Rigorous Elementary Mathematics

Existsforall Academy presents a 4-volume series 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 algebra, counting, number theory, and geometry. 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.

You can purchase paperbacks on Amazon websites or download PDFs below.

The tables of contents are provided below as a preview. 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: Algebra

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: Counting

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

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
Volume 4: Geometry

ISBN 978-1-7389501-3-3

  1. Lines
  2. Angles
  3. Transformations
  4. Dot Product
  5. Polygons
  6. Circles I
  7. Classifications
  8. Circles II
  9. Area
  10. Cevians
  11. Triangle Centers
  12. Conics
  13. Cross Product
  14. Three Dimensions