Rigorous Elementary Mathematics

Existsforall Academy presents a trilogy of books: Rigorous Elementary Mathematics, authored by Samer Seraj. The theme of the books is the rigorous proof of many results that are often learned by students without proof in elementary mathematics, specifically in algebra, counting, and number theory. The books were written for students of mathematical contests and olympiads who want to add a rigorous foundation to their understanding of the theorems that they use, and for teachers and coaches who want to solidify their own fundamentals. The cumulative number of letter-sized pages is 750. The documents are available for free from this webpage. The download links and tables of contents are provided below. To stay up to date on further projects, please sign up for the academy’s Newsletter. You might also be interested in my research Publications.

Volume 1:

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ISBN 978-1-7389501-0-2

  1. Sets and Maps
  2. Arithmetic
  3. Indexed Objects
  4. Equality and Order
  5. Special Functions
  6. Closed Forms
  7. Trigonometric Functions
  8. Complex Numbers
  9. Quadratics
  10. Polynomials
  11. Multivariable Inequalities

Volume 2:

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ISBN 978-1-7389501-1-9

  1. Addition and Subtraction
  2. Pigeonhole Principle
  3. Multiplication and Division
  4. Double Counting
  5. Algebraic Counting
  6. Principle of Inclusion-Exclusion
  7. Distributions
  8. Graph Theory
  9. Probability
  10. Classic Recursions
  11. Linear Recurrences
  12. Group Theory

Volume 3:
Number Theory

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ISBN 978-1-7389501-2-6

  1. Divisibility
  2. Primes
  3. Arithmetic Functions
  4. Modular Arithmetic
  5. Diophantine Analysis
  6. Linear Diophantine Equations
  7. Base Representations I
  8. Combinatorial Expressions
  9. Modular Exponentiation
  10. Base Representations II
  11. Modular Power Residues
  12. Special Forms of Integers
  13. Difference and Sum of Powers