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 🙂

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100 Responses

  1. Tibees says:

    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.

  2. juan guevara says:

    Chapter 1.3 of Roads to Geometry caused me to really understand what math really is.

  3. Simon Jackson says:

    Forever undecided. Raymond S… The recursive universe. William poundstone.

  4. Gerald McMullon says:

    The best "book" for a particular course is often a copy of the notes taken by a student in the previous year. If the topics of the course are unchanged then you can fill in gaps in the notes without having to take down all the equations and steps and concentrate better on the lecturer. Not all lecturers hand out printed notes. Even fewer make these available the week before – either handed out or online.

    Some courses go a step further and have material produced by more than one person – the lecture given by one, the handouts drafted by another and an online version of the lecture given by a third. This certainly worked well for the early stages of a degree course.

  5. username says:

    is she mathematician or physicists ? i am confused .

  6. B Marq says:

    video is great resource in todays day and age of growing numbers of people who are autodidactic. I studied a decent amount of mathematics for Chemical Engineering in college, might have to look into a few of these books for recreation. Math is a great gateway into a solid foundation for Philosophy in its entirety. It is both one of the hard sciences, as well as a very specific epistemological framework. Grounded in this practice, one has a vantage point to analyze different fields of philosophy in a way many others might not have (Objective vs Subjective for simple example). 

    Furthermore, as far as the Problem of Induction and the ambiguity of Linguistics goes within the sciences, I thinks it's fair to say that math is the closest to "being deductive" as any science can be. It other words, in Logic it can be considered #1 within any epistemology regarding science and 'proven' truths, followed shortly by physics and chemistry, which both fundamentally rely on mathematics. Math has by all appearances the least amount of axioms, and much of them seem self-evident to even a child. Outside of that, much of it is definitions, "it follows by definition," that are laid out right up front, which is very much different than the multiplicity of sometimes semi-unconscious axioms other sciences need to even exist, which arguably exponentially growths with time. And almost all of the mathematics that follows this can be based on this simpler foundation, thus it has the highest degree of reasonableness in its claims to epistemological truth outside of deductive reasoning, (if in fact it really is outside of deductive). 

    I'd also suggest looking into Chaos Theory once those who have a firm grasp of at least Multivariable Calculus. Not that that it is inherently necessary per se, on the other hand, one could say more is needed, or less. Either way, Chaos Theory is a fascinating topic which did something very similar to what Einstein's General Theory of Relativity did, but I'll let you figure out the rest on your own. Does mathematics have any knowledge, on its own, in regard to the weather, possibly even Climate Change? Perhaps the weather had something it wanted to teach Math? (Also, it apparently does not have a very high opinion of Newton; very temperamental about him, the weather is)……..

  7. Javier Hernandez says:

    What about Soviet math books?

  8. Jean-Baptiste Dupont says:

    You are SO cool, and so is your channel! And here is my little contribution to your list: "Mathematics for the Million" by Lancelot Hogben – this book made me appreciate math years after having any interest in it destroyed by painful school lessons.

  9. Kikki Hiiri says:

    I wish there were books that actually explained the proofs and more importantly reason why selected methods/steps are applied

  10. Khoa Nguyen says:

    Sound like British accent.

  11. John Mc Keown says:

    how can she recommend a partial differentiation book when she has not recommended a book on ORDINARY differentials first ? Ridiculous.
    Try Tannenbaum and Pollard for that [Dover].
    Start Complex Analysis using Needham, really ?
    You would need to study a bit geometry before that.
    The list is dumb.

  12. Android Game says:

    Ma'am I have a question.. In which university did you study?

  13. Ritesh Arora says:

    Most important book before getting into pure math like algebra, analysis is " How to Prove it". Most of students will appreciate pure Math after reading this

  14. Nikita Sid says:

    Top intro stuff (IMhO):
    1. Ian Stewart "Concepts of modern mathematics" – issued in 1980, now a bit forgotten.
    2. Courant/Robbins "What is mathematics" – absolute classics.
    3. "Infinitely large napkin" – google it.
    4. "Concrete mathematics" by Knuth & gang – more toward discrete math.

  15. Val Markov says:

    Morris Kline's books would make an excellent addition to these lists. My favorites are "Mathematics for the Nonmathematician" and "Calculus: An Intuitive and Physical Approach." Must-reads for those who've dreaded math since childhood.

  16. La El says:

    Is there a book to learn more about you?

  17. xyzct says:

    Mathematics: Its Content, Methods and Meaning
    by: A. D. Aleksandrov, A. N. Kolmogorov, M. A. Lavrent'ev

  18. DW yes DW says:

    Start with rd Sharma

  19. mentatphilosopher says:

    I was taught by Mary Boas. It is pronounced “bow as”.

  20. SimulatedReality says:

    For anyone else who started Calculus Early trascendentals 8th edition, I found the answers in this page: https://www.slader.com/textbook/9781285741550-stewart-calculus-early-transcendentals-8th-edition/ it was of great help to me, mainly to verify if I did the excersise right or wrong.

  21. John Barrett says:

    I need a good book to refresh my knowledge of Vector Calculus

  22. fatsole says:

    Thank you Rapunzel.

  23. Pritam Mishra says:

    you are cool

  24. Amrit Singh Chauhan says:

    The best book is Rd sharma

  25. David Santosuosso says:

    How to think like a Mathematician is very good.

  26. StYle says:

    is reading these books a good idea for a high school student ? or are they higher lvl?

  27. Joseluis ohe says:

    Visual Complex Analysis!
    What a great book! thanks for your recommendation

  28. Raghu Seetharaman says:

    witch ones r used for laerning mafs ?

  29. epenies says:

    You’ve grown to be one of my favorite youtubers.

  30. saikat93ify says:

    "Talking about Maths books is one of the most boring things you can do."

    I couldn't disagree with you more.

  31. Hani Singh says:

    I think walter rudin is quite advance and will tough job to understand for beginners. So for basics, I usually gone through are as follows

    1.Calculus Vol 1 and Vol 2 by  J.H. Heinbockel
    2. Calculus Early Transcendentals
    3. Calculus by Gilbert Strang.

  32. KROG SPY says:

    Merci beaucoup !

  33. orestesdd says:

    Yeah! Calculus by Micheal Spivak is what I used back in 1975-76 during my first year in college at the university of Havana Cuba. Other books I used mostly in Cuba were: Mathematical Analysis by Tom M. Apostol, and Calculus On Manifolds: A Modern Approach To Classical Theorems Of Advanced Calculus by Micheal Spivak too. I wish I could remember more books since I do remember to spent one semester, a real semester or about 5 months, learning topics about geometry with an introduction to lineal algebra, and then I took at least 2 or 3 semesters of pure lineal algebra and group theory, but I do not remember the books I used. By the way, I was never able to complete my BA in Cuba, but I did finish it here; however, I can tell that I learned more Math in all my 5 semesters in Cuba than my last 4 short semesters of college, semesters which were no more than 4 months, to obtain my BA in Math.

  34. Arriaga Two says:

    There are a lot: "50 shades of prime numbers", "How to break the Da Vinci code", "To kill a mockingThird", "The tangent in the rye", etc

  35. Angel Saint Thoss says:

    I luv u for this!

  36. Jota Mudo P says:

    For me, the calculus book is N. Piskunov, it has good problems and a quite complete theory, for anyone that has a good basis it's very doable, for the dense ones check spivak, courant and apostol

  37. aditya gupta says:

    I need your brain 😂😂

  38. Dustin Ransom says:

    This is my pick for book of the year 2018 for mathematical thinkers, by one of the smartest fellas who ever lived (a self-taught International Master in chess who improved faster than anyone else in history to the title): https://www.amazon.com/Applying-Logic-Chess-Erik-Kislik/dp/1911465244 For those who like strategy and logical thinking and have a bit of knowledge of chess, this book is an absolute must.

  39. John B says:

    Your Vids are very nice, I'm about to take a calculus class and my initial plan is basically to review Algebra (maybe just Algebra 2 so by doing that I will be reviewing Algebra 1 also) and Geometry and a small a small amount of Trig to, and then maybe I'll be ready to take the calc class, btw the calc class is calculus w/analytic geometry, as it is called here at the college I'm going to, I know it's going to take some hardwork but maybe one day I'll get to Calculus 3, wish me luck :).

  40. Debajyoti Datta says:

    Which book would you recommend for real analysis ? I have decent knowledge in calculus, linear algebra and stochastic process and its allies due to majoring in EE. But never really have tried RA. Actually I want to learn stochastic geometry for wireless research, which takes Measure theory as a prerequisite and measure theory and as I have read in some blogs, measure theory is next step of real analysis.

  41. Ginger Stalin says:

    thank youu

  42. Dayna Holgate says:

    I am studying maths in college and I still struggle I find books easier to understand and my teachers recommended me some books for it.

  43. Harrison Ndalama says:

    "Math Major's Drug Deal"
    I just wanna take a moment to appreciate that combination of words 😂

  44. zadran zadrani says:

    Thank you

  45. brock lesnerd says:

    Pam from The Office

  46. aa888zz says:

    No Rudin? No Ahlfors? wew

  47. Ernesto Huaman says:

    Let A be a student forced to pass a calculus course X and with motivation M such that mes(M)=0, any suggestion?

  48. Ali Esquembre Kucukalic says:

    I'm so happy that you mentioned Needham's Visual Complex Analysis 🙂

  49. Notna Никип says:

    "Математика в техническом университете"
    " Смирнов. Курс высшей математики."
    " Зорич. Математический анализ"

  50. Thomas Hodges says:

    I strongly recommend Trudeau's Introduction to Graph Theory. It would be especially beneficial to a clever high school kid who is not challenged or bored by his math courses.

  51. Thomas Hodges says:

    Also – I worked as a mathematics teacher for 10 years. I strongly recommend Polya's How To Solve It. I always composed and conducted my best lessons after rereading/reviewing that book.

  52. Marc-Andre Rousseau says:

    you are gorgeous!

  53. David López says:

    This girls is an alien robot from the future to make human race improve.

  54. George O'Reilly says:

    This is weird. I'm in to the vid to the second book and i read both. And I read 1 book per year. Now calculate the odds…

  55. Tom H. says:

    Wow suddenly i am extremely interested in math and physics.

  56. Kriegerdammerung says:

    The Argentinian Adrian Paenza was named in 2014 the "best maths divulger of the world", some of his videos are mindblowing, I can only imagine his books

  57. Waking Up says:

    Have you looked at the mathematics of the flat earth?

  58. John DeBord says:

    What do you think of Khan?

  59. Simon Says says:

    how come this is left out: https://en.wikipedia.org/wiki/Boris_Demidovich?

  60. salaa23 says:

    Looks and smarts

  61. ambivalent ambiguity says:

    For analysis, I'd recommend G.H. Hardy's book, A Course in Pure Analysis. Although it's notations and certain usages are by now somewhat antiquated, it's spirit of "math for math's sake" is eternal. But I must share my personal experience with Riemann and Abel. Everytime I've struggled cluelessly through Riemann's and Abel's original papers, I've emerged clear-eyed, and from them on, math that previously had perplexed me became transparent. Bottom line: Read the masters.

  62. LonleyGuns says:

    Math starts above linear algebra imo, at topics where drawing a chart or schematics isn't possible 😀 great vid 🙂

  63. Bharat Penumutchu says:

    After all these years i have realised that advanced Math cannot be taught without Physics and vice versa. If context is provided to Mathematics tools the motivation behind inventing those tools then it would be easier to perceive…

  64. David Shechtman says:

    Oystein Ore. "Number Theory and it's History"!

  65. Sachin Singh says:

    IIT MATHEMATICS BY TMH

  66. gamer cat says:

    I left math two years ago.
    I left physics just a few months back.😅

  67. hardlyb says:

    Hardy was completely wrong about number theory being useless…

  68. hardlyb says:

    Rudin's book Real and Complex Analysis is a gem. It's admittedly complex analysis from the point of view of an analyst (which means someone who mostly studies PDE's, etc.) but I liked the book (although I did PDE's until I became a full-fledged real-world applied mathematician, so I'm biased). Complex analysis seems weird until you realize that it's harmonic functions, and then (if you have my background) it starts making sense. And then, when you start looking at the stuff for several complex variables, it stops making sense again (and I never got far enough into the subject to understand it).

  69. jesuisravi says:

    I think it's pretty important what sort of texts you get even early on in math studies. I have a precalculus text that seems to have been written by a man who has learned to write Greek with English words. He may be (or have been) a good mathematician but has no knack at all for clarifying the concepts involved.

  70. suspended suplex channel says:

    Where is R D Sharma

  71. Kirke182 says:

    Perhaps mathematics books may be boring but, as Darwin once said, "Doing what little one can to increase the general stock of knowledge is as respectable an object of life as one can, in any likelihood, pursue." So let us not worry about such trifles as whether mathematics is boring. It is truth and that is enough.

  72. Joseph Schaffer says:

    This is like ASMR

  73. TRIBHUWAN K. says:

    In University of Cambridge graduation all mathematics book

  74. TRIBHUWAN K. says:

    First second and third year book graduate mathematics

  75. Brian Aragón says:

    Loved it. A new subscriber here. Keep the good job !!!

  76. 0 1 says:

    I am now in the beginning of studying college level math by myself. Do you think that brilliant is a good tool for that? (as a substitute for books and lectures such as on MIT ocw)

  77. khrap says:

    Bit of a random point, but I love the way your videos start IMMEDIATELY, without intro or other nonsense.

  78. Gustavo Martinez says:

    Great!!! Books!!! Love them!!!

  79. Raja Sarkar says:

    green iris

  80. china cat says:

    l like you video

  81. Martin Mitchell says:

    A really good book, albeit at a somewhat lower level than Toby was aiming at, is 'Maths: A Student's Survival Guide' by Jenny Olive, published by Cambridge. It's very broad ranging and very friendly. Very good for those who need some fairly serious maths for their course, but are perhaps not maths specialists.

  82. Mohamed Abo Shama says:

    the calculus take a lot of time to study is that mean I am a loser in mathematics

  83. Dhananjay Sharma says:

    Apna to RD SHARMA hi Sahara hai

  84. Kenneth Howell says:

    Thank you!

  85. Kenneth Howell says:

    Let me see. I am pretty smart really. My IQ is about .005 and your IQ is about 1,000,000.

  86. Kageyama Recordings says:

    Wonderful!!!

  87. FREE MINDED Launda says:

    We should understand Maths . Study maths not for exams only but also to improve thinking power 💪 and for satisfaction and confidence as it's a magic and very interesting thing. #respectMaths

  88. FREE MINDED Launda says:

    its not your fault to have weak maths. its your fault to keep it weak

  89. tubernovoice says:

    When I think about mathematics I think of Good Will Hunting. Some of it will remain beyond our ability to understand. Have a good day 🙂

  90. Deb Jyoti Mitra says:

    Good & Informative Video

  91. Varun Shrivastava says:

    Flatland is good one

  92. João Wiciuk says:

    REMEMBER: All numbers are imaginary!

  93. P DB says:

    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.

  94. Silly Penguin says:

    I like her eye color.

  95. bathed in black says:

    Wtf you remind me of that Anne frank girl that had the 6 MacBooks lmaoooo

  96. Hamoody Mdii7 says:

    Very nice your explanation learn from you has been the work of Envolo and will support you for your knowledge: Professor Mohammed Professor of Mathematics and researcher all what Shi scientific Thank you

  97. sirolsh2 says:

    K. A Stroud wrote a lot mathematics books.

  98. Brecht Geeraerts says:

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

  99. James Groccia says:

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

  100. LITE says:

    tibeeeeeeeeeeeeeeeeees is a female who possess great power and will do great things in her lifetime

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