Teaching Yourself Mathematics: A Guide from Start to Finish

Introduction

• Video outlines how to self-study mathematics from the basics to advanced levels
• Focus on learning proofs, logic, and discrete mathematics as a foundation
• Recommended order of learning and resources/books to use

Starting Point: Discrete Mathematics and Proofs

• Books Recommended:
  • Discrete Mathematics with Applications by Susannah Epp
    • Good beginner book, no algebra required
    • Covers logical implications, truth tables, and basic mathematical logic
  • Discrete Mathematics and its Applications by Coleman, Busby, and Ross
    • Beginner-friendly, covers logic, sets, and proof writing
• Proof Writing Books:
  • Mathematical Proofs: A Transition to Advanced Mathematics by Chartrand, Polimeni, and Zhang
  • The Art of Proof: A Companion for Discrete Mathematics by Bond and Keane
    • Teach proof writing, key skill for math majors

Pre-Algebra and College Algebra

• Pre-Algebra Books:
  • AGS Pre-Algebra
  • Fieron's Pre-Algebra
• College Algebra:
  • College Algebra by Kaufmann
  • College Algebra by Blitzer
  • Courses include intermediate algebra but focus more on problem-solving

Precalculus and Trigonometry

• Recommended Book:
  • A Graphical Approach to Algebra and Trigonometry by Hornsby, Lial, and Rockswold
    • Covers conic sections, matrices, and algebra problems

Calculus

• Books Recommended:
  • Calculus by James Stewart
    • Covers Calculus I, II, and III
  • Calculus by Michael Spivak
    • More advanced, requires knowledge of logic and proof writing

Differential Equations

• Books Recommended:
  • Differential Equations by Dennis Zill
  • Ordinary Differential Equations by Larry Andrews
    • Requires understanding of integration methods

Linear Algebra

• Books Recommended:
  • Elementary Linear Algebra by Howard Anton
  • Linear Algebra by Friedberg, Insel, and Spence
    • Proof-based learning

Advanced Mathematics

• Subjects to Explore:
  • Mathematical Statistics: Requires calculus knowledge
  • Complex Variables/Complex Analysis
  • Real Analysis: Considered challenging, requires understanding of proofs
  • Abstract Algebra: Study of groups, rings, and fields
  • Topology
  • Combinatorics
  • Set Theory
  • Functional Analysis
  • Graph Theory

Graduate-Level and Specialized Mathematics

• Books:
  • Real Analysis by Royden
  • Real and Complex Analysis by Rudin (Papa Rudin)
  • Finite Dimensional Vector Spaces by Paul Halmos

Additional Resources

• Additional Books:
  • Linear Algebra by Serge Lang
  • Calculus Made Easy by Thompson
  • Geometry and other supplemental texts

Conclusion

• Encourage self-study, explore interests, and enjoy learning
• Mathematics is challenging but rewarding
• It's okay to not understand everything immediately and move on to other topics
• The aim is to enrich mathematical knowledge and pursue enjoyment in learning