# Books for Learning Mathematics

Hi everyone I recently did a video about books for learning physics and as a follow-up to that I guess I wanted to do a video about books that I’d recommend for learning mathematics. Now there are so many math books out there, textbooks and otherwise, and so the ones that I’m going to recommend to you are just the ones that I have encountered and that I’ve read and that I didn’t think were terrible but there are going to be so many others out there that might be just as good or even better. It’s really hard to have a grasp of all math literature that’s out there, but it’s sort of my opinion that, especially if you’re just looking to learn mathematics, then it doesn’t really matter too much which introductory book you pick up, or, on any of these topics I’m going to mention, pretty much the best book for you to get is going to be the one that’s available at your local library, or that you can get access to. Don’t go too far out of your way to try and get a particular book because I think concepts like calculus or linear algebra, you know, they’re so broad that any book that claims to cover them will probably do a decent job Now as a place to start I want to recommend a reading list put out by Cambridge University – and it’s their mathematical reading list. This is a document that’s intended for students wanting to go to Cambridge and study mathematics, and it’s, I think, a really good list that contains books about maths and it really contains everything. It includes books about how to study maths well; books about the history of mathematics to give you a lot of context for the idea as you’re going to learn; there are some books about theoretical physics and the maths behind that; and there are what they class as readable textbooks. So, all the books on this list – which I’m going to link first in the description – I think are really good for someone who’s trying to learn math and wants a well-rounded idea of what math is and how you study it and how you get into the mind set of learning math. So I think that’s really good, check that out. And I do have a list here of some of my more specific recommendations And yeah, I’ll start off again with some general or ‘just for fun’ math themed books which are for… in case you’re looking for maybe a few books to get inspired about the idea of maths or you just want some fun reading. So… One of my fun reading math books would be Fermat’s Last Theorem by Simon Singh and I think that Simon Singh is a really good science communicator He also wrote a book about the mathematics behind some Simpsons episodes and he’s got some astronomy and physics books too. It’s a really good book into a little bit of a history of mathematics, but also some of the fun of math, and about proving things and about numbers themselves Also, I’m going to mention a book called Flatland by *Edwin Abbott and this is like a real old classic book and it’s not really about maths apart from the fact that it’s set in a mathematical universe on a ‘flatland’ which is like a 2D landscape where the main character is a square and it’s talking about mathematical ideas like lines and polygons and shapes, and those are all the characters, but it’s actually I guess commenting on politics and social constructs all through the lens of like a very nerdy sort of mathematical view. I thought that’s just… I’ll put it on there as a fun book. Another one that I’ll mention in this category would be ‘A Mathematicians Apology’ by G.H. Hardy and this isn’t really a book as much of it as an essay and It’s just something that I encountered multiple times during my math studies because a lecturer would recommend you read it or something, and it’s like an essay from this mathematician talking about what it’s like to be a mathematician – the mindset of being creative in that context – and I guess how to be in the mindset. It’s like his reflections on his career as a mathematician. I think it’s just maybe a useful thing if you’re hoping to head in that direction. Okay, so moving on to my list here, starting with calculus. So again, like I said I think pretty much any calculus textbook is going to be fine if you’re just wanting to learn calculus. The specific textbook I use during university like for second year studies was ‘Early Transcendentals’ by James Stewart, and this was just like the standard textbook for us at my university, and I thought it was fine. It covered like all the concepts that I really needed to know from my course. It was bearable to understand. So I guess I’d recommend it. I actually remember buying my copy of this book from one of those like textbook exchange sites and I met up with this random stranger at the stairs of my university library. He pulled out the textbook from his bag I pulled out some cash and we exchanged in I guess what was like a math majors drug deal. Another book often recommended in this category is ‘Calculus’ by Michael Spivak. So moving on to the category of linear algebra. The one I read and I guess I recommend because I thought it was good was ‘Elementary Linear Algebra’ by Howard Anton. Maybe I have a very biased liking of this book because I remember using it to study for a linear algebra test that I did really well in. But I also thought that of all the linear algebra books I encountered, this one seemed to do things reasonably intuitively, and I know it covered ideas by first giving clear definitions of them so you were never too confused. OK Differential equations is a really big category in learning math. You’ve gotta start with learning ordinary differential equations, then partial differential equations But what I’ve found in all these differential equation courses is the lecturer it tends to sort of write up their own course notes. Which you can probably learn from, completely self-sufficiently without needing a textbook. It just seems to be that since differential equations are such a big area that if you’re doing a course on them, often you will have some sort of at least recommended reading from your lecturer or they’ve written their own set of notes to go along with it. But some books that I’d say are alright are ‘Partial Differential Equations – An Introduction’ by Walter Strauss. And also another online resource, which is called ‘Mathematical Tools for Physics’ and it’s by James Nearing and I’ll give the link to this but it’s actually a completely free online PDF of this guy who’s written up a bunch of notes on math used for physics and because differential equations are so often used in physics that’s a big part of this like online textbook. I’d say check that out if you’re looking to learn not only differential equations, but a lot of these more physics or application based ideas in maths. One more book I specifically want to mention is for complex analysis, and it’s called ‘Visual Complex Analysis’ by Tristan Needham, and this book claims to give a very intuitive explanation of complex analysis – more so than I’ve seen anywhere else. I guess complex analysis is maybe an inherently unintuitive topic sometimes because you’re dealing with the imaginary numbers and you’re dealing with all these ideas and results and theorems that come out of imaginary numbers, and they can seem really sort of strange and like they just came out of nowhere. So this book, I’d say, is worth reading if you want to actually understand imaginary numbers, and not just be satisfied with saying ‘oh, they are weird and crazy, they’re imaginary’, but actually I guess understanding that area of maths. Now in my last video about the books for learning physics some of you guys left really awesome comments detailing like further book recommendations you have, and even some recommendations for math. So I’m going to read out some of the most recommended books that I saw in the comment section of that video. These are books I haven’t personally read myself, but I know I trust you guys that if you’re recommending them, they’re probably good. So a shout out to ‘Principles of Mathematical Analysis’ by Walter Rudin; ‘Analysis One’ by Terence Tao, the famous mathematician there; ‘Algebraic Topology’ by Allen Hatcher; ‘Mathematical Methods and the Physical Sciences’ by Mary Boas, and I think that’s actually possibly the only book I recommended on here and even in my last video written by a woman, which is kind of sad but at least I’ve got one to include; ‘Abstract Algebra’ by Dummit and Foote. I think that’s actually quite a classic book for abstract algebra; ‘Discrete Math and its Applications’ by Kenneth Rosen; and ‘How to Think Like a Mathematician’ by Houston. And I saw that that last one was recommended if you’re wanting to learn more about formulating proofs and the idea of proofs. And that said, those are some of my recommendations, But I’m sure you guys have even more recommendations than I do, so I’d love you to just sort of talk amongst yourselves in the comments and share some of your further experiences with these books. Also good places to get like really specific book recommendations or reviews are on the math subreddit or even on like math Stack Exchange. People are often talking there about their experiences with certain textbooks, and like I said, there are so many textbooks out there, and they’re often so large and take so long to read that for one person to understand all that’s available is pretty hard Thanks for watching this video, I feel like talking about math textbooks must be inherently one of the most boring things someone can do so, I mean, I’m really grateful that you guys are interested in this content and I’m always open to more ideas. So let me know what you think in the comments and hope you have a good day 🙂

My list is by no means exhaustive, they are just the books that I have encountered. The hardest part of making a video like this is trying to pronounce all the author's names correctly so my apologies in advance for any that I said wrong 😛 Also in Australia we tend to say 'maths' instead of 'math' and there is very little consistency in which I choose to use.

Thank you so much for these videos and information therein. I am seriously considering returning to school in order to begin a career in mathematics.

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K. A Stroud wrote a lot mathematics books.

Love the videos! Is there a book you could advise for precalculus?

Riemannian geometry by Do Carmo is a great book for learning differential geometry.

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Nearing was my professor at UM

What's your opinion on sexagesimal system. In short it's a system where radix is 60 instead of 10. Because of the number 60 it is easier to handle for example in division 60/1, 60/2, 60/3, 60/4, 60/5, 60/6. It was used by ancient Sumerians, Babylonians and Accadians in time, lenght and volume calculations. I'm studing on a mechanical engineer line and when I plan my machines I usually try to use it in measuring and in the machines structures planning. Is this good or not or am I just a lunatic? 60, 120, 180, 240, 300, 360, 420, 480 and so on. Kinda makes sense in 360 it's a full circle?

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I'm gonna read How to think like a mathematician by Houston

. Do you think you could do a video on distance online education? Thank you very much Tibees, you're helping a lot.

Discrete Mathematics, by Norman L. Biggs.

It's a step by step introduction to math, proofs, and other concepts like graphs and algorithms

Is also stupid-proof as it has a lot of step by step exercises

I recommend 'Calculus' by L. V. Tarasov

It is really a good for a beginner.

Thanks for reading.

Please share with us your way of learning/studying maths, elaborate please. thanks

One of the hottest mathematician I've ever seen in YouTube

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When you say one it is a vector of magnitude one. Similarly two and three etc. But if you take a straight line and represent one or two or three etc then it is a vector. Numbers show irrationality when you hit boundaries of vector representations. Like root two etc. In physics we always have weird worlds. Nobody does vectors integral. Maths has only scalar numbers and scaler number systems. Something like water is a scalar representation of vector representations of ice.

I've got (and have worked through) the Houston book and have found it very useful. His in-text and end-of-chapter questions are very good and even inspiring (a bit unusual for maths textbook questions!), often providing excellent jumping-off points for further investigation and generalisation.

While on the subject of inspiration, has anyone here read 'A Mathematicians Lament' by Paul Lockart, available both online and as a short hardcopy book.

Maybe should of started the book list with preAlgebra -> algebra -> geometry -> precal -> so on

Rudin is a great book, but you won’t understand anything if you don’t already know how to write proofs and know a bit of metric space topology

Linear Algebra – Hoffman and Kunze, Steven Roman (actually Roman has several pretty solid books on everything from mathematical finance to writing excel macros).

Differential Geometry – Lee

Topology – Munkres -> Hatcher

Algebra – Herstein (or how to teach someone algebra by brutally punishing them)

Analysis – There are no good analysis books. I work in an analysis field. I have yet to find a non-trash book. Bass is terrible too but at least it's free and concise. One day an analyst will figure out that you're allowed to draw pictures and it will be amazing.

What really got me into Mathematics. Is when I actually started to take time to sit down and study it. In my early twenties and then realizing. Because I really didn’t have a favorite subject in school. Never really took the time to study until I realized if I hope to do anything in life I have to probably study something in college. So I really officially decided that I want to go for my associates in math when I manage to get into community college.

Naive Set Theory by Paul Halmos.

I used Ron Larson’s book on Calculus and really like it. Next semester I hope to take vector calculus ?

The two texts that got me through my engineering degree and are keepers for me are… Erwin Kreyszig's Advanced Engineering Mathematics, and… Howard Anton's Calculus.

Other keepers are Engineering Thermodynamics by William Reynolds and Henry Perkins (my professor 🙂 ), and for psych Henry Gleitman's Psychology.

What introductory book do you recommend for Algebraic K-theory?

How can you be always smiling? I'm sorry, I'm not trying to criticise.

Spivak > Stewart

Textbooks are expensive and to address this the University of Minnesota has organized the Open Textbook Library (https://open.umn.edu/opentextbooks) that has many of the books you mentioned in pdf form. I recommend everyone take a look at the project.

Here are some of my favorites. "Calculus" by Gil Stang is an MIT standard (it's downloadable from OpenTextbooks). . My personal preference is "Thomas' Calculus, Early Transcendentals." Also an MIT standard. Very readable, lots of examples and student solution manuals are available. Boyce and Diprima's "Elementary Equations and Boundary Value Problems" is an engineering school standard and very readable. Strauss' "Partial Differential Equations" you recommended is also up on OpenTextbook for download.

My favorite linear algebra book is "No Bullshit Guide to Linear Algebra" by Savov. It's great. It's especially good for people in non-math disciplines since it spends a good amount of time in the beginning setting a baseline of math needed to get into linear algebra. For Complex Variables I don't think you can do better than Schaumn's Outline "Complex Variables." Most people end up using it no matter what textbook is required for their course. Lots of solved problems.

"Real Analysis" by Cummings is good if you're not a pure math major and are coming at the subject from an engineering background. Rather than a theorem-proof presentation there's more commentary,

My favorite recreational/overview read is "Elements of Mathematics: from Euclid to Godel" by Stillwell. A bedtime read that will make you think!

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Are there books or courses that take a radically different approach to the ultra-dry usual textbooks and videos? I kind of feel like one day we will realise the way we are teaching maths currently only works for a certain section of the population, and that there are other more intuitive and less abstract ways to do it. People often think they are "just not a maths person", when in fact they may "just not be a person who learns from the standard way maths is taught".

Any books for Analytical Geometry ?

nice tips on complex analysis

i have nothing to do with mathematic, i just come to watch you,you are relaxing and attractive in a different way

Ooh i really want to learn about mathematics. all my math teachers were jerks who loved torturing their students both mentally and physically Their method of teaching had succesfully put me off maths during my school years.

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Cassandra or Sophitia? win either way

Anything for nonlinear Dynamics and chaos youd recommend?

pretty, what basically the meaning of physics? what's the basis

“Technical Calculus with Analytic Geometry” by Judith Gerstig is very basic, but has good practical physical problems! and best of all ishe only uses SI units!

who else attracted to her

Best book I encountered is Stroud 'Engineering Mathematics' A very large book because it starts at the very beginning and works as a tutorial as it explains; gives examples, tests on one question and answers that question before moving on to the next part of the chapter. If you, like me came into mathematics at a later age then this is highly recommended.

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what is graph of equation of square and cubic volume of cube and cuboid and square and it's diagonal

this will deeply shows that basic nature of natural resources and phenomenon unevenly universally

if a mirror beak, which patterns will it following, so we can predicted it's lines of breaking before it's break down and found the results with our predictions

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It starts at (2:43)

Hello! Thanks for the recommendations, they're great… Do you know any pre calculus books that are great?

My mathematical ambitions are humble, but an understanding of trigonometry would shed light on my interests. Do any of your book recommendations cover trigonometry? I shall have to rewatch your Joy of Math video on logarithms to get a grasp of log.

I recommend Game of Logic as a recreational read.

Some mathematical disciplines are distinct from others which means that maths are plural.

Might you make a video about one or more of Euler's mathematical contributions?

Great teacher

I'm really interested in this reading list and am currently looking into it. However, could you make a video about cell phone apps for math? I understand that this means a lot of your time going through them but for us plebs, going through all the math apps out there and reading through the reviews in the app store is of little help. I trust YOUR judgement on what constitutes a good app for maths way more then the poorly written reviews on the store. Thank you for your time 🙂

In Italy while I was studying all books written by Giuseppe Zwirner for Calculus and Differential Equations were a golden bible

What did you talk about? ??

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Mathematical methods in the physical science by m l.boas is one of the worst book,it contains only cheap and poor theory and problems.I am shocked, you recommend this book.

You are an inspiration to many. Keep it up! My best wishes! ?

Not so many books by women mathematicians? What about Frances Kirwan (Algebraic curves), Ingrid Daubechies ( Ten lectures on wavelets) they wrote classics!

I'm old now but when I was 15 Sylvanus P Thompson Calculus Made Easy really turned me on.

Books are for crooks

Rules are for ghouls

Math is a curse

You all got fooled

Thanks

Coool .. Where to get a brain ?

Hi you might like 'Emmys Noethers wonderful theorem'. She wasnt allowed to be a prof because of being a woman! But Noether symmetry is now central to all of physics. No exaggeration.

Why does YouTube keep recommending math bro wtf

You recommended Dr. Nearing's book! He was my advisor when I was an undergraduate. He was a very good instructor. He really loved showing us mathematical tools he liked.

He wrote a mechanics text, and you can find a free PDF on his faculty page.

Raymond Redheffer wrote a fantastic introductory differential equations text: Differential Equations – Theory and Applications.

I wish I can get them in Peru… Please, send me the books in pdf, this is my e-mail:

[email protected]

Thanks Miss Tibees.

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Can you make it a series story like thesses math, physics math,……

Unlike math, beauty is subjective. That said, I think you’re gorgeous!

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I think I have only one of those books.

Will have to stop playing video games, and get back to GCSE Maths….

I was placed in a Learning disabled class until 9th grade. This did a number on me looking back on it as I am convinced that my self extreme took a hit. Never got any further than Geometry am working on improving whatever foundations whenever I can. Thank you for you mathematical recommendations. as I may dig in again when I have a free moment. In college I did have to take statistics I forgot about that but It was for Psychology….Think my first reading will be How to think like a mathematician

thanks this video is inspiring me

If you're intimidated by math, read 'A Mind For Numbers' the first 4 chapters are about neural plasticity and the einsteholm effect which is going over something several times thinking that you'll find something new but if you follow the eyes, you're just repeating the exact steps that led you to the wrong answer. It's about the psychology of math.

Does anyone recommend anything for young teens that are fast learners

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for more advanced students, two great universal classics "Functional Analysis", Walter Rudin and "Algebra", Serge Lang

For me the best book and the easiest was james Stewart.

Where are the math jobs?

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Tibees

Thanks Tibees madam

Which book is best for arithmetic math

Tank you so much for your help ?

A good text book which was recommended along with Calculus by Michael Spivak, was, Advanced Engineering Mathematics by Erwin Kreyzig.

WHAT ABOUT GEB ?

Really enjoyed the video but thought I would give my thoughts one something.

Perhaps I missed it (please someone say if she did) but I wish you added a note that this list for the most part is maths for physics and engineering rather than maths in general, while someone who did a maths degree would cover most of the things mentioned here they will also cover much more, from various types of geometry (ie differential, projective) to rigirous approachs to anaylsis (not just complex anaylisis) to topology to many areas of statistics to number theory etc etc. Even just things like how to write mathematics (which learning maths just to solve physics problems doesnt teach you)

imo this list is for someone that wants to learn maths to do physics rather than someone who has a broad interest in doing maths as a subject at a higher level (these 2 things couldn't be more different unless you are interested in mathematical physics)

Warning to everyone thinking of buying abstract algebra by Dummit & Foote. This book is awful for learning ring theory.

So this book is written by David dummit and Richard Foote. You can clearly tell when the book change author. The group theory part of the book is really good and quite easy to get the head around the field. But the ring theory part is awful. There is no continuity between the two chapters and the format in the group chapter have been thrown out in the ring chapter.

If you want to learn group theory, field & Galois theory and algebraic geometry, this is the book for you. If you want to learn ring theory and modules, I would recommend Ring theory by Louis Halle Rowen and commutative ring theory by H. Matsumura and miles Reid.

The abstract algebra book will just confuse you more than you need to be.

(Btw, if the course have pensum from abstract algebra, the you should by the book)