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Ok, I'll take a crack at this: Up to high-school level: 1. Precalculus: Precalculus: A Prelude to Calculus - Axler 2. Calculus: The Calculus Tutoring Book -

by mikevm 9y ago

Ok, I'll take a crack at this: Up to high-school level: 1. Precalculus: Precalculus: A Prelude to Calculus - Axler 2. Calculus: The Calculus Tutoring Book - Ash. College: 3. Preparation for Collegel-level maths: 3a. General prep for high level maths: How to Study as a Mathematics Major - Alcock 3b. Proof writing: How to Prove It - A Structured Approach - Velleman OR Book of Proof (2nd ed) - Hammack (it's free!) 4. Mathematical Analysis: 4a. Good prep for Analysis: How to Think About Analysis - Alcock 4b. Understanding Analysis (2nd ed) - Abbott OR Yet Another Introduction to Analysis - Bryant (has full solutions) OR The How and Why of One Variable Calculus - Sasane OR Mathematical Analysis - A Straightforward Approach (2nd ed) - Binmore (has full solutions) 5. Discrete Mathematics (a combination of set theory, combinatorics, a bit of discrete probability and graph theory): Discrete Mathematics - Chetwynd, Diggle 6. Linear Algebra: Linear Algebra - A Modern Introduction (4th ed) - Poole 7. Probability: Introduction to Probability - Blitzstein, Hwang + online course https://projects.iq.harvard.edu/stat110 https://projects.iq.harvard.edu/stat110 8. Statistics: (for Bayesian) Statistical Rethinking - A Bayesian Course with Examples in R and Stan - McElreath + online course https://www.youtube.com/playlist?list=PLDcUM9US4XdM9_N6XUUFrhghGJ4K25bFc https://www.youtube.com/playlist?list=PLDcUM9US4XdM9_N6XUUFr... Usually you'll be doing courses on #4, #5, and #6 simultaneously.

dvddgld 9y ago
Two quick things I can recommend without hesitation, which focus on an intuitive understanding of concepts: 1. Essence of Linear Algebra mini-series - https://m.youtube.com/watch?v=kjBOesZCoqc https://m.youtube.com/watch?v=kjBOesZCoqc 2. Better Explained website - https://betterexplained.com https://betterexplained.com YouTube has a lot of high quality math content, it definitely helped through university. It's also worth mentioning the Stanford U courses. The main takeaway I have for you is learn the concepts intuitively first, then spend the time to play around with them on paper until they sink in. Some things will be easy, some will be frustrating, much like programming you will walk away from a frustrating problem and have an epiphany while doing something completely different. All the best and have fun!
- Entalpi 9y ago
  My two cents are whenever something seems hard/impossible/infuriating/etc, take a break then seek dofferent sources on the material. A lot of times I have been hung up on something only to find that things make much more sense when approaching it from a different viewpoint. :)
  - dvddgld 9y ago
    Absolutely! Not having to hit your head against a wall helps prevent burnout as well as just plain being more effective
    - chris_wot 9y ago
      That’s how I got to grips with trigonometry... I tried to understand why sine, cosine, tangent, cotangent, secant and close can’t we’re named like they were... then I found a bunch of stuff on the unit circle. Never looked back!
  - Bizarro 9y ago
    No matter what I'm learning, I always refer to at least 3 different sources for any concept.
RossBencina 9y ago
To the above I would add: How to Think Like a Mathematician - Kevin Houston (an excellent book to read before starting) How to Read and Do Proofs - Solow The Keys to Advanced Mathematics: Recurrent Themes in Abstract Reasoning - Solow Calculus - Spivak (Actually a Real Analysis book, not a Calculus book, see e.g. https://math.stackexchange.com/questions/1811325/spivaks-calculus https://math.stackexchange.com/questions/1811325/spivaks-cal... ) Linear Algebra Done Right - Axler (Intended for a second course in Linear Algebra, but I found it helpful during my first course.) And for something from left-field: Visual Group Theory - Carter http://web.bentley.edu/empl/c/ncarter/vgt/ http://web.bentley.edu/empl/c/ncarter/vgt/ There are many many many books on every mathematics topic under the sun. Finding books that speak to you is important. I have had mixed success buying books upon other people's recommendation. You would be best to get access to a library.
- evanb 9y ago
  There's also How To Solve It by Polya. Small, accessible, casually written.
  - vram22 9y ago
    That Polya book is very good. [1] [2] I had read it many years ago. It may be influential beyond what people know. There is a version inspired by it, for programming, called How to Solve it by Computer [3], by R. G. Dromey, who, IIRC, was/is a professor at an Australian university (Wollongong?). I had the Dromey book. It is not exactly parallel to the Polya book, because it shows the details of how to come up with a solution, either in pseudocode or in a Pascal-like language, while the Polya book, IIRC, is more about principles and techniques for general problem-solving. [1] https://press.princeton.edu/titles/669.html https://press.princeton.edu/titles/669.html [2] https://en.wikipedia.org/wiki/How_to_Solve_It https://en.wikipedia.org/wiki/How_to_Solve_It [3] https://en.wikipedia.org/wiki/How_to_Solve_it_by_Computer https://en.wikipedia.org/wiki/How_to_Solve_it_by_Computer I'm pasting below the first few paragraphs from the URL [1] above: [ A perennial bestseller by eminent mathematician G. Polya, How to Solve It will show anyone in any field how to think straight. In lucid and appealing prose, Polya reveals how the mathematical method of demonstrating a proof or finding an unknown can be of help in attacking any problem that can be "reasoned" out—from building a bridge to winning a game of anagrams. Generations of readers have relished Polya's deft—indeed, brilliant—instructions on stripping away irrelevancies and going straight to the heart of the problem. First published in 1945. George Polya (1887–1985) was one of the most influential mathematicians of the twentieth century. His basic research contributions span complex analysis, mathematical physics, probability theory, geometry, and combinatorics. He was a teacher par excellence who maintained a strong interest in pedagogical matters throughout his long career. Even after his retirement from Stanford University in 1953, he continued to lead an active mathematical life. He taught his final course, on combinatorics, at the age of ninety. John H. Conway is professor emeritus of mathematics at Princeton University. He was awarded the London Mathematical Society's Polya Prize in 1987. Like Polya, he is interested in many branches of mathematics, and in particular, has invented a successor to Polya's notation for crystallographic groups. ] The John Conway mentioned is the one who invented the Game of Life. https://en.wikipedia.org/wiki/John_Horton_Conway https://en.wikipedia.org/wiki/John_Horton_Conway https://en.wikipedia.org/wiki/Conway%27s_Game_of_Life https://en.wikipedia.org/wiki/Conway%27s_Game_of_Life
PyComfy 9y ago
for linear algebra, may i add http://immersivemath.com/ila/index.html http://immersivemath.com/ila/index.html
chris_wot 9y ago
I find there are things I’m missing. Does the precalculus stuff cover geometry and polynomials?
- mikevm 9y ago
  Precalc basically covers algebra and trigonometry. From my experience as a CompSci major, euclidean geometry is not really necessary.
  - chris_wot 9y ago
    True, but CompSci isn’t the motivation of this poster.
hudibras 9y ago
> Statistics: (for Bayesian) Statistical Rethinking - A Bayesian Course with Examples in R and Stan - McElreath + online course Because it's a relative newcomer to the statistics scene, McElreath's book isn't as well known as the classic textbooks that many of us used back in the day. But it's steadily becoming one of the mainstays of graduate level statistics programs. A must-read.
hackernewsacct 9y ago
This is a great list. Thank you.
sidyapa 9y ago
Hi, OP here, thank you so much for the list and effort.
kingkongjaffa 9y ago
6,7,8 can be done alongside 2 there's no reason to leave those 4 foundation topics to late IMHO.

by mikevm 9y ago

dvddgld 9y ago

Two quick things I can recommend without hesitation, which focus on an intuitive understanding of concepts: 1. Essence of Linear Algebra mini-series - https://m.youtube.com/watch?v=kjBOesZCoqc https://m.youtube.com/watch?v=kjBOesZCoqc 2. Better Explained website - https://betterexplained.com https://betterexplained.com YouTube has a lot of high quality math content, it definitely helped through university. It's also worth mentioning the Stanford U courses. The main takeaway I have for you is learn the concepts intuitively first, then spend the time to play around with them on paper until they sink in. Some things will be easy, some will be frustrating, much like programming you will walk away from a frustrating problem and have an epiphany while doing something completely different. All the best and have fun!

Entalpi 9y ago
My two cents are whenever something seems hard/impossible/infuriating/etc, take a break then seek dofferent sources on the material. A lot of times I have been hung up on something only to find that things make much more sense when approaching it from a different viewpoint. :)
- dvddgld 9y ago
  Absolutely! Not having to hit your head against a wall helps prevent burnout as well as just plain being more effective
  - chris_wot 9y ago
    That’s how I got to grips with trigonometry... I tried to understand why sine, cosine, tangent, cotangent, secant and close can’t we’re named like they were... then I found a bunch of stuff on the unit circle. Never looked back!
- Bizarro 9y ago
  No matter what I'm learning, I always refer to at least 3 different sources for any concept.

RossBencina 9y ago

To the above I would add: How to Think Like a Mathematician - Kevin Houston (an excellent book to read before starting) How to Read and Do Proofs - Solow The Keys to Advanced Mathematics: Recurrent Themes in Abstract Reasoning - Solow Calculus - Spivak (Actually a Real Analysis book, not a Calculus book, see e.g. https://math.stackexchange.com/questions/1811325/spivaks-calculus https://math.stackexchange.com/questions/1811325/spivaks-cal... ) Linear Algebra Done Right - Axler (Intended for a second course in Linear Algebra, but I found it helpful during my first course.) And for something from left-field: Visual Group Theory - Carter http://web.bentley.edu/empl/c/ncarter/vgt/ http://web.bentley.edu/empl/c/ncarter/vgt/ There are many many many books on every mathematics topic under the sun. Finding books that speak to you is important. I have had mixed success buying books upon other people's recommendation. You would be best to get access to a library.

evanb 9y ago
There's also How To Solve It by Polya. Small, accessible, casually written.
- vram22 9y ago
  That Polya book is very good. [1] [2] I had read it many years ago. It may be influential beyond what people know. There is a version inspired by it, for programming, called How to Solve it by Computer [3], by R. G. Dromey, who, IIRC, was/is a professor at an Australian university (Wollongong?). I had the Dromey book. It is not exactly parallel to the Polya book, because it shows the details of how to come up with a solution, either in pseudocode or in a Pascal-like language, while the Polya book, IIRC, is more about principles and techniques for general problem-solving. [1] https://press.princeton.edu/titles/669.html https://press.princeton.edu/titles/669.html [2] https://en.wikipedia.org/wiki/How_to_Solve_It https://en.wikipedia.org/wiki/How_to_Solve_It [3] https://en.wikipedia.org/wiki/How_to_Solve_it_by_Computer https://en.wikipedia.org/wiki/How_to_Solve_it_by_Computer I'm pasting below the first few paragraphs from the URL [1] above: [ A perennial bestseller by eminent mathematician G. Polya, How to Solve It will show anyone in any field how to think straight. In lucid and appealing prose, Polya reveals how the mathematical method of demonstrating a proof or finding an unknown can be of help in attacking any problem that can be "reasoned" out—from building a bridge to winning a game of anagrams. Generations of readers have relished Polya's deft—indeed, brilliant—instructions on stripping away irrelevancies and going straight to the heart of the problem. First published in 1945. George Polya (1887–1985) was one of the most influential mathematicians of the twentieth century. His basic research contributions span complex analysis, mathematical physics, probability theory, geometry, and combinatorics. He was a teacher par excellence who maintained a strong interest in pedagogical matters throughout his long career. Even after his retirement from Stanford University in 1953, he continued to lead an active mathematical life. He taught his final course, on combinatorics, at the age of ninety. John H. Conway is professor emeritus of mathematics at Princeton University. He was awarded the London Mathematical Society's Polya Prize in 1987. Like Polya, he is interested in many branches of mathematics, and in particular, has invented a successor to Polya's notation for crystallographic groups. ] The John Conway mentioned is the one who invented the Game of Life. https://en.wikipedia.org/wiki/John_Horton_Conway https://en.wikipedia.org/wiki/John_Horton_Conway https://en.wikipedia.org/wiki/Conway%27s_Game_of_Life https://en.wikipedia.org/wiki/Conway%27s_Game_of_Life

PyComfy 9y ago

for linear algebra, may i add http://immersivemath.com/ila/index.html http://immersivemath.com/ila/index.html

chris_wot 9y ago

I find there are things I’m missing. Does the precalculus stuff cover geometry and polynomials?

mikevm 9y ago
Precalc basically covers algebra and trigonometry. From my experience as a CompSci major, euclidean geometry is not really necessary.
- chris_wot 9y ago
  True, but CompSci isn’t the motivation of this poster.

hudibras 9y ago

> Statistics: (for Bayesian) Statistical Rethinking - A Bayesian Course with Examples in R and Stan - McElreath + online course Because it's a relative newcomer to the statistics scene, McElreath's book isn't as well known as the classic textbooks that many of us used back in the day. But it's steadily becoming one of the mainstays of graduate level statistics programs. A must-read.

hackernewsacct 9y ago

This is a great list. Thank you.

sidyapa 9y ago

Hi, OP here, thank you so much for the list and effort.

kingkongjaffa 9y ago

6,7,8 can be done alongside 2 there's no reason to leave those 4 foundation topics to late IMHO.