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High school student here...

Recently I was told I could go to the book store and pick out any math books I want. (2 or 3)

Does anyone have some good suggestions? I'm comfortable with anything involving calculus and I am currently studying linear algebra. It doesn't really even need to be a textbook just something that will help me learn and build mathematical maturity.

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Have you seen What is Mathematics? by Courant and Robbins? It is by far the best introduction to mathematics that you can get your hands on, in my opinion. Here's a short form of the table of contents for your reference:

  1. Chapter I: The Natural Numbers
  2. Chapter II: The Number System of Mathematics
  3. Chapter III: Geometrical Constructions. The Algebra of Number Fields.
  4. Chapter IV: Projective Geometry. Axiomatics. Non-Euclidean Geometry.
  5. Chapter V: Topology
  6. Chapter VI: Functions and Limits
  7. Chapter VII: Maxima and Minima
  8. Chapter VIII: The Calculus
  9. Chapter IX: Recent Developments
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  • $\begingroup$ Topology before calculus? This definitely seems worth checking out. $\endgroup$ – CaptainAmerica16 Aug 5 '18 at 18:08
  • $\begingroup$ @CaptainAmerica16 Here are the subsections in the Topology chapter: 1. Euler's Formula for Polyhedra; 2. Topological Properties of Figures; 2.1 Topological Properties; 2.2 Connectivity; 3. Other Examples of Topological Theorems; 3.1 The Jordan Curve Theorem; 3.2 The Four Color Problem; 3.3 The Concept of Dimension; 3.4 A Fixed Point Theorem; 3.5 Knots; 4. The Topological Classification of Surfaces; 4.1 The Genus of a Surface; 4.2 The Euler Characteristic of a Surface; 4.3 One-Sided Surfaces $\endgroup$ – Brahadeesh Aug 5 '18 at 18:13
  • $\begingroup$ @CaptainAmerica16 The chapter ends with an appendix: 1. The Five Color Theorem; 2. The Jordan Curve Theorem for Polygons; 3. The Fundamental Theorem of Algebra. $\endgroup$ – Brahadeesh Aug 5 '18 at 18:13
  • $\begingroup$ @CaptainAmerica16 So, definitely a very interesting set of topics :) $\endgroup$ – Brahadeesh Aug 5 '18 at 18:14
  • $\begingroup$ It's official, this will be the first book I look for. Thanks! $\endgroup$ – CaptainAmerica16 Aug 5 '18 at 18:15
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Two suggestions:

  • Discourses on Algebra, by Igor Shafarevich
  • Geometry: Euclid and Beyond, by Robin Hartshorne
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  • $\begingroup$ +1 for Hartshorne's book. A very good suggestion. $\endgroup$ – Brahadeesh Aug 5 '18 at 18:06
  • $\begingroup$ @Brahadeesh Thank you. $\endgroup$ – José Carlos Santos Aug 5 '18 at 18:07
  • $\begingroup$ You like a lot Shafarevitch José... $\endgroup$ – Isham Aug 5 '18 at 18:09
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    $\begingroup$ @Isham I like the book Discourses on Algebra a lot. $\endgroup$ – José Carlos Santos Aug 5 '18 at 18:12
  • $\begingroup$ Everyone seems to really like these suggestions. I'll keep them in mind 😊 $\endgroup$ – CaptainAmerica16 Aug 5 '18 at 18:19
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If you wanna study linear algebra, then I'd like to suggest - I) Linear Algebra (Friedberg, Insel, Spence) II) Introduction to Linear Algebra (Gilbert Strang) III) Linear Algebra (Kwak and Hong) IV) Linear Algebra (Kenneth Hoffman, Ray Kunze)

For abstract algebra, go for - I) A First Course in Abstract Algebra (J.B. Fraleigh) II) Abstract Algebra (Dummit and Foote) III) Contemporary Abstract Algebra (Joseph A. Gallian)

For Analysis, study - I) Mathematical Analysis (Tom M. Apostol) II) Real and Complex Analysis (Walter Rudin) III) Real Analysis (Royden , Fitzpatrick) IV) Introduction to Real Analysis (Bartle, Sherbert)

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  • $\begingroup$ Thank you! Abstract algebra is high on my list of further interests. $\endgroup$ – CaptainAmerica16 Aug 5 '18 at 18:11
  • $\begingroup$ I'd say that Rudin requires quite a bit of mathematical maturity in the reader. I wouldn't recommend it to a high school student... $\endgroup$ – Brahadeesh Aug 5 '18 at 18:16
  • $\begingroup$ Then what should he study for analysis ?? @Brahadeesh $\endgroup$ – Anik Bhowmick Aug 5 '18 at 18:26
  • $\begingroup$ Perhaps Tao's Analysis I and II, or Sherbert and Bartle's Introduction to Real Analysis would be my recommendations. But Apostol is good too! $\endgroup$ – Brahadeesh Aug 5 '18 at 18:29
  • $\begingroup$ Yeah, thanks @Brahadeesh, I forgot to mention Bartle-Sherbert. $\endgroup$ – Anik Bhowmick Aug 5 '18 at 18:30
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Vector Analysis by Murray Spiegel is one good book with lots of exercises. Also I googled "Cambridge University Maths Reading List" and found this. I am sure others exist if you look.

I read Fraleigh's "First Course in Abstract Algebra" when I was at school, and found it accessible and good.

Körner's "Calculus for the ambitious" is a quirky approach - but geared to building understanding in the bridge between "calculus" at school and "analysis" at university.

And Feynman's Lectures on Physics are worth reading, too.

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  • $\begingroup$ +1 For the reading list. Thanks :) $\endgroup$ – CaptainAmerica16 Aug 5 '18 at 19:01
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V. Prasolov's Problems and Theorems in Linear Algebra will not only teach you everything you need to know about linear algebra, but also give you germs of ideas that will prove very useful later if you study more advanced mathematics where linear algebra occurs in a way or another.

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  • $\begingroup$ Wow, thank you so much for the link! This will be really helpful. $\endgroup$ – CaptainAmerica16 Aug 5 '18 at 18:23
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I have a favorite author. Peter J. Cameron. I love the way he tells the math-story for you. He has some lecture notes that you may find on his homepage for both linear algebra and abstract algebra.

Beside that I recently faced a cute book for Linear Algebra which I highly recommend. The approach of the book is based on Linear Maps (and not matrices) and try to not use Determinant concept directly. The name of the book is "linear algebra done right".

Also, for real analysis I recommend to read Real Mathematical Analysis by Charles Pugh. The book is compact and lovely with really nice exercises.

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    $\begingroup$ Thank you for the recommendations. The lecture notes are perfect :) $\endgroup$ – CaptainAmerica16 Aug 6 '18 at 16:05

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