MindMax2000
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I'm curious to learn more maths at uni level for the sake of learning, and that it will help me in the sort of scientific field I want to go into.
Although I appreciate most of the maths in science will be applied maths, I want to learn more about the problem solving skills mathematicians generally use in case they are useful in the research I want to go into (essentially, I'm looking to do the work for the equivalent of a maths undergrad and post grad degrees without getting the creditation of one).

Firstly, what books do you recommend someone who has studied A Level Maths, should start off with at uni level?
If I go into Further Maths at A Level, which books should I not look into within the list?

I'm currently looking for uni level math courses on platforms like udemy, but like anything, the courses can vary in quality and level of difficulty.

The books that were recommended to start from basic to advanced from an American YouTuber were the following:
Discrete Maths with Applications by S Epp
Discrete Maths Structures by Kolman, Bubsy, & Ross
Mathematic Proofs by Chartrand, Polimeni, & Zhang
Abstract Maths by Bond and Keane
Pre-Algebra by AGS
Pre-Algebra by Fearon
College Algebra by Kaufmann
College Algebra by Blitzer
Graphical Approach to Algebra and Trigonometry by Hornsby, Rockswold, and Lial
Calculus by Stewart
Calculus by Spival
A first course in differential equations by Zill
Ordinary differential equations with applications by Andrews
Elementary linear algebra by Anton
Linear Algebra by Friedberg, Insel, & Spence
Mathematical statistics with applications by Mendenhall, Scheaffer, & Wackerly
A first course in probability by Ross
Fundamentals of Complex Analysis by Saff and Snider
Complex Variables and Applications by Brown and Churchill
Analysis 1 and 2 by Terrance Tao
Principles of Mathematical Analysis by Rudin
Advanced Calculus by Fitzpatrick
Elementary Analysis by Ross
Abstract Algebra by Saracino
Contemporary Abstract Algebra by Gallen
Introdction to Topologu nu Gamelin and Greene
Applied Cominatronics by Tucker
Naive Set Theory by Holmo
Functional Analysis by Kreyszig
Graph Theory by Gould
Real Analysis by Royden
Real and Complex analysis by Rudin
Linear Algebra by Hoffman and Kunze
Linear Algebra by Lang
Algebra by Artin
Calculus Made Easy by Thompson
Geometry by Jurgensen
Linear Algebra by Schaum
Finite Differential Vector Spaces by Malmor
Linear Algebra by Strang

Have you used any of the above books, and would you recommend any of them? Do you have recommendations of your own in addition or in place of the above?

Thanks
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artful_lounger
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A fairly extensive resource on such matters is the Chicago Undergraduate Mathematics Bibliography, made by several mathematics students (then later PhDs) at the University of Chicago, where they surveyed various textbooks and wrote their thoughts on the relative strengths of each. You can find it here: https://www.ocf.berkeley.edu/~abhishek/chicmath.htm

I would also note that there is a lot of overlap among the above books in terms of the topics they cover, and you really wouldn't need (or want) to go through them all in order. For example for analysis you could probably do well enough starting with Spivak, then going to either/both Rudin books. Also one not listed which I've heard a lot of good things about is Linear Algebra Done Right by Sheldon Axler. I never used it as such but looked through it after I was no longer doing maths-y things and it looked pretty nicely done, which since I generally never found a LA book that I liked before seemed notable to me.
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zetamcfc
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(Original post by MindMax2000)
The books that were recommended to start from basic to advanced from an American YouTuber were the following:
Discrete Maths with Applications by S Epp
Discrete Maths Structures by Kolman, Bubsy, & Ross
Mathematic Proofs by Chartrand, Polimeni, & Zhang
Abstract Maths by Bond and Keane
Pre-Algebra by AGS
Pre-Algebra by Fearon
College Algebra by Kaufmann
College Algebra by Blitzer
Graphical Approach to Algebra and Trigonometry by Hornsby, Rockswold, and Lial
Calculus by Stewart
Calculus by Spival
A first course in differential equations by Zill
Ordinary differential equations with applications by Andrews
Elementary linear algebra by Anton
Linear Algebra by Friedberg, Insel, & Spence
Mathematical statistics with applications by Mendenhall, Scheaffer, & Wackerly
A first course in probability by Ross
Fundamentals of Complex Analysis by Saff and Snider
Complex Variables and Applications by Brown and Churchill
Analysis 1 and 2 by Terrance Tao
Principles of Mathematical Analysis by Rudin
Advanced Calculus by Fitzpatrick
Elementary Analysis by Ross
Abstract Algebra by Saracino
Contemporary Abstract Algebra by Gallen
Introdction to Topologu nu Gamelin and Greene
Applied Cominatronics by Tucker
Naive Set Theory by Holmo
Functional Analysis by Kreyszig
Graph Theory by Gould
Real Analysis by Royden
Real and Complex analysis by Rudin
Linear Algebra by Hoffman and Kunze
Linear Algebra by Lang
Algebra by Artin
Calculus Made Easy by Thompson
Geometry by Jurgensen
Linear Algebra by Schaum
Finite Differential Vector Spaces by Malmor
Linear Algebra by Strang

Have you used any of the above books, and would you recommend any of them? Do you have recommendations of your own in addition or in place of the above?

Thanks
I like The Math Sorcerer (for those who don't know https://www.youtube.com/channel/UCr7...3PZnBw3bezl-Mg ) he's quite fun. I think starting with, Naive Set Theory by Halmos is a gentle introduction. Then as the prophet of Spivak above said, you could use Spivak's Calculus then continuing with Baby and Papa Rudin. Although I hear that the Terry Tao analysis books are good. Never used a book for LA but I assume most of them are ok. For Algebra I prefer R.B.J.T. Allenby's book Rings, Fields and Groups.
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MindMax2000
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(Original post by zetamcfc)
I like The Math Sorcerer (for those who don't know https://www.youtube.com/channel/UCr7...3PZnBw3bezl-Mg ) he's quite fun. I think starting with, Naive Set Theory by Halmos is a gentle introduction. Then as the prophet of Spivak above said, you could use Spivak's Calculus then continuing with Baby and Papa Rudin. Although I hear that the Terry Tao analysis books are good. Never used a book for LA but I assume most of them are ok. For Algebra I prefer R.B.J.T. Allenby's book Rings, Fields and Groups.
I'm surprised you were able to tell so quickly. How did you managed to deduce the above? Google search? You've seen the video recently?

Thanks for the recommendations.

Just out of interest, what do you also think of the following YouTube channels?:

  • Numberphile
  • Eddie Woo
  • Stand-Up Maths
  • NancyPi
  • Krista King
  • BlackPenRedPen
  • MindYourDecisions
  • 3Blue1Brown
  • Mathologer
  • Zach Star
  • Michael Penn
  • Tibees


I'm trying to find more examples of stuff and problems you will get at undergrad or postgrad level. Although I will be doing A Level maths and further maths, I want to look into things that are more advanced as well.
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zetamcfc
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(Original post by MindMax2000)
I'm surprised you were able to tell so quickly. How did you managed to deduce the above? Google search? You've seen the video recently?

Thanks for the recommendations.

Just out of interest, what do you also think of the following YouTube channels?:

  • Numberphile
  • Eddie Woo
  • Stand-Up Maths
  • NancyPi
  • Krista King
  • BlackPenRedPen
  • MindYourDecisions
  • 3Blue1Brown
  • Mathologer
  • Zach Star
  • Michael Penn
  • Tibees


I'm trying to find more examples of stuff and problems you will get at undergrad or postgrad level. Although I will be doing A Level maths and further maths, I want to look into things that are more advanced as well.
Saw the video. Numberphile is good. Eddie Woo would be great as your teacher. BlackPenRedPen presents solutions well. Mathologer is quirky. Idk the rest.


I can give you some harder stuff that is on YouTube but will be way above you, though might be nice to look at.



https://www.youtube.com/channel/UCI2...SbQGb3GdKx_ixg Good lectures from the best people, some good interviews.



https://www.youtube.com/channel/UCHW...4WvE5AHmNVXziw Very good at explaining things, undergrad/graduate/research level.



https://www.youtube.com/channel/UCXl...rvjyLwAR-Xh9pQ He's a bit strange but a very good lecturer, his history of maths series is worth a look.



https://www.youtube.com/user/OpenLektorium/featured In Russian but there are some lectures in English



https://www.youtube.com/channel/UC4R...s_qlWKTm9pT82Q IHES the place in France to do mathematics.



https://www.youtube.com/channel/UC8a...weiS50pvCmo0pw IAS very good research talks.



https://www.youtube.com/channel/UCkE...QhfXqbL1btKBmw Brilliant lecturer, would recommend his Complex Analysis lectures on this and his old channel.



https://www.youtube.com/user/matsciencechannel/videos Lectures and talks.



https://www.youtube.com/channel/UC6S...Op31Vo8cQG1oXw Another good lecturer, he has the best boards I've seen with the way he arranges things.



https://www.youtube.com/channel/UCpu...MPXiqlkwrxCbNA Quality Seminars.



https://www.youtube.com/channel/UCgK...eLzge35EAZnUBQ Nice student talks and informative talks.



https://www.youtube.com/channel/UCZ_...NpVVKvfT9tuCOg Steven Miller's other channel.
Last edited by zetamcfc; 5 months ago
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0le
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Some other books:

Calculus by Strang
Calculus by Apostol
Linear Algebra by Sheldon Axler
Proofs by Velleman
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