Books for Learning Mathematics

GENERAL

CALCULUS

LINEAR ALGEBRA

DIFFERENTIAL EQUATIONS:

COMPLEX ANALYSIS:

OTHER

Some Amazon affiliate links have been included (I get a small reward from Amazon but it costs you no extra). I encourage you to check your library first.

[tibees]

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 :P Also in Australia we tend to say 'maths' instead of 'math' and there is very little consistency in which I choose to use.

[kostiskavalos]

When I wanted to learn English and practise for my listening exam I was listening to her videos!

[lordoftheflings]

"Calculus Made Easy" by Silvanus P. Thompson is one of the best calculus books. Reads like a novel and is over 100 years old and can be downloaded for free now, but its also still in print.

[drumchefhubdad]

Not boring in the least! I'm 48 years old and I have just started my studies in math. The things that I once found boring are now all I want to do. I am a self taught and well rounded drummer like Buddy Rich. Not quite as great as Buddy but as close as I could possibly get and am still striving to get there. Drums are all about patterns and once you master a particular pattern it opens up a wormhole to other galaxies of rhythms and syncopations and I will be forever be learning these patterns. It is the same for mathematics for me. Drums and math are two of the same and the list that you just introduced is going to open up a whole other Universe for me to absorb. Thank you for all the videos you post! You are a wealth of knowledge and you speak with such clarity so please keep posting videos and I will keep watching. Thank you!

[nicksm]

The most important math books for a modern student are Velleman 'How to Prove It' and Bloch 'Proofs and Fundamentals'. Work these books out and you will never have serious issues with math. Bloch has also written 'The Real Numbers and Real Analysis' which is a perfect book on the foundations of real analysis, it's so perfect that I wanna cry.

[JoachimderZweite]

My best math teacher was a mad crazy Englishman who taught me math at a high school in France. His name was Mr. Baugh and he taught us calculus, algebra, analytical geometry, trig and probability. He referred to us as the blockheads - pas douée pour le math . He always said he wanted students who had no mathematics but who were extraordinarily good at Latin (sic loquitur). He spoke in a manner that proved he was completely mad and one girl said it sounds like broken French with an English accent but I can't understand a word he says. Mr. Baugh inspired me to swot - hit the books - so I was totally prepared before the lessons as I knew his language was incomprehensible. It worked until we hit probability. In order to understand probability one needs good clear language so I did not do well in that aspect of math as his musings were gibberish to me. I was not alone, except for Gilbert who seemed to understand him. Later on in life I studied Latin and I understood why Mr. Baugh had loved its precision and exactness - "know the correct ending, " After this high school college was a breeze and I shuffled through with middling grades and no effort. Later in life however, when faced with difficult choices and unsolvable problems I reached back across time to the insane muttering of that mad, crazy bastard who came to my aid down the years and in my mind I saw him put the numbers on the board and say " prove to me you lot are not hopeless." He is long dead. When he marched into heaven he read the sign put up by the students - " she was always the fastest of ships (said of a very clever girl in class).

[rauleduardohernandezaraujo]

Tibees you’re really an inspiration for any student, you just make the school seem, not easy, but worthy. I like your personality and I’m looking forward to be like you. Greetings from Mexico and keep it going 🇲🇽

[andrewmayer]

The key to any good maths textbook is to have a picture on its cover completely unrelated to the subject it's discussing.

[sciencepetr]

This is a great list.
The best resource i've seen for understanding linear algebra is the video series The Essence of Linear Algebra by 3blue1brown. Absolutely amazing, intuitive way to get the concepts, rather than just rote-learn. So good.

[darrylwade]

I love "Measurement" by Paul Lockhart. It gives a great introduction to thinking about geometry and calculus by guiding you to great exercises without giving you the answers. It's like having a great teacher guiding you.

[RockitMan-eytx]

If you are looking for a single book to start with on math that covers the basics of the entire spectrum of subject in math, I can suggest "Mathematics For The Millions: How To Master The Magic Of Numbers" by Lancelot Hogben. It really helped me understand how each area of mathematics is related to solving problems in the real world and the writer assumes the reader has little or no background in the subject matter. First published in 1937, it is the older books and their authors that I believe had a talent for reaching students interested in complex subjects like Math.

[WalidsChannel]

just your voice calms me down and your videos reduce my stress. Thanks Tibees!

[theflaggeddragon]

"Introduction to Manifolds" By Loring Tu is super accessible and absolute GOLD, same with Munkre's "Topology"

[harrisonsonandalama]

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

[nikitasid]

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.

[owenhughes]

I recently drove across the country, and stopped by every used bookstore that I saw in order to check the mathematics section. I really enjoy taking a look at the way that different books expose mathematics differently, and I'm also always looking to expand my collection.
Based upon this, as well as my experience as a math major, I have several comments on what books should be used to learn mathematics. The key point is this: the right book to use is highly dependent upon what you want to learn, what parts of it you want to learn, and what interests you.
Several of the books that are mentioned in the video are what I would call "applied textbooks", which will give good methods for solving individual problems, or classes of techniques for applying mathematical tools to real world problems. "elementary linear algebra" and "early transcendentals" would fall into this category. These have almost nothing in common with some of the books here, like "principles of mathematical analysis", Dummit and Foote, or, to some degree, spivak's "calculus", which I would call "pure mathematics texts". These texts are based upon developing mathematical techniques through proving things. Most of so-called "pure math" is based around making mathematical statements and proving them. Spivak's calculus covers similar material to "early transcendentals", but each topic that is introduced will be rigorously defined (to some degree) and proved. The latter will teach you how to find the integral of e^2x, while the former will teach you how to show that taking that integral is possible, and examine the properties of integration (e.g. how do I know that the antiderivative of a continuous function is continuous, let alone differentiable). "principles of mathematical analysis" goes beyond this, wherein the backbone of the book is theorems, and its main goal is developing techniques for proving other things. Dummit and Foote probably wouldn't look very much like mathematics to the high-school reader, because it presumes that the reader is already very well versed in mathematical arguments.
However, there is a third group of books, which aren't talked about all that frequently. These apply rigorous mathematical techniques to applied fields. Most of these books become accessible once someone has a background in linear algebra and multivariate calculus. For example, many books on optimization are fairly proof-based, however they have tangible examples that help to keep the exposition grounded. The book that convinced me to be a math major wasn't an abstract algebra textbook, it wasn't number theory, or non-euclidean geometry. The book is called "optimization in function spaces" by Amol Sasane, and it is accessible to anyone with a good grasp of linear algebra and abstract vector spaces, and multivariate calculus. Finite element analysis textbooks (a common method of dealing with PDEs in engineering) often have these properties as well, but a thorough understanding of these books requires a good understanding of differential equations first. Game theory, mathematical biology, and many other fields have books like these, which delicately introduce a new reader to the idea of rigorous proof without scaring them half to death.

What I have discovered is that I really do love rigorous proof, I just find most of mathematics endlessly dry until I find something to apply it to. For me, the third category of books is perfect. For some people (like my roommate), the math is enough in and of itself, so the second "pure math" category is for them. Others will never catch the "rigorous proof" bug, and that's fine. There will always be books in the first category, and the math you'll learn from them will be tremendously useful in whatever field has driven you to learn math.

[NicleT]

Thank you so much, I finished _Fermat’s last theorem_ today. It was a very good entertaining math book, I really enjoyed it.

[dominikcl]

I'm currently a junior high school student that has exhausted all the high school maths curriculum that my school teaches so now I bought the book Mathematical methods for physics and engineering by Riley, Hobson, and Bence. The book was recommended to me by a great YouTuber called Simon Clark. Absolutely love it, I've already bought it since and study maths with the book and with free online lectures. I'll soon start Calc 3 (multivariate calculus), then linear algebra and after that differential equations. Thank you very much +Tibees for inspiring me to study maths and physics. Love you!

[jfposada]

Nice job, Tibees. One of my favorite books is "e: The Story of a Number". I read it in 2008 and loved it. Will read it again.

[Materialismodialecticohoy]

Your videos are very nutritive for advanced students in high schools or students in first year of mathematics and surely they have changed the live of many of them. I congratulate you.

I wanted to attach my list because I think that for mathematicians and physicists your list is necessary but not enough:

1.- Elementary Classical Analysis by J. Marsden or Mathematical Analysis by L. Kudriavstev or Calculus by D. Zill. 4:43

2.- Linear Algebra by S. Friedberg A. Insel L. Spence or Linear Algebra by V. Voevodin or Foundations of Linear algebra by A. Maltsev or Linear Algebra by S. Lang. 5:36

3.- Ordinary Differential Equations by V. Arnold or Differential Equations with Applications and Historical Notes by G. Simmons. 6:51

4.- Introductory Complex Analysis by R. Silverman or The Theory of Analytical Functions: A brief course by A. Markushevich. I would like to add that T. Needham has covered of glory!! 7:40

5.- Introductory Real Analysis by A. Kolmogorov and S. Fomin or Mathematical Analysis by T. Apostol or Hilbert Spaces with Applications by L. Debnath and P. Mikusinski. 8:54

6.- Abstract Algebra by I. Herstein or Introduction to Algebra by A. Kostrikin. 9:21

7.- Reason in Revolt by Ted Grant and A. Woods. 9:32

Your work about science diffusion will be rewarded by many students of science in the future because your work is love, and, as López Obrador says “The love is paid with love”.