🤗🤗🤗This is a comprehensive collection of mathematics resources designed to support self-study from high school to graduate-level topics. Starring this repository will enable it to reach larger audience! 👍👍
American Mathematical Society (AMS) has the most comprehensive file on the classification of mathematics. It is about 230 pages.
Khan Academy is the best resource to learn grade 1 to 12 mathematics topics. It is also the official platform to prepare for SAT, as well good (not fully sufficient) for IGCSE, AP exam, IB exam and A-level exam. Depending on your mathematics goals, if you are someone (e.g undergraduate, working professional) that quickly want to revisit your old mathematics knowledge, Khan Academy is your surest bet.
If you prefer books, you can download these books via the websites given below:
- Cambridge IGCSE, AS & A level coursebooks
- Haese Mathematics coursebooks for IB exam
- AP books
- Calculus AB and BC: Barron's AP Calculus, Princeton Review's Cracking the AP Calculus AB & BC Exams
- Statistics: Princeton Review AP Statistics Prep, Barron’s AP Statistics and 5 Steps to a 5: AP Statistics
- Pauls Online Math Notes - an excellent online resource!
- MIT 18.01 Single Variable Calculus* - This introductory calculus course covered differentiation and integration of functions of one variable, with applications.
- MIT 18.02 Multivariable Calculus - Covers vector and multi-variable calculus. Topics include vectors and matrices, partial derivatives, double and triple integrals, and vector calculus in 2 and 3-space.
- Calculus, 4th edition [by Michael Spivak] - One of the most popular calculus book for undergraduates.
- Calculus Early Transcendentals, 11 edition [by Anton, Bivens, Davis] - Covers both single variable, multivariable variable and vector calculus. It is easy to understand, with a lot of excercises.
- Calculus of a Single Variable, 10th edition [by Ron Larson] - A top-notch book for single variable calculus.
- Calculus Early Transcendentals, 9th edition [by James Stewart, Saleem Watson, Daniel Clegg] - Similar to Anthon-Bivens-Davis book.
- Vector Calculus, 4th edition [by Susan Jane Colley] - Similar to Anton boo, but slightly mathematically more intense.
- Vector Calculus, 6th edition [by Anthony Tombra, Jerrold E. Marsden] - Very similar to Susan Jane book.
- Mathematical Proofs: A Transition to Advanced Mathematics, 4th edition [by Ping Zhang, Albert D. Polimeni, Gary Chartrand] - Covers set theory, mathematical induction, logical reasoning, equivalence revaltions, functinos, nuber theory, combinatorics, calculus, group theory, ring theory , linear algebra, etc. It is highly recommended.
- How to Prove It, 3rd edition [by Velleman Daniel] - A very nice book.
- Introduction to Proof Writing - A full 12-hour YouTube video by MathMajor.
- Book of Proof [by Richard Hammack] - A great book and less advanced compared to Mathematical Proofs by Ping Zhang. Here is the YouTube playlist for the book.
- Introduction to Mathematical Thinking - A popular Coursera course by Dr. Keith Kelvin from Stanford University.
- Proofs [by Jay Cummings] - Very intuitive and fun!
- Haese Mathematics HL (Option): Sets, relations and groups [by Catherine Quinn, Robert Haese, Michael Haese] - A beginner-friendly intrdouction to abstract algebra.
- A First Course in Abstract Algebra, 8th edition [by John B. Freleigh, Neal Brand] - A very rigorous introduction to abstract algebra(groups, rings & fields, in that order) and briefly covers advanced concepts like Galois theory.
- Contemporary Abstract Algebra [by Joseph A. Gallan] - One of the highly recommended books for beginners in abstract algebra. It is rich both in theory and calculation exercises. Kimberly Brehm covers the group parts on her YouTube channel.
- Abstract Algebra [Thomas W. Judson] - Covers the theoretical aspects of grops, rings, and fields. Moreover, it treats their applications in coding theory and cryptography.
- Visual Group Theory - Taught by Professor Macauley from Clemson University and should very conducive to self-study. This is the course website.
- Abstract (Modern) Algebra Course - Taught in Spring of 2018 by Professor Bill Kinney at Bethel University. It contained 68 videos on group, ring, field and Galois theories.
- Abstract Algebra I - Taught by Professor James Cook in Fall 2016. This is the course website which contained solutions to numerous questions.
- Introduction to Linear Algebra, 5th edition [by Gilbert Strang] - This is a golden book for introductory linear algebra course. Professor Gilbert is a legend and famously konown for this course globally. Check MIT 18.06SC Linear Algebra and the YouTube playlist for his lesson videos.
- Elementary Linear Algebra, 8th edition [by Ron Larson] - A very beginner-friendly introductory to Linear Algebra.
- Linear Algebra Done Right, 4th edition [by Sheldon Axler] - A rigorous introductory course, most suitable after finishing Gilbert's or Ron's book.
- Linear Algebra [by Stephen H. Friedberg, Arnold J. Insel, Lawrence E. Spence] - An excellent and elaborate introductory book but demands sufficiently high mathematical maturity from its readers.
- Theory of Infinite Sequences and Series [by Ludmila Bourchtein, Andrei Bourchtein] - Covers a lot of theprem extensively on series and sequences than a standard single variable calculus book
- Real Analysis via Series and Sequences, 2015 edition [by Charles H.C. Little, Kee L. Teo, Bruce van Brunt] - Requires the mastery of single variable calculus and foundational proof writing.
- Real Analysis: A Long-Form Mathematics Textbookb [by Jay Cummings] - The author has a knack for explaining advanced concepts in funny and intuitive ways.
- Principles of Mathemamtical Analysis [by Walter Rudin] - Colloquially known as "PMA" or "Baby Rudin". It is a famous book but requires rich mathematical maturity!
- MIT 18.100A Real Analysis - Contains MIT lecture notes, videos, assignments and examinations.
- A First Course in Complex Analysis with Applications by Dennis G. Zill - Its only prerequisite is multivariable and vector calculus.
- Lebesgue Measure and Integration by Presidomath - Useful playlist to learn lebesgue measure and integration.
- Introduction to mathematical logic, 6th edition [by Elliott Mendelson] - Covers first-order logic, number theory, axiomatic set theory & computability.
- Mathematical Logic [by Ian Chiswell and Wilfrid Hodges] - Covers natural deduction, propositional logic, quantifier-free logic, first-order logic.
- A friendly introduction to mathematical logic [by Christopher C. Leary and Lars Kristiansen] - covers the central topics of first-order mathematical logic such as Incompleteness theorems, computability theory, completeness and compactness.
- Elements of Set Theory [by Herbert B. Enderton] - Highly respected book but might not be the best for self studying.
- A first course in mathematical logic and set theory [by Michael L. O'Leary] - covers the necessary basics in a very friendly approach.
- Topology [by James R. Munkres] - The standard undergraduate book in most universities.
- General Topology by Presidomath - An excellent playlist to learn the basics of general toplology.
- Elementary Topology Problem Textbook [by Viro, Ivanov, Netsvetaev, Kharlamov] - Really good for improving problem solving skills in topology.
- Undergraduate Topology: A Working Textbook [by Aisling McCluskey, Brian McMaster] - Short and wonderful book to sharpen your problem solving skills.
- Differential Geometry of Curves and Surfaces, 2nd edition [by Manfredo P. Do Carmo] - Well-respected book that covers differential geometry from a classical approach.
- Elementary Differential Geometry, 2nd edition [by Barret O'Neill] - It uses differential forms approach and requires only the knowledge of calculus and linear algebra courses.
- John Lee's 3 book series: (In ascending order of study) Introduction to Topological Manifolds, Introduction to Smooth Manifolds, and Introduction to Riemannian Manifolds - Good for beginning-level graduate study.
- Introduction to Probability Models, 10th edition [by Sheldon Ross] - A golden textbook which covers basic probability theory, markov chains, renewal theory, queuing theory, etc.
- Introduction to Probability [by Dimitri P. Bertsekas and John N. Tsitsiklis] - Quite rigorous and should not be used as a first exposure to probability unless you've built an excellent mathematical maturity. Its online version can be found on MITx.
- Probability & Statistics for Engineers and Scientists, 9th edition [by Ronald E. Walpole, Raymond H. Myers, Sharon L. Myers, Keying Ye] - Very easy to understand and really suitable for beginners in engineering/statistics field.
- Elementary Number Theory [by David Burton] - A good book to start your number theory journey.
- Methods of Solving Number Theory Problems [by Ellina Grigorieva] - Doesn't require more than high-school mathematics.
- Elementary Number Theory [by Thomas Koshy] - An excellent introductory book with less emphasis on rigorous proofs.
- Problems in Algebraic Number Theory [by M. Ram Murty Jody Esmonde] - Contains a collection of 500 problems with solutions.
- Essential Discrete Mathematics for Computer Science [by Harry Lewis] - Beginner-level book with interesting topics like set theory, logic, graph theory, automata theory, probability, cryptography.
- Discrete Mathematics with Applications [by Kenneth H. Rosen] - Doesn't require any prerequisites other than sufficient maturity to use it.
- The Elements of Statistical Learning (ESL) [by Trevor Hastie, Robert Tibshirani, Jerome Friedman] - A must-read book if you are serious about machine learning.
- Introduction to Statistical Learning (ISL) - Written by same authors, you can start with ISL if ESL is too advanced for you. ISL has both Python and R editions.
- Introduction to Probability for Data Science [by Stanley H. Chan] - Highly recommended and contains most needed knowledge for a typical machine learning/data science job, coupled with Python and MATLAB codes.
- Elementary Differential Equations and Boundary Value Problems [by Boyce, Diprima , Meade] - Very explanatory and suitable for beginners. It also covers basics of PDEs. Its online version can be found via MIT OCW
- Understanding Engineering Mathematics [by John Bird] - Started from middle-school mathematics up the way to PDEs, Fourier and Laplace transforms.
- Advanced Engineering Mathematics [by Dennis G. Zill] - Covers differential equations, PDEs, Laplace & Integral transforms, numerical analysis, vector and matrix calculus, complex analysis.
- Mathematical Methods for Engineers and Scientists (Book 1,2 and 3) [by K.T Tang] - Covers vector analysis, ODEs, Laplace transforms, matrix analysis, complex analysis, tensor analysis, Fourier analysis, Sturm-Liouville theory and special functions, PDEs.
It's simple using the internet. See an example: "Prerequisites of functional analysis AAA" on any search engine. AAA should be replaced with Reddit, Quora, Math stack exchange or Math oveflow. In addition, you should always read the preface of any mathematics book you are studying.
Here is a list of famous prestigious mathematics competitions you can participate in.
- International Mathematics Olympiad (IMO) - The most prestigious international competition for high school students!
- International Mathematical Kangaroo - For grade 1 to grade 12 students.
- Harvard MIT Mathematics Tournament - Occurs yearly in Februarys and Novembers for high school students.
- International Mathematical Modelling Challenge - For middle and high school students.
- Mathematical Contest in Modeling - For undergraduates.
- International Mathematics Competition for University Students (IMC) - Similar to Williams Putnam exam's style.
- Mental Math World Cup & Global Mental Math Olympiad
- SIMIODE Challenge Using DifferentiaL Equation Modeling - For both high school and undergraduate studeents.
- William Lowell Putnam Mathemmatical Competition - The most prestigious competition for undegraduates in USA and Canada.
- Alibaba Global Mathematics Competition - A global mathematics competition for EVERYONE, with a prize pool of $300,000+.
- Simon Marais Mathematics Competition - Similar to Williams Putnam exam's style.
- Vojtech Jarnick International Mathematics Competition - For undergraduates.
- Open Mathematical Olympiad for University Students - For both bachelor and master's students.
- North Countries Universities Mathematical Competition - Students from the rest of the world can also participate.
- International Student Team Competition in Mathematics - For bachelor’s, master’s and postgraduate students.
- Imperial Cambridge Mathematics Competition - For undergraduate and master's students in United Kingdom.
- South Eastern European Mathematical Olympiad for University Students (SEEMOUS)
NOTE: Excellent resources to prepare for some of these competitions are AoPs, Mathematics Olympiads Discord Server, et cetera.
Open-Source Softwares for Mathematics contains a list of free mathematics software packages you can install on your computer devices to aid your learning.
- Math Overflow - An advanced mathematics community mainly for professional mathematicians and graduate students. Of course, anyone is free to join.
- Mathematics Stack Exchange - Question & Answer community for people studying math at any level and professionals in related fields.
These are some of the websites to download millions of books freely!
- PDF Drive
- Z Library: This requires you to sign up using either your email address or gmail acount. Next, follow these steps to download books of your choice:
- Join the discord server or click on "click here".
- Type the correct name of the book or the author in the search bar and click on the search button.
- Scroll down a little bit and copy the code in front of "Request Code".
- Open the discord and paste the code in either "book-request" or "book-request-2" channel.
- Click on the generated link it brings and finally download your book.
- NOTE: Click on "Mirror" or "Proxy" in case the download option didn't work directly.
- PDFCOFFEE
- Internet: At times, all you need to do is to append the word "pdf" to the title of the book and paste on internet search tools (e.g Google Search, Firefox, etc).