richard Courant taught me to think at a higher level of abstraction. I read it after I realized the parts of SICP (which recommendation here I obviously second) I liked most were the math-related parts and I think it is fair to call it a SICP for mathematics, at least I don't know a book that comes closer. Then I also used Courant's "Differential and Integral Calculus" and Spivaks "Calculus" with his very detailed answer book, and this way I self-taught myself enough material to finally be able to do some reasonably serious math, e.g. proofs.

After reading "Compilers: Principles, Techniques, and Tools" I wrote my own implementation of grep with state machines, then a compiler for a simple language and finally understood what a programming language really is. "Programming Language Pragmatics" was a very useful book here, too, thanks to it teaching me a range of different possible semantics for common concepts in programming languages I was able to learn new languages much more easily and easily spot bugs that I would otherwise spent hours on.

"The Mindful Way through Depression" and the accompanying CD with guided meditations taught me to meditate and meditation forever changed the way I react do difficult situations.

"Starting Strength" taught me correct exercise technique that completely changed the outcomes of my strength training.

WallWextraonDec 4, 2014

I got started on "real" math with Spivak's Calculus. Some people start with Topology by Munkres, which is not a difficult book but is very abstract and rigorous so makes a good introduction. If you feel like you have ok calculus chops, maybe Real Mathematical Analysis by Charles Pugh. Other good books are Linear Algebra Done Right by Axler, or the linear algebra book by Friedberg, Insel, and Spence. Maybe even learn linear algebra first. It's so useful.

Do plenty of exercises in every chapter, and read carefully. Count on about an hour per page (no joke). Plenty of math courses have their problem sets published, so you can google a course which uses your chosen book and just do the exercises they were assigned.

If you don't feel comfortable with basic algebra and other high school math, there's Khan Academy, and some books sold to homeschoolers called Saxon Math.

If you haven't had a course in calculus before, maybe you should skim a more intuitive book before or alongside reading Spivak. I don't know of any firsthand, but I heard Calculus for the Practical Man is good. Scans are freely available online (actually, of all these books) and Feynman famously learned calculus from it when he was 12.

del_operatoronMay 21, 2020

I remember entering high school and picking up this book from the library in 9th grade. It was honestly not my favorite. It being called “Calculus Made Easy” had me feeling frustrated. I went to put it back and there was a book next to it called Calculus and Pizza. Within the introduction I got the intuitive feel for differentials and limits. I was able to play with the difference formula and take a much more enthusiastic high level trip through both differential and integral calculus. A big bonus was Calculus and Pizza allowed me to check my algebra in the back of the book and forward those questions to my algebra teacher. That was a few years before I was able to officially take calculus, so I kept re-reading chapters because they were fun.

A book with an easy approach really needs to make the chapters super easy to restart and spiral through concepts. Stories and characters didn’t hurt to have too though.

Arun2009onJuly 27, 2018

In a previous life, I used to tutor students who had to cover high-school mathematics for their university courses.

What I found was that going through a proper high-school textbook was the best way to cover all the topics systematically and in a focused manner. If you can get your hands on some such books (such as a text for the International Baccalaureate Higher Mathematics or the UK Advanced Levels), that would be the ideal solution.

You can also look at Schaum's series at this level (search them on Amazon). Some useful books are Schaum's Basic Mathematics, Intermediate Algebra, Precalculus, and Calculus. These have the advantage that many problems are solved and the text is completely waffle-free. I myself enjoyed working through Schaum's Calculus whenever I had to brush-up my calculus skills in the university.

Yet another option is to go through the texts by "Art of Problem Solving" (https://artofproblemsolving.com/). From what I have seen so far, these are beautiful texts that stress on improving your problem solving skills along with acquiring technical knowledge. However, I haven't taught from these, so I can't vouch for how the learning experience with them will be like.

EnderMBonJune 15, 2015

I have a degree in Computer Science, but I am terrible with Maths. It's surprising how far someone can go without ever doing calculus if they're able to fit enough answers in their head.

Despite already having my degree, I've felt for a long time that I've wanted to REALLY learn this stuff, at least to a point where I can read through Introduction to Algorithms and "get it".

My base level of knowledge is probably the start of Algebra 1, so I've been going through Khan Academy to build myself up. I'm halfway through Algebra 1 and I've already come across a ton of stuff that I barely ever covered in my GCSE's. I can't vouch for Khan Academy enough. It's been a far better teacher for me than any I've had.

I've given myself around two years to complete the following in my own time:

* Algebra 1 and 2

* Calculus

* A read through of Knuth's Concrete Mathematics

* A read through of Introduction to Algorithms and TAOCP.

I'm part-way through the first one, and I'm hoping that if I stay consistent (an hour of Khan Academy a day, and maybe a bit more on the weekends) I'll be able to work my way through this list.

cabacononApr 12, 2014

I clicked on this topic in order to make this suggestion. To expand on it a little bit, I was a math major and read Spivak's 'Calculus' after I had already taken real analysis. I found it delightful - it really approaches the topics from first principles and unlike many calculus textbooks actually goes through the effort of presenting proofs of the theorems. Highly recommended.

As some recreational reading, less suiting the original request, I very much enjoyed David Foster Wallace's 'Everything and More: A Compact History of Infinity' (http://www.amazon.com/Everything-More-Compact-History-Infini...). DFW is not for everyone, but I enjoyed it a lot. Maybe just check it out of the library first to see if it's for you.

eshvkonJune 21, 2012

Since you have done math previously, the best way to ramp up rapidly is not through fluff material geared towards beginners and to focus on stuff that will actually force you to learn. You might have noticed that the only way to acquire mathematical intuition is to solve lots and lots of problems.

One possible path to follow:

1. Start off with Sheldon Axler's Linear Algebra done right. This is a more theoretical book (than Strang) but should help keep you challenged and motivated. Work through most of the problems. The best way to attack the proofs is to do them yourself.

2. Feller is the best probability book barring none. This is the kind of stuff that Persi Diaconis went through. Solve as many problems as possible but remember that trying to finish it all will take you years.

3. An excellent introductory stats book that doesn't assume you are an immature child is Freedman's book on statistics. This focuses less on the math and more on what statistics really means. Techniques in stats are fairly trivial but using them right is hard.

4. Calculus is useful stuff. As you go through your probability education, you will eventually hit the world of continuous probability which requires a good amount of calculus to go through. Spivak is an awesome book which should prepare you for that.

5. Learn some real analysis. Real analysis from the machine learning perspective is useful because a lot of measure theoretic arguments in research papers have underpinnings here.

btillyonFeb 19, 2019

The original version of Calculus by Newton used "fluxions". That doesn't correspond to any number system that we use today.

Leibniz's (re?)invention of Calculus used infinitesmals. Infinitesmals as understood by mathematicians then do not correspond to any numbers we use today. (Yes, yes, something else called infinitesmals do show up in nonstandard analysis and the notation deliberately looks the same. But the underlying concepts are more..complicated.)

skywal_lonMay 3, 2019

Lagrange Optimizer are in his LinAlg Book? Isn't it in his Calculus book?

vector_spacesonFeb 24, 2020

I like books by Ron Larson, particularly his Trigonometry and Applied Calculus books -- the applied calculus title (intended for social science and business majors) vs his "Calculus" book (intended for math, physics, and engineering majors) is much gentler for people seeing this material for the first time. Although I do highly recommend his Calculus book once you have the other book down.

Gelfand also has some nice texts on Algebra, Trig, and Geometry that are reasonably cheap, especially if used.

I'm older and went back to school later in life to study math, and these are the books I learned that material (for the first time -- I flunked math thru high school) from.

Here are the exact titles and ISBN-10s:

Ron Larson, Calculus: An Applied Approach, ISBN: 0618218696

Ron Larson, Trigonometry, ISBN: 1133954332

Israel Gelfand, Trigonometry, ISBN: 0817639144

Israel Gelfand, Geometry, ISBN:1071602977

Israel Gelfand, Algebra, ISBN: 0817636773

And as others have mentioned, Khan Academy is pretty good, although I tend to prefer patrickJMT's explanations a bit more: http://patrickjmt.com/

hyperpallium2onNov 14, 2020

So (assuming a similar mechanism in humans) when you are taught, it primes memory formation, but it's only when you are tested that the memory forms.

This is, "tests", far from being merely evaluative or diagnostic, are educative.

This effect is well-known, but I feel it's easier to act on with a physical model.

If you work hard to learn something but don't then test (or use/recall) it, then all that priming was wasted.

The mechanism sounds like it's struggling to form memories, but it may also play a role in only remembering what's needed (not filling your head with useless learning, hence the need for the contrived "use" of tests) It could even be a separately evolved filter.

(pre-emptive qualification: there's some learning before the test else you couldn't recall it.)

PS I've just finished Spivak's Calculus chapter 1 problems (about one a day), and he introduces more new material there than in the body. There is tremendous reuse of previous problems (at times to a ridiculous extent, see 6(d) in the answer book) - I noticed that this consolidated my memory of them, but the physical model additionally tells me I'd best "test" myself on all the problems - especially the later ones which haven't yet been reused.

venomnertonJuly 15, 2016

I am planning on using "How to Prove it" by Daniel J. Velleman to prepare for Tom Apostol's Calculus Vol 1. How does "Book of Proof" compare to "How to Prove it" .

impendiaonDec 28, 2013

> Spivak's Calculus is used as a first-year calculus textbook at lots of schools

Umm... where? Not at Stanford, where we used a mainstream, much easier book. So does Princeton. Harvard is famous for having developed a "touchy-feely" calculus book.

Perhaps abroad? It is typical of calculus courses in the US that the students come with fairly weak backgrounds, and a major purpose is to expose and patch holes in the students' backgrounds in algebra and trigonometry.

susamonSep 21, 2016

At my previous job ( https://www.rsa.com/ ), where I worked for 7 years, I've often had to use various mathematical techniques to do my work as a software engineer.

Here are some concrete examples from my career.

* Combinatorics and probability theory - Entropy analysis of authentication schemes, algorithms for enumeration and analysis of related combinations and permutations, etc.

* Calculus and probability theory - Implementation of bloom filter for space-efficient indexing, tests and analysis of false-positive rates, etc.

* Statistics - Adaptive authentication schemes, performance measurements and predictions, network event correlation, etc.

* Discrete mathematics and algorithms - Tree traversals, graph search algorithms, etc. were useful in a variety of situations, e.g. data retrieval in tree-based distributed databases.

* Asymptotic analysis, Big O notation, etc. - Data deduplication in distributed databases.

* Modular arithmetic - I didn't really implement any cryptography algorithms; I only used them. But the understanding of modular arithmetic was essential to understanding some of the limitations of cryptographic algorithms based on modular arithmetic, e.g. why the message size cannot exceed the size of the modulus, as well as for software engineering tasks like decoding certificates found in network traffic.

* Formal language theory, DFAs, etc. - Parser generators for network event parsing, SQL engine development and related query optimizations, etc.

So these are 7 examples from 7 years. There may be more examples but I cannot remember them right now.

I have felt from my personal experience that although most of the work does not involve mathematics, once in a while an interesting problem comes up that can be solved efficiently with the knowledge of mathematics. It is not easy to predict when such a problem would come and it is not always easy to recognize whether the problem at hand can be solved or analyzed efficiently by known mathematical techniques. So it is good to have someone in the team or be the person who has a good breadth of knowledge in mathematics, so that when such problems do come up, they are addressed efficiently. Fortunately for me, I love mathematics and there were people around me who also loved mathematics, so we got a lot of interesting work done.

thomanqonApr 4, 2018

Links as per the original comment [0]

Calculus Revisited: Single Variable Calculus | MIT https://ocw.mit.edu/resources/res-18-006-calculus-revisited-...

Calculus Revisited: Multivariable Calculus | MIT https://ocw.mit.edu/resources/res-18-007-calculus-revisited-...

Complex Variables, Differential Equations, and Linear Algebra | MIT https://ocw.mit.edu/resources/res-18-008-calculus-revisited-...

Linear Algebra | MIT - https://www.youtube.com/watch?v=ZK3O402wf1c&list=PLE7DDD9101...

Introduction to Linear Dynamical Systems |Stanford https://see.stanford.edu/Course/EE263

Probability | Harvard https://www.youtube.com/playlist?list=PL2SOU6wwxB0uwwH80KTQ6...

Intermediate Statistics | CMU https://www.youtube.com/playlist?list=PLcW8xNfZoh7eI7KSWneVW...

Convex Optimization I | Stanford https://see.stanford.edu/Course/EE364A

Math Background for ML | CMU https://www.youtube.com/playlist?list=PL7y-1rk2cCsA339crwXMW...

[0] https://news.ycombinator.com/item?id=14581962

aportnoyonOct 24, 2019

Calculus by Michael Spivak

tzsonApr 5, 2012

> Everyone has a favorite teacher, but no one has a favorite textbook

I disagree. I bet if you went over to Reddit and asked on /r/math or /r/physics, you'd find plenty of people who have a favorite textbook. For example, I'd say Apostol's "Calculus", volume 1, is a favorite textbook of mine. I've read it 3 or 4 times over the last 30 years. The Feynman Lectures on Physics are another favorite of mine.

> 4x more expensive/less accessible/same form factor

(that's from their infographic, comparing textbooks in the '60s to now. It was on three lines there, which I've marked with slashes to fit the quote on one line)

Based on inflation, they should be about 6x more expensive, so if that 4x figure is right textbook prices have improved since the '60s. However, I suspect that they are a little low in their estimate here. I think prices have gone up faster than inflation.

I don't see how text books have become less accessible since the '60s.

> There’s one other major concern: textbooks are just flat-out terrible products. They’re ineffective pedagogical tools: dense collections of long-form text that fail to engage students’ wide range of learning styles

This may be true for the less technical fields, but I have yet to see anything better for, say, a rigorous upper level math course.

mdkessonOct 7, 2013

Man, don't let people discourage you.

I guess the thing that you should realize though is that this isn't a degree in computer programming, or IT - it's a more or less a mathematics degree. As the saying goes, "Computer Science is no more about computers than astronomy is about telescopes."

If you want to get ready for serious computer science, I'd recommend a few things which are what I think I got out of my undergraduate degree:

1. A solid understanding of algorithms and data-structures. To this end, topcoder.com/tc is invaluable and some serious study will quickly bring you up to speed. CLRS (Introduction to Algorithms) is a great resource, as is train.usaco.org.

2. A basic understanding of theoretical computer science. To that end, I found this a really useful book: http://www.amazon.com/dp/0321455363

3. Basic understanding of networking and operating systems. Not sure the best route here, there must be online courses. Not too many great self-study books in this area, unfortunately. So find some online courses.

4. A decent math background: linear algebra, calculus, combinatorics, and probability. For self study:

    Calculus: Stewart's Calculus is great.

Linear Algebra: I've yet to meet a linear algebra text I liked, so not sure here.

Probability: A First Course in Probability is an outstanding textbook.

Other: Concrete Mathematics by Knuth is an incredible book, very VERY hard and took me a long time to get through, but packed with useful and interesting information. I'd recommend it after the rest of these.

5. Read Snow Crash and watch Hackers.

Also, keep writing lots of code. Daily practice is the secret to everything.

RossBencinaonMar 11, 2018

To the above I would add:

How to Think Like a Mathematician - Kevin Houston (an excellent book to read before starting)

How to Read and Do Proofs - Solow

The Keys to Advanced Mathematics: Recurrent Themes in Abstract Reasoning - Solow

Calculus - Spivak (Actually a Real Analysis book, not a Calculus book, see e.g. https://math.stackexchange.com/questions/1811325/spivaks-cal... )

Linear Algebra Done Right - Axler (Intended for a second course in Linear Algebra, but I found it helpful during my first course.)

And for something from left-field:

Visual Group Theory - Carter http://web.bentley.edu/empl/c/ncarter/vgt/

There are many many many books on every mathematics topic under the sun. Finding books that speak to you is important. I have had mixed success buying books upon other people's recommendation. You would be best to get access to a library.

te_plattonDec 21, 2007

My experience in school indicates that the quality of the teacher makes quality of the lesson. Also different teachers with different styles have different effects on different students. What I would like to see is series of video lessons by various teachers on a given topic in a neatly organized environment. For example, I would love a site that had links for History, Math , Science, etc. Under Math there would be links for Algebra, Geometry, Calculus, etc. Under Calculus there would be links like
Differential Calculus in 38 lessons by Dr. Math Guy.
Calculus in 3 lessons by Dr. Talks To Fast
Calculus in 7362 lessons for slow people

Anyone know of such a place?

laichzeit0onOct 7, 2016

Without getting bewildered I would suggested going through these 3:

1. Book of Proof by Hammack (http://www.people.vcu.edu/~rhammack/BookOfProof/)

2. Calculus by Spivak

3. Linear Algebra Done Right by Axler

Be prepared to work through all (or at the very least only the odd numbered) exercises. If you can't stomach that or find that life gets in the way of you completing even these very basic books, you do not have the time or discipline required to advance in mathematics.

gmsonFeb 3, 2008

The book 'Calculus' by Michael Spivak.

In the same way that SICP transforms you from a high-schooler into a wise adult when it comes to programming, so too does Calculus when it comes to maths. If you find the book to be heavy going, then read whatever preliminary material you need, and go back to it.

Edit: I should also stress that maths requires a fair amount of discipline (a lot more than programming), so it's really hard to study maths while also having a day job.

dsaccoonMar 17, 2018

> Remembering 3 pages of content per day (especially in math where concepts build on each other) is really not hard.

I disagree. Or rather, I think that's unsustainable. Any given three consecutive pages from Spivak's Calculus are probably doable on a daily basis. But is would be legitimately hard for most people to go through three pages of Rudin's Principles of Mathematical Analysis each day and consistently retain that information. Axler's Linear Algebra Done Right is very readable, but Halmos' Finite-Dimensional Vector Spaces will start getting just as dense as Rudin. These are difficult textbooks even when students are well-prepared for them with prerequisite courses. Terence Tao wrote two books to cover (with better exposition) what Rudin did in one. I think it would be pretty hard to read consistently three pages of Tao's Analysis I each day, before he even gets to limits.

I think you're underestimating the intellectual effort here. In my opinion, even if you're reading a math book targeted to your level, committing to reading and understanding three days of material each day would become exhausting. A typical semester is 15-16 weeks, with lectures 1 - 3 times a week, and most undergraduate courses do not actually work through the entirety of a 300 page textbook. Even at that slower pace it's not typical for most people to ace the course. If you read three pages a day and had a solid understanding of it, you'd be absolutely breezing through math courses.

In my experience students need to really step away from the material and let it percolate for a bit every so often in order to solidify their understanding. I really don't think you can partition the material into equal, bite-sized amounts each day. The learning progression doesn't tend to be that consistent or predictable.

nicklovescodeonNov 13, 2011

Calculus - Michael Spivak

Salt: A World History - Mark Kurlansky

jordighonAug 29, 2016

Yeah, it's an old chestnut:


Spivak has a version of it in his Calculus book, phrased as 17 (heh!) professors who must resign if a flaw is found in their published work (hehehe), and all have a flaw in their papers, known to each other except each author.

maxkwallaceonOct 1, 2018

For undergraduate mathematics (semi-opinionated, not exhaustive):

Calculus: Apostol and Spivak, take your pick

Linear Algebra: Valenza

Abstract Algebra: Artin

Multivariate Calculus:

  - Vector Calculus, Linear Algebra, and Differential Forms by the Hubbards

- Calculus on Manifolds by Spivak

For other fields and classes I am not recommending a book because either (1) I don't think there is a clear winner, e.g. as in the case of Real Analysis, or (2) I'm not familiar enough with the books in that field.

If you have other recommendations, please add them! These books changed my life in the best of ways.

eggyonSep 28, 2015

I don't think it is bikeshedding IMO from a pure mathematics point of view. In set theory you begin with the empty set as zero, and the next integer is N+1. I find it interesting that the Julia devs have gone with APLs style of slicing, and did not move to 0-based indexing. Ken Iverson, who created APL and later the J programming language, chose 0-based indexing for J. APL has a system variable QUAD-IO that allows you to choose your indexing value of 0 or 1, which leads to many issues. Iverson's books on math (Arithmetic, Calculus, and the Concrete Math Companion (Knuth's book done in J)) all to demonstrate mathematics in J, all begin at Chapter 0. I think the choice of index in APL is a bad idea. I have run other people's code, and realized it doesn't work because they set their indexing to 1 vs. 0 for me.
People and programmers can work around most things, but I think if you are a functional programmer, and mathematics is your thing, you are more comfortable with 0-based indexing. This of course does not address scientists, for whom pure mathematics is not at the forefront of whatever they are working on. I do think it leads to problems down the road for 1-based indexing.

albertsunonMay 10, 2010

After arithmetic, the typical order of classes (The ones I took, and the one I believe most often taught in American schools) goes like this:

More Advanced Algebra

I don't have any book recommendations and I don't know of any books that would cover all those topics. There are many books on algebra however, and I don't think it will particularly matter which one you use. It should get you up to speed very quickly.

The other topics are probably less important for intro CS. I don't think I ever used calculus or trig for any CS classes. Very basic geometry might be used as examples.

usgrouponMar 10, 2019

I’d call what you want to do “teach yourself maths as a foreign language”. It can very much be done because I did it.

If I could do it all again, I’d start with an old school Calculus course. E.g “Calculus” by Binmore . It’s a decent and well explained introduction to Calculus and Linear Algebra and it’s useful maths. You have to get a hang of thinking in maths and doing maths you find boring.

From here it depends on what you want to do. There’s more maths out there than there is time so you need goals.

XichekolasonJan 13, 2010

I'm not sure how you could squeeze the education I got into 10 months, but I think, properly motivated and with good books, you could self-study an equivalent to a solid BS in 18-24 months.

Topics that come to mind (that my CS bachelors had):

Boolean algebra and gate level stuff (one semester)

Calculus and Linear Algebra (this totaled 4 semesters)

Discrete Math (had one semester on this)

Fairly deep knowledge of at least one traditional language and at least one functional language (as part of other classes)

Basic computing theory (DFA/NFA/Regex/Push-down Automata/Grammars/Turing Machines)

Data structures (the more the merrier)

Algorithms and algorithm analysis (had three semesters on this)

AI (lots of interesting stuff is happening here, and AI is totally not what you think it is)

Architecture and Assembly (had two semesters in this area)

Operating Systems and Assembly (two semesters)

Compilers and other practice at large engineering tasks with programming languages

If you search over at searchyc.com, you'll find the topic of good CS books has come up here over and over. You'll find lots of good reads in those threads.

asharkonNov 20, 2016

For most technical, scientific, and mathematic topics, it's usually very easy to find the next best book to read with a google search or two. Starting on or strengthening long-forgotten Calculus? Probably get Spivak. Et c. This might get tougher as you approach a field's state-of-the-art, but by then "what to read next" will often come from a journal, not a book.

Literature and verse? Harold Bloom's western canon list, whatever its faults, is pretty damn good and could keep one busy for a lifetime. If you want to mix in more works by e.g. women or more asian works or whatever, there are very good lists for that, too:


(above includes Bloom's list)

Want to learn science and math from classics? The list from How to Read a Book should help, and St. John's College's reading list is public: https://www.sjc.edu

For a given topic, there's often a subreddit with a decent reading list in the FAQ.

Which topic(s) are you having trouble with?

angryprofessoronOct 28, 2007

For 4, try Calculus on manifolds, then Differential Geometry (both by Michael Spivak).

tzsonSep 5, 2019

> How different is my Calculus 1 textbook Ed. 17 from this year's Ed. 18?

There's almost certainly no significant mathematical difference as far as the calculus it teaches goes.

If the target audience for the book includes students who aren't really interested in calculus, such as students who are just taking it to fulfill a requirement, and don't expect to use calculus much after they finish the class, then I'd expect the new edition to update exercises and examples to try to make them interesting and relevant to today's youth.

If the target audience is students who actually want to learn calculus, either because it is interesting to them per se, or because they know it is useful for things that do interest them, then there is probably little or no need for frequent new editions.

For example, a few top schools use Apostol's two volume text, "Calculus", either as their main calculus text, or for the more advanced track if they have multiple calculus tracks. Apostol volume 1 is currently all the way up to 2nd edition, which came out in 1967. Volume 2 is also on its 2nd edition, which came out in 1969.

Another example is Spivak's "Calculus", also used at several top schools, which is on its 4th edition, which came out in 2008. According to the preface, "Although small changes have been made to some material, especially in Chapter 5 and 20, this edition differs mainly in the introduction of additional problems, a complete update of the Suggested Reading, and the correction of numerous errors". The preface to the 3rd edition says that the biggest change was the addition of a chapter on planetary motion. It also rearranged quite a bit of material, and added problems. It looks like 2nd edition was a pretty substantial upgrade over 1st edition.

Spivak 1st edition was 1967, 2nd edition 1980, 3rd 1994, and 4th 2008.

wfnonAug 27, 2013

If you're motivated / curious enough / have enough time, I recommend Spivak's Calculus as a way of learning calculus the, as you say, 'proper (whatever that means) way.' It's basically a treatment of calculus from the perspective of real analysis, having the 'no ad-hoc rule teaching' policy at its foundation. You prove things, build up a non-fragmented edifice slowly, and end up being introduced to analysis such that you can then pick up other things (it's a rather extensive treatment of calculus indeed.) Or so goes my narrative in the midst of frustrations regarding self-motivating to continue individually progressing through the book. :)

The 3rd and maybe the 4th editions can be found online by doing an internet search for pdf/djvu files.

jcnnghmonAug 13, 2012

I agree with what you are saying. I've been contemplating the reasons why I don't understand math the way I understand programming for some time, and opened up my old Calculus book a few days ago and started rereading it from the beginning. I intend to work through the entire book over the next couple of months. I believe because of the way math is taught, I never developed a truly intuitive understanding of what was going on after algebra. When I taught myself how to program, I learned then immediately applied what I was learning, and continued applying it continuously in all the software I wrote after. Algebra and arithmetic are pretty analogous in that regard.

For me, it is necessary to develop an intuitive understanding of something before I can really appreciate it, and more importantly, manipulate and apply it to arbitrary situations. The way math is taught, intuition is never really delivered. In programming, it's possible to look at an algorithm and have trouble understanding what it does. However, I have never implemented and debugged an algorithm or data structure and not developed a thorough understanding of it in the process. In Calculus, the rules were given, but there was never any effort spent to foster an understanding of why they are the way they are, or the bigger picture. To me, this would be like learning merge sort by running through the steps, but never actually implementing the algorithm as a whole to truly understand what is going on. In going back through it, I intend to relearn it the way I learned programming, so I can truly apply and reason around such a powerful tool.

jggonMay 2, 2010

(Aside: the possible exception to this is Calculus and other advanced math classes, unless you are one of those people who absorb that with a minimum of effort; you either understand Calculus (et. al.), or you don’t, you can’t really just push your way through those one. YMMV.)

First day of my Calculus class: "This is a proof-based Calculus class! This means you must memorize these proofs for the test...".

Actually understanding Calculus is hard work. If you don't believe me, pick up Michael Spivak's Calculus book sometime. Regurgitating the exact formula for integration by parts or the difference quotient will get you nowhere in solving real mathematics problems, which assume that knowledge as a base-step and require you to apply critical thinking and creativity to actually get somewhere. But universities are run like a business: the more money, the better. Students who become frustrated and quit because they're too lazy and/or stupid to actually think do not give them money, so it's easier just to make the tests easy for idiots who spend all their time memorizing things without analyzing their meaning or worth.


sepaonJune 19, 2009

Calculus by Tom Apostol

RBerenguelonJuly 7, 2014

Calculus is incredibly complicated. Keep in mind it wasn't properly formalised until the late 19th century. It depends a lot on the book and the student, but essentially, to read Calculus I+2\varepsilon you need a solid foundation in Calculus I+\varepsilon. So, usually most of the skipping is because the text assumes the prior is known and digested (or it is just a bad book.)

jfaucettonMar 3, 2018

I've done exactly what you are talking about, so I think I have some insight that might be helpful.

For me, the biggest aid was finding good books, ones with exercises and that explained the material very well. Then it was just a matter of reading, actually doing all the exercises, and struggling with the material until I could fully understand it; then I moved on to the next chapter/book/etc.

I've worked through a lot of books by now but here's the short list of ones that I think are great for getting started(especially, if you do what I described above), also when these have solutions manuals I would advise getting them as well so you can ensure you understand how every problem works.

1. Calculus 4th ed. by Smith and Minton

2. Introduction to Linear Algebra (Gilbert Strang)

3. Introduction to Probability Theory (Hoel, Port, Stone)

4. Discrete Mathematics and its applications (Rosen)

5. Introduction to Automata Theory, Languages, and Computation (Hopcroft)

6. Introduction to Algorithms (Cormen, Leiserson, Rivest, Stein)

It will take you at least a year if not two to work through all these in your spare time, but the advantage is that after that you'll have the skills to be able to approach just about any topic in computer science (even highly theoretical ones) and not have much difficulty understanding them (at least that was my experience).

coliveiraonAug 8, 2015

Everything can be considered fluff if you're familiar with the topic already. For example, I respect Richard Dawkins and agree with his position, but if you read two or three of his books, everything else is just "fluff". Similarly, if you already know Calculus, every other book on Calculus will be just "fluff". So, this basically means that you need to be reading books on topics that you don't know. The main difference I would make here is not in terms of having fluff or not, but if the book is well written.

throwaway2245onNov 13, 2020

I have tutored people taking Calc 1, 2 and 3 at several universities in the US.

There's no exact standard that I'm aware of, but they tend to closely align at different universities. Dividing James Stewart's Calculus into three parts is a reasonable guide: you can see the contents on Amazon preview.

I'd expect solving first order linear differential equations would be solidly in Calc 2.

* https://www.amazon.co.uk/Calculus-Transcendentals-Internatio...

eggyonAug 10, 2015

That's why so many people point to Spivak's Calculus book, since it has been hinted it should be called an analysis text. Strang's Calculus, as do others, start with real-world examples. Personally, I had to do the Strang route, and then come back to Spivak, since I had to admit to myself, I don't have that ability to think 100% abstractly; I need motivation from the real world.

whacked_newonFeb 3, 2008

Have you ever seen one of those REA problem solvers books? They are thick books full of problems and solutions (and explanations) and are organized from beginner to advanced.

Do you think this would lead to a more solid foundation (from less frustration), for self studying, than reading from a thorough but dense text? I don't know Spivak's Calculus, but some reviewers on Amazon compare it to Apostol, which I found so abstract, and so unpractical, that I promptly forgot everything. It is now on my to-read list, but like you said, I won't be starting until I can dedicate myself to studying it, and now that I have seen the REA book, I wonder if it would be better to work on that book, as a refresher and foundation builder.

Oh yeah, dwaters, if you happen to be interested in Apostol, and want a study buddy, I nominate me.

erikonJune 14, 2009

I remember Stewart's Calculus as a reasonably good textbook. Though they seemed to publish a new edition every two years. Chapter and problem set numbers would get shuffled around, and so you needed the newest edition in order to do assigned homework.

How much has undergraduate level calculus really changed in the last 200 years? The constant updates disrupted the used textbook market and drove sales of new books.

dannygarciaonJune 19, 2020

Here are a few notation resources I've found helpful when teaching myself computer science:

- Mathematics for Computer Science: https://courses.csail.mit.edu/6.042/spring17/mcs.pdf

- Calculus Made Easy: http://calculusmadeeasy.org

Not directly related to your question but useful for interviews and programming puzzles nonetheless:

- Algorithms and Data Structures, The Basic Toolbox: https://people.mpi-inf.mpg.de/~mehlhorn/ftp/Mehlhorn-Sanders...

- Basic Proof Techniques: https://www.cse.wustl.edu/~cytron/547Pages/f14/IntroToProofs...

gerbillyonJuly 22, 2016

I'm always in awe of what the ancient Greeks managed to figure out with nearly no mathematical notation at all.

Or Calculus in Newton's book. The techniques were sound, but you'd have to be a superman to work the way he did.

Mathematical notation allows for better "chunking" and reduces cognitive load.

[1] https://en.wikipedia.org/wiki/Chunking_(psychology)

laichzeit0onJune 10, 2016

Step 1: Read Lockhart's Lament: https://www.maa.org/external_archive/devlin/LockhartsLament....

Step 2: Download the Book of Proof: http://www.people.vcu.edu/~rhammack/BookOfProof/ You read through it and do all the odd numbered exercises (the solutions are at the end of the book).

Step 3: Get a book called Real Mathematical Analysis by Charles Pugh and you work through that and attempt as many problems as you can, with a view not to rush through it, but to expand your mind through each problem.

Step 4: Pick any of these books that interest you the most and do the same:

- Calculus by Spivak

- Algebra: Chapter 0 by Paolo Aluffi

- Linear Algebra Done Right by Axler

By then you should have enough mathematical maturity to know what to do next.

feklaronAug 2, 2016

If I remember correctly this method was advised in the comments of an HN article about Concrete Mathematics by Knuth et all. I used this for undergraduate texts, mainly in math that I had collected over the years and never got around to reading, like Calculus (Apostol), Algebra: Chapter 0 (Aluffi), Computer Systems: A Programmer's Perspective (Bryant/O'Halloran), a 1000+ page book I flew through as I found I knew some of the material already and used the exercises to test what I had forgotten so I could re-read that material. I also discovered by doing this, subjects I thought I knew and would've skipped, I didn't actually remember correctly and other subjects I thought I didn't know, I could solve without having to read the chapter as previous exercises were helpful enough to understand the problem or I had picked up the information somewhere else.

Aluffi's book also helpfully tells you which exercises require solving previous exercises.

pixelperfectonDec 4, 2014

Thanks so much for your insightful post.

I was working through the first couple of chapters in Spivak's Calculus recently, and was struck by 1) what a great book it was, and 2) what a time commitment it would take to complete it properly! If I could choose a book to take to a tropical island for a year, Spivak might be it. But is it worth spending hundreds of hours working through Spivak and Pugh from the standpoint of developing a professional skill? For someone like OP already out of college and wanting to learn to think mathematically to apply it to programming/electronics, is working through these books as a basis to pursue further mathematical studies overkill? Or worth it?

katovatzschynonNov 2, 2010

   Anybody have insight into how to actualize these 
nuggets into some semblance of a self-learning course?

Buy Calculus by Micheal Spivak. Solve at least one problem every day. Make it ritual and a daily requirement. Watch MIT lectures for corresponding chapter you are on.

To learn this, don't trouble over the path and reason at present. Buy the book and start. Right now.


Buy it. To learn this- buy it and start. Right now.

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