Rigorous Elementary Mathematics

In mathematical contests and olympiads, there are four subject areas: algebra, counting, number theory, and geometry. These areas are addressed in the four books of the Rigorous Elementary Mathematics series, authored by Samer Seraj. While there are many books that are geared towards helping students to learn how to solve problems on contests, those books tend to assume prerequisite knowledge without proofs, and they instead focus on applying theorems to problems. This 4-volume series fills the gap in the literature by proving many of the relevant theorems in a logically sequenced framework from the ground up. Throughout this series, three ideas are reinforced: writing the same object in multiple ways, breaking up equality using antisymmetry, and using equivalence relations. It is suggested that the books be read by those who have experience with mathematical proofs and problem-solving. The ideal audience consists of teachers of mathematics who want to solidify their own fundamentals, and outstanding students who want a second look at the material.

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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

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