Books for high school students starting on college math

Question

I am a student beginning high school from India. I have recently developed a taste for physics and mathematics. I am doing Lagrangian and Quantum Mechanics in Physics. But my mathematics is not too good. I have just finished the high school course (basic calculus and stuff). Could you help me out and suggest some books for the following topics:

1. Algebra (Linear and Abstract)
2. Analysis (Real and Complex)
3. Combinatorics
4. Geometry and Topology.
5. Number Theory.

Answer 1

If you've completed the equivalent of first-year University-level Calculus and want to learn more math for further studies in physics, the following might be helpful:

Mathematical Methods in the Physical Sciences, 3rd. by Mary L. Boas

The table of contents is here.

At the minimum you should cover this book before moving on to more advance material.

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For more advance reading material / textbooks, you might find the following also useful:

How to learn math and physics by Prof. John Baez;

The Road to Reality: a complete guide to the laws of the universe by Prof. Roger Penrose, FRS

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For textbook suggestions, there are many good ones available on the market, so it will depend on your own likes / dislikes and approach to learning / studying; but some of the well-known and recommended are:

1. Algebra:

A. Linear Algebra:

• S. Lang, Introduction to Linear Algebra, 2nd. and Linear Algebra, 3rd.

B. Abstract Algebra:

• G. Birkhoff and S. MacLane, A Survey of Modern Algebra, 3rd.

• I. N. Herstein, Topics in Algebra, 2nd.

2. Analysis:

A. Real:

• W. Rudin, Principles of Mathematical Analysis, 3rd.

B. Complex:

• W. Rudin, Real and Complex Analysis, 3rd.

• L. Ahlfors, Complex Analysis, 3rd.

3. Combinatorics:

• P. J. Cameron, Combinatorics: Topics, Techniques, Algorithms

• J. H. van Lint and R. M. Wilson, A Course in Combinatorics, 2nd.

4. Geometry and Topology:

This is a vast area with many excellent textbooks in various sub-branches.

A. Geometry:

For basic geometry I'm not sure if any "standard" / well-known work / textbook is popular or recommended.

B. Topology:

For General Topology (i.e. point-set):

J. Munkres, Topology, 2nd.

5. Number Theory

Many good introductory books exist, among them:

• G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers (a classic; possibly outdated)

• I. Niven, et al., An Introduction to the Theory of Numbers, 5th.

• W. J. LeVeque, Elementary Theory of Numbers

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Additionally, Schaum's Outlines are excellent supplements / study aids / problems and solutions to all the above subjects.

Answer 2

For Analysis, check out the books by walter rudin , u have two versions of the same book , based on sizes.

also u can try thomas's calculus by thomas finney.

If u got ample time, may be you can even try out the books by apostol.

go to complex analysis only once you are done with real analysis.

for complex analysis, i would suggest the book by L V Alfors ( First fields medal winner)

Number theory , use the book - Number theory an introductory course by Ivan Niven, Zuckerman

There are many books available for number theory in the market but this one stands apart.