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Stage 1
Advanced Calculus & Analysis
Rigorous treatment of calculus with proofs and abstract foundations.
Key Topics
Real number systemSequences and seriesContinuity and limitsDifferentiationRiemann integrationMetric spaces
4 books
~16 weeks
Pure Mathematics
Mathematics in its purest form
Advanced
4 stages
~52 weeks
15 books
Dive deep into abstract mathematics: analysis, algebra, and topology. For those who love math for its own beauty.
Prerequisites: Complete the foundations path first.
Skill Tree
Visualize your learning journey. Click on any stage (branch) to see the recommended books for that subject. Topics are shown as leaves on each branch.
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Learning Stages
This path is divided into 4 stages covering 23 topics. Work through each stage in order for the best learning experience.
Recommended Books for Stage 1
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Understanding Analysis
Stephen Abbott
Intermediate
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Introduction to Real Analysis
Bartle & Sherbert
Intermediate
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Principles of Mathematical Analysis
Walter Rudin
Advanced
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Advanced Calculus
Buck
Advanced
2
Stage 2
Abstract Algebra
Groups, rings, and fields - the algebraic structures underlying mathematics.
Key Topics
Group theoryRing theoryField theoryGalois theoryModule theory
4 books
~16 weeks
Recommended Books for Stage 2
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Abstract Algebra: A First Course
Dan Saracino
Beginner
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A First Course in Abstract Algebra
John B. Fraleigh
Intermediate
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Contemporary Abstract Algebra
Joseph Gallian
Intermediate
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Abstract Algebra
Dummit & Foote
Advanced
3
Stage 3
Topology
The study of spaces and their properties under continuous deformation.
Key Topics
Topological spacesContinuityConnectednessCompactnessSeparation axiomsQuotient spaces
4 books
~12 weeks
Recommended Books for Stage 3
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Introduction to Topology
Gamelin & Greene
Intermediate
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Introduction to Topology
Bert Mendelson
Intermediate
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Topology
Munkres
Advanced
Schaum's Outline of Topology
Schaum's
Intermediate
4
Stage 4
Number Theory
The study of integers and their fascinating properties.
Key Topics
DivisibilityPrime numbersCongruencesQuadratic reciprocityDiophantine equationsContinued fractions
3 books
~8 weeks
Recommended Books for Stage 4
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Elementary Number Theory
Underwood Dudley
Intermediate
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Number Theory
George Andrews
Intermediate
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Elementary Introduction to Number Theory
Long
Intermediate
All Books in This Path
Complete list of 15 recommended books across all stages. You don't need to buy them all — pick what works for you.
Understanding Analysis
by Stephen Abbott
Very wonderful text for learning basic undergraduate analysis.
Introduction to Real Analysis
by Bartle & Sherbert
Really easy book! Great supplement with alternate explanations. Highly recommended.
Principles of Mathematical Analysis
by Walter Rudin
Famous 'Baby Rudin'. Very dry, hard, rigorous. Classic that takes lots of effort.
Advanced Calculus
by Buck
The favorite! Starts with multivariable approach, includes n-dimensional space. Tons of math!
Abstract Algebra: A First Course
by Dan Saracino
Easier than Fraleigh and Gallian. Better for true beginners in abstract algebra.
A First Course in Abstract Algebra
by John B. Fraleigh
Recommended by former professor. Great book for beginners though not the easiest.
Contemporary Abstract Algebra
by Joseph Gallian
The book of examples! Has many examples not found in other abstract algebra books.
Abstract Algebra
by Dummit & Foote
Better index than Artin, great reference. Lots of math and exercises.
Introduction to Topology
by Gamelin & Greene
Awesome! Has FULL solutions to all exercises including proofs! Dover book, very affordable.
Introduction to Topology
by Bert Mendelson
Older classic book. Standard treatment. Probably quite affordable.
Topology
by Munkres
Standard book for advanced undergrads and beginning grad students. Lots of mathematics.
Schaum's Outline of Topology
by Schaum's
One of the best Schaum's! Very standard topics you'll see in courses. Great resource.
Elementary Number Theory
by Underwood Dudley
Had this one the longest. Great book, smells wonderful! Classic number theory text.
Number Theory
by George Andrews
Excellent choice for learning number theory. Any of these books will work.
Elementary Introduction to Number Theory
by Long
Another excellent option for number theory fundamentals.
What's Next?
Complete this path and continue your mathematical journey.