Recommendations for Good Math Books?

[Michael DeBellis]

Sorry a bit off topic but a friend of mine has a son who is starting undergrad at Berkeley and has decided to study math. She asked me for good books on mathematics and except for Gilbert Strang whose books are the best I've seen for Linear Algebra, I couldn't really think of any. I still have all my text books for Calculus, statistics, etc. but never thought that any of them were particularly good.  If anyone has any recommendations for things like: Best book on Calculus or Statistics, Topology, etc.  Please let me know. 

Happy holidays,

Michael

https://www.michaeldebellis.com/blog

[Mayukh Bagchi]

The following texts are quite helpful @ UG level:

1. Calculus Vol. 1 and Vol. 2 by Nikolai Piskunov

2. Mathematical Analysis by Walter Rudin

3. Naive Set Theory by Paul Halmos.

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[Michael DeBellis]

Excellent! Thanks.

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[John F Sowa]

Textbooks are useful, but they only give a view of one topic at an introductory level.  I strongly recommend the four volume World of Mathematics, which has a wide range of readings on the history, developments, and applications of every branch of mathematics.  All the readings are written by professional mathematicians or by professionals in other fields who are using math for their applications. 

When it was first published in 1956, I was in 11th grade in high school, and I let "Santa Claus"  know that it would make a good present.  In those days, the 4-volume set cost $19.95.   It's still in print, and a new copy (identical to the original) has a list price over $500.  But Dover has paperback versions for less than $20 for each volume.

My father had studied chemical engineering at Carnegie Tech, and I was considering chemistry for my major.  But that reading led me to choose math at MIT, and that was an excellent basis for my future work in AI. 

John

 

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From: "Mayukh Bagchi" <mbagchi....@gmail.com>

[Michael DeBellis]

Thanks John. That sounds great. 

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[Dan Gillman]

Topics in Algebra - Herstein

Dan Gillman

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[James Davenport]

Biased, but my father's book: The Higher Arithmetic.

My workload obliges me to work, and send e-mails, outside working hours, but I don’t expect recipients to respond outside their working hours 

 

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Subject: Re: [ontolog-forum] Recommendations for Good Math Books?

 

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[Alex Shkotin]

Michael,

I couldn't resist the pleasure of asking Gemini 3.5/Thinking.[1]

Alex

[1] RAW ANSWER

This is an exciting milestone! For a math major at a top-tier university like UC Berkeley, the "standard" textbooks used in lower-division courses (like Stewart’s Calculus) are often replaced or supplemented by more rigorous, proof-based texts.

Berkeley's math department is famous for its rigor, and many of its faculty members have written the definitive books in their fields. Below is a curated list of the "gold standard" textbooks that are either frequently used at Berkeley or highly regarded by math majors for their clarity and depth.

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1. Calculus & Real Analysis (The "Transition")

In a math major’s journey, Calculus transitions into Real Analysis (the study of why calculus works).

• For Rigorous Calculus: Calculus by Michael Spivak. This is not a typical calculus book; it is an introduction to mathematical thinking. It is challenging, witty, and focuses on proofs rather than just "plug and chug" problems.

• The Berkeley Favorite: Real Mathematical Analysis by Charles Pugh. Pugh is a Professor Emeritus at Berkeley. His book is famous for its "Real Proofs" and its visual approach to complex concepts. It is often used for Berkeley’s Math 104 course.

• The "Gold Standard": Principles of Mathematical Analysis by Walter Rudin (often called "Baby Rudin"). It is incredibly dense and elegant, though beginners often find it helpful to have a "friendlier" companion like Stephen Abbott’s Understanding Analysis.

2. Linear Algebra (Beyond Strang)

While Gilbert Strang is excellent for engineers and applied math, math majors at Berkeley often move toward a more abstract, coordinate-free approach.

• The Modern Classic: Linear Algebra Done Right by Sheldon Axler. This is actually the official department-selected text for Berkeley's Math 110. It avoids using determinants until the very end to focus on the geometry and structure of linear operators.

3. Abstract Algebra

This is usually the first "true" pure math course students take, focusing on groups, rings, and fields.

• For Intuition: A Book of Abstract Algebra by Charles C. Pinter. This is a Dover book (very affordable) and is celebrated for how it eases students into abstract thinking with clear prose.

• The Comprehensive Reference: Abstract Algebra by Dummit and Foote. This is a "tome" that almost every math major keeps on their shelf for years. It covers everything and has excellent exercises.

4. Topology

Topology is often called "rubber-sheet geometry." It is a fundamental pillar of modern mathematics.

• The One and Only: Topology by James Munkres. This is universally considered the best introductory book on the subject. It is exceptionally clear and builds the subject from the ground up (Set Theory) to advanced concepts (the Fundamental Group).

5. Statistics & Probability

For a math major, the "plug-in-the-numbers" stats books are usually replaced by those that focus on the underlying probability theory.

• The Berkeley Classic: Statistics by Freedman, Pisani, and Purves. David Freedman was a legendary Berkeley professor. While it is less "math-heavy" in terms of symbols, it is considered the best book for teaching a student how to think like a statistician and avoid logical fallacies.

• For Mathematical Rigor: Statistical Inference by Casella and Berger. This is the standard for a mathematically mature undergraduate who wants to see the proofs behind the statistical methods.

6. Mathematical "Culture" & Problem Solving

To help him get into the "Berkeley headspace," these books are highly recommended:

• How to Solve It by George Pólya: A classic on the heuristics of mathematical problem-solving.

• The Art and Craft of Problem Solving by Paul Zeitz: Excellent for students who want to participate in math competitions or just sharpen their logic.

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Summary Table for Quick Reference

Subject	Recommended "Best" Book	Why it's great
Calculus	Calculus (Spivak)	Transforms calculus into a rigorous art form.
Linear Algebra	Linear Algebra Done Right (Axler)	The standard text for Berkeley's Math 110.
Real Analysis	Real Mathematical Analysis (Pugh)	Written by a Berkeley Prof; very visual and rigorous.
Topology	Topology (Munkres)	The undisputed "Bible" of introductory topology.
Abstract Algebra	Abstract Algebra (Dummit & Foote)	The definitive encyclopedic reference.
Statistics	Statistics (Freedman)	Teaches the deep intuition of the "Berkeley school."

Would you like me to look up the specific syllabus or required reading for any of his first-semester courses (like Math 1A/1B or Math 53/54) to see exactly what they are using this year?

вт, 23 дек. 2025 г. в 03:20, Michael DeBellis <mdebe...@gmail.com>:

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[Michael DeBellis]

Alex, why didn't I think of that! That's great. Thanks. We're giving him more than enough to read as it is so no need to delve deeper. Thanks also to everyone who contributed.

Michael

https://www.michaeldebellis.com/blog

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