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How to learn Mathematics from the ground up

A guide to the intrepid adventurer

Description

Imagine the following scenario, you are a middle age engineer that studied applied mathematics in the context of a certain field of engineering and would like to remember everything again, or better to learn everything again from the ground up! With your knowledge of having done the path once, and experience to know what is a good book, what would be the best path to revisit everything again, or to structure the best a path to a friends children or a family member life journey? This will be a journey from the most basics mathematics, all the way to pure mathematics, a real adventure along 50 books :-D

Important and free resource

One of the most valuable resources in math from kinder garden to college math.

Khan Academy https://www.khanacademy.org/

Motivational books on Mathematics

Video - The Map of Mathematics https://www.youtube.com/watch?v=OmJ-4B-mS-Y
The Math Book - Big Ideas Simply Explained by DK
The Man Who Loved Only Numbers: The Story of Paul Erdos and the Search for Mathematical Truth by Paul Hoffman
Logicomix: An epic search for truth by Apostolos Doxiadis, Christos Papadimitriou

First years in math

This is a series that focus on giving you a problem solving mentality, with this book series the intent is to teach you how you to solve problems with mathematics as a tool. The solution books have worked out problems, so you can rely on them for self study.

The Art of Problem Solving: Prealgebra by Richard Rusczyk, David Patrick, Ravi Boppana Text: 608 pages. Solutions: 224 pages.
The Art of Problem Solving: Introduction to Algebra, 2nd Ed by Richard Rusczyk Text: 656 pages. Solutions: 312 pages.
The Art of Problem Solving: Introduction to Counting & Probability, 2nd Ed by David Patrick Text: 256 pages. Solutions: 120 pages.
The Art of Problem Solving: Introduction to Geometry, 2nd Ed by Richard Rusczyk Text: 557 pages. Solutions: 226 pages.
The Art of Problem Solving: Introduction to Number Theory by Mathew Crawford Text: 336 pages. Solutions: 144 pages.

Intermediate years in math

The Art of Problem Solving: Intermediate Algebra by Richard Rusczyk and Mathew Crawford Text: 720 pages. Solutions: 336 pages.
The Art of Problem Solving: Intermediate Counting & Probability by David Patrick Text: 400 pages. Solutions: 208 pages.

Pre-Calculus

At this point you have three good options see what adjusts better to you.

The Art of Problem Solving: Precalculus, 2nd Ed by Richard Rusczyk Text: 528 pages. Solutions: 272 pages.
Precalculus: Mathematics for Calculus 7th ed by James Stewart, Lothar Redlin, Saleem Watson
Precalculus by Jay Abramson https://openstax.org/details/books/precalculus

University or college mathematics

Linear Algebra

You will learn Linear Algebra with examples in code (Python and Matlab) without calculus.

Linear Algebra: Theory, Intuition, Code by Mike X Cohen

Calculus and friends

This is a book to motivate you to go further in the most beautiful way!

Calculus Made Easy by Silvanus P. Thompson, Martin Gardner

Then to my knowledge there are 3 similar good paths that you can follow, but with increasing depth in mathematics and Calculus.

First path

Engineering Mathematics, 5th Ed by Prof Anthony Croft, Dr Robert Davison, et al.

Second path

Modern Engineering Mathematics, 6th Ed by Glyn James, Phil Dyke
Advanced Modern Engineering Mathematics, 5th Ed by Glyn James, David Burley, Dick Clements, et al.

Third path

Mathematical Methods for Physics and Engineering: A Comprehensive Guide 3rd Ed by K. F. Riley
Student Solution Manual 1st Ed for Mathematical Methods for Physics and Engineering 3th Ed by K. F. Riley

Fourier Series and Transforms

Fourier Analysis: An Introduction by Elias M. Stein and Rami Shakarchi
Fast Fourier Transform and Its Applications 2th Ed by E. Brigham

Probability and Statistics

Then you need to learn about Probability and Statistics the following are two nice books with a companion book with the solutions.

Probability: For the Enthusiastic Beginner by David J. Morin
Introduction to Probability, Statistics, and Random Processes by Hossein Pishro-Nik http://www.probabilitycourse.com/preface.php
Student's Solutions Guide for Introduction to Probability, Statistics, and Random Processes by Hossein Pishro-Nik http://www.probabilitycourse.com/preface.php
All of Statistics: A Concise Course in Statistical Inference by Larry Wasserman

Optimization

Then you will need to learn about Optimization, two good books, the first with code in Julia.

Algorithms for Optimization by Mykel J. Kochenderfer, Tim A. Wheeler Note: See the book PDF site link on the authors page. https://mykel.kochenderfer.com/textbooks/
Convex Optimization by Boyd, Vandenberghe https://web.stanford.edu/~boyd/cvxbook/

Discrete Mathematics

Discrete Mathematics with Applications 5th Ed by Susanna S. Epp

Numerical Analysis and Computational Mathematics

Numerical Methods for Engineers 8th Ed by Steven Chapra, Raymond Canale
Numerical Recipes 3rd Edition: The Art of Scientific Computing by William H. Press
Numerical Methods in Physics with Python by Alex Gezerlis
Computational Physics: Problem Solving with Python 3rd Ed by Rubin H. Landau, Manuel J Páez, Cristian C. Bordeianu
Applied Computational Physics by Joseph F. Boudreau, Eric S. Swanson
Hans Petter Langtangen - Various writings http://hplgit.github.io/
Hans Petter Langtangen - Last versions https://library.oapen.org/discover?rpp=10&etal=0&query=Langtangen%2C+Hans+Petter&scope=&group_by=none&page=1

Information Theory

A Mind at Play: How Claude Shannon Invented the Information Age by Jimmy Soni, Rob Goodman
Information Theory, Inference and Learning Algorithms by David J. C. MacKay Note: In the author site you have de book and the video lectures. http://www.inference.org.uk/mackay/itila/

Error Correction Codes

Error-Correction Coding and Decoding: Bounds, Codes, Decoders, Analysis and Applications by Martin Tomlinson, Cen Jung Tjhai, Marcel A. Ambroze, Mohammed Ahmed, Mubarak Jibril Note: Book on open access. https://www.springer.com/gp/book/9783319511023
Error Correction Coding: Mathematical Methods and Algorithms 2nd Ed by Todd K. Moon

Problems with answers

Schaum's 3,000 Solved Problems in Calculus by Elliott Mendelson
Schaum's Outline of Calculus, 6th Ed by Frank Ayres, Elliott Mendelson
Schaum's Outline of Advanced Calculus, 3rd Ed by Robert Wrede, Murray Spiegel
Schaum's Outline of Advanced Mathematics for Engineers and Scientists by Murray Spiegel
Schaum's Outline of Probability and Statistics, 4th Ed by John Schiller, R. Alu Srinivasan, Murray Spiegel
Schaum's Outline of Discrete Mathematics, 3rd Ed by Seymour Lipschutz, Marc Lipson
Schaum's Outline of Complex Variables, 2th Ed by Murray Spiegel, Seymour Lipschutz, John Schiller, Dennis Spellman
Schaum's Outline of Differential Equations, 4th Ed by Richard Bronson, Gabriel B. Costa
Schaum's Outline of Partial Differential Equations by Paul DuChateau, D. Zachmann
Vector Analysis, 2nd Ed by Murray Spiegel, Seymour Lipschutz, Dennis Spellman
Schaums Outline of Tensor Calculus by David Kay

Very good synthesis of all around Mathematics

The Princeton Companion to Mathematics by Timothy Gowers, June Barrow-Green, Imre Leader
The Princeton Companion to Applied Mathematics by Nicholas J. Higham, Mark R. Dennis, Paul Glendinning, Paul A. Martin, Fadil Santosa, Jared Tanner

To go further into Pure Mathematics

This book section specific recommendation come from the wonderful video about learning pure mathematics, see the video and the video description for more details.

Aleph 0 - How to learn pure mathematics on your own: a complete self-study guide https://www.youtube.com/watch?v=fo-alw2q-BU

Real Analysis

Calculus, 4th Ed by Michael Spivak, Michael Spivak
Combined Answer Book For Calculus Third and Fourth Editions, 1th Ed by Michael Spivak
Understanding Analysis by Stephen Abbott.

Linear Algebra

Linear Algebra Done Right by Sheldon Axler

And for the problems.

Linear Algebra by Insel, Freidberg, and Spence

Topology

Topology through Inquiry by Su and Starbird

Differential Equations

Differential Equations with Boundary Value Problems by Zill and Cullen

Complex Analysis

A Friendly Approach to Complex Analysis by Sara Maad and Amol Sasane
Visual Complex Analysis by Tristan Needham

Abstract Algebra

Contemporary Abstract Algebra by Gallian

Differential Geometry

A Geometric Approach to Differential Forms by David Bachman
Introduction to Manifolds by Loring Tu

The great men and women behind mathematics

Men of Mathematics by E.T. Bell

All my other guides

The links to all my guides are in: Guides on Linux - Programming - Embedded - Electronics - Aeronautics https://github.com/joaocarvalhoopen/Guides_Linux-Programming-Electronics-Aeronautics

Have fun!

Best regards, João Nuno Carvalho