textbook · v0.1.0 · free & open

Foundational Mathematics

by rokurooooo · first release

A small textbook that grows from counting to change.

Foundational Mathematics — First Edition spans 19 chapters plus a bonus: from the real number system, functions, and algebra, through trigonometry, calculus, and differential equations, to vectors, graph theory, logic, and computing with R. Four chapters are already readable as web previews below; the complete text ships in the print edition and PDF on 20 Sept 2026.

This is v0.1.0 — releasing 20 Sept 2026. Free to read, share, remix under CC BY-SA 4.0.

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20 Sept 2026, 00:00 (Malaysia time). After that this box flips to “Released” by itself — no update needed.

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Tip: open the print edition (all chapters, one page) and press Ctrl+P to Save as PDF any time.

Contents

  1. Chapter 1 - The Real Number System
  2. Chapter 2 - Functions
  3. Chapter 3 - Polynomials
  4. Chapter 4 - Inequalities
  5. Chapter 5 - Matrices
  6. Chapter 6 - Trigonometry
  7. Chapter 7 - Linear Law
  8. Chapter 8 - Circles and Conic Sections
  9. Chapter 9 - Probability and Statistics
  10. Chapter 10 - Complex Numbers
  11. Chapter 11 - Hyperbolic Functions
  12. Chapter 12 - Differential Calculus
  13. Chapter 13 - Integral Calculus
  14. Chapter 14 - Differential Equations
  15. Chapter 15 - Sequences and Series
  16. Chapter 16 - Vectors
  17. Chapter 17 - Graph Theory
  18. Chapter 18 - Logic and Proof
  19. Chapter 19 (bonus) - Computing with R

Synopses for every chapter are listed below. Web previews available now: Arithmetic · Number System · Limits and Continuity · Differential Equations · Why Mathematics?

Web preview — read now

Four chapters are already live on this site. The complete 19-chapter First Edition ships in the print edition and PDF below.

Ch. 1complete

Arithmetic

Operations, properties, primes, modular arithmetic.

open
Ch. 2complete

Number System

Naturals to complex numbers and why each step matters.

open
Ch. 3complete

Limits and Continuity

The bridge between algebra and calculus.

open
Ch. 4complete

Differential Equations

Growth, decay, motion, models of change.

open

Read the book (PDF)

The complete First Edition lives in one PDF, typeset in LaTeX. It opens right here on this page — or grab it for offline reading.

Your browser cannot preview PDFs here. Download the PDF instead.
  1. Download: foundational-mathematics.pdf (always the latest First Edition build).
  2. Web previews: four chapters are also readable as pages — Arithmetic · Number System · Limits and Continuity · Differential Equations.
  3. Print edition: single-page HTML → Ctrl+P → Save as PDF, if you ever want a browser-printed copy.

Chapter synopses

  1. Chapter 1 — The Real Number System. Rational and irrational numbers, set and interval notation, indices, surds, and logarithms — the notation and algebraic vocabulary used throughout the book.
  2. Chapter 2 — Functions. Domains and ranges, common families, transformations, composite and piecewise functions, and inverses — the correspondence between algebraic rules and graphs.
  3. Chapter 3 — Polynomials. Arithmetic, division, factorisation, the remainder and factor theorems, and partial fractions (needed again in integral calculus).
  4. Chapter 4 — Inequalities. Quadratic, cubic, rational, and absolute-value inequalities.
  5. Chapter 5 — Matrices. Operations, determinants, inverses, and systems of simultaneous linear equations — foundations of linear algebra for graphics and machine learning.
  6. Chapter 6 — Trigonometry. Angle measure, the basic functions, graphs, identities, equations, and inverses.
  7. Chapter 7 — Linear Law. Turning non-linear relationships into straight-line form: scatter plots, best fit, gradients and intercepts in models.
  8. Chapter 8 — Circles and Conic Sections. Parabolas, ellipses, hyperbolas, and parametric equations.
  9. Chapter 9 — Probability and Statistics. Permutations and combinations; the binomial, normal, Poisson, and exponential distributions.
  10. Chapter 10 — Complex Numbers. Algebraic, polar, exponential, and geometric forms, with De Moivre's theorem and Euler's formula — numbers as points with geometric structure.
  11. Chapter 11 — Hyperbolic Functions. Identities, graphs, and inverses, alongside comparisons with the circular functions of Chapter 6.
  12. Chapter 12 — Differential Calculus. Limits and first principles, the standard rules, and applications to rates of change and optimisation.

Chapter synopses, continued

  1. Chapter 13 — Integral Calculus. Integration as the inverse of differentiation: methods, definite integrals, accumulated quantities and areas.
  2. Chapter 14 — Differential Equations. First-order equations and models of growth, decay, mixing, and continuous change.
  3. Chapter 15 — Sequences and Series. Convergence, binomial and power series, Taylor and Maclaurin expansions.
  4. Chapter 16 — Vectors. Two and three dimensions: operations, products, relative motion, vector-valued differentiation and integration.
  5. Chapter 17 — Graph Theory. Vertices and edges, paths, cycles, trees, planar graphs, colouring, shortest paths, spanning trees — discrete maths for CS and networks.
  6. Chapter 18 — Logic and Proof. Propositions, truth tables, implications, quantifiers, valid arguments, fallacies, and elementary proof methods.
  7. Chapter 19 (bonus) — Computing with R. Vectors, data frames, importing and visualising data, probability distributions, one-sample tests, and matrix calculations — applying Chapters 5 and 9 computationally.

Afterword: Why Mathematics? — a personal reflection on why this book exists.

Cite and reuse

License: CC BY-SA 4.0 for the book text. Site code: MIT (see LICENSE).

rokurooooo. Foundational Mathematics, v0.1.0. 2026-09-20. https://rokurooooo01.github.io/foundational-mathematics.html

Release

First Edition - First release of the book. 19 chapters plus a bonus chapter on computing with R. v0.1.0, 20 Sept 2026.

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