by rokurooooo - 2026-09-20 - CC BY-SA 4.0 - free to share and remix
Press Ctrl+P and choose Save as PDF for a clean offline copy.
Chapter 1 - The Real Number System
Rational and irrational numbers, set and interval notation, indices, surds, and logarithms. Establishes the notation and algebraic vocabulary used throughout the book.
Full text ships in the First Edition, 20 Sept 2026.
Chapter 2 - Functions
Domains and ranges, common families of functions, transformations, composite and piecewise functions, and inverse functions. Algebraic rules and their graphical representations.
Full text ships in the First Edition, 20 Sept 2026.
Chapter 3 - Polynomials
Arithmetic operations, division, factorisation, the remainder and factor theorems, and partial fractions - needed again in integral calculus.
Full text ships in the First Edition, 20 Sept 2026.
Chapter 4 - Inequalities
Inequalities involving quadratic, cubic, rational, and absolute-value expressions.
Full text ships in the First Edition, 20 Sept 2026.
Chapter 5 - Matrices
Operations, determinants, inverses, and systems of simultaneous linear equations. Foundations of linear algebra for graphics and machine learning.
Full text ships in the First Edition, 20 Sept 2026.
Chapter 6 - Trigonometry
Angle measurement, the basic functions, graphs, identities, equations, and inverse functions.
Full text ships in the First Edition, 20 Sept 2026.
Chapter 7 - Linear Law
Transforming non-linear relationships into straight-line form: scatter plots, lines of best fit, gradients and intercepts in models.
Full text ships in the First Edition, 20 Sept 2026.
Chapter 8 - Circles and Conic Sections
Parabolas, ellipses, hyperbolas, and their parametric equations.
Full text ships in the First Edition, 20 Sept 2026.
Chapter 9 - Probability and Statistics
Permutations and combinations; the binomial, normal, Poisson, and exponential distributions.
Full text ships in the First Edition, 20 Sept 2026.
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.
Full text ships in the First Edition, 20 Sept 2026.
Chapter 11 - Hyperbolic Functions
Identities, graphs, and inverse functions, compared with the circular functions of Chapter 6.
Full text ships in the First Edition, 20 Sept 2026.
Chapter 12 - Differential Calculus
Limits and first principles, the standard rules of differentiation, rates of change and optimisation.
Full text ships in the First Edition, 20 Sept 2026.
Chapter 13 - Integral Calculus
Integration as the inverse of differentiation: methods, definite integrals, accumulated quantities and areas.
Full text ships in the First Edition, 20 Sept 2026.
Chapter 14 - Differential Equations
First-order equations and models of growth, decay, mixing, and continuous change.
Full text ships in the First Edition, 20 Sept 2026.
Chapter 15 - Sequences and Series
Convergence, binomial and power series, Taylor and Maclaurin expansions.
Full text ships in the First Edition, 20 Sept 2026.
Chapter 16 - Vectors
Two and three dimensions: operations, products, relative motion, vector-valued differentiation and integration.
Full text ships in the First Edition, 20 Sept 2026.
Chapter 17 - Graph Theory
Vertices and edges, paths, cycles, trees, planar graphs, colouring, shortest paths, spanning trees. Discrete maths for CS and networks.
Full text ships in the First Edition, 20 Sept 2026.
Chapter 18 - Logic and Proof
Propositions, truth tables, implications, quantifiers, valid arguments, fallacies, and elementary proof methods.
Full text ships in the First Edition, 20 Sept 2026.
Chapter 19 (bonus) - Computing with R
Vectors, data frames, importing and visualising data, probability distributions, one-sample tests, matrix calculations. Applies Chapters 5 and 9 computationally.
Full text ships in the First Edition, 20 Sept 2026.
Web previews (read now)
Four chapters are already readable on this site. The complete First Edition ships 20 Sept 2026.
Web preview - Arithmetic (Ch 1 material)
fundamental ideas
Arithmetic is the language of calculation.
Definition
Arithmetic is the branch of mathematics concerned with the
basic operations on numbers: addition, subtraction, multiplication, and
division. In a broader sense, it also includes exponentiation, roots, and
logarithms. Put simply, arithmetic is the part of mathematics that studies
how numbers behave under calculation.
Core operations
Arithmetic deals with how numbers can be combined, transformed, and compared.
The most important operations are addition,
subtraction, multiplication, and
division. These operations let us describe quantities,
compare values, and model change. From these basic rules, more advanced ideas
in algebra and calculus are built.
Examples
A few familiar examples are:
7 + 5 = 12
9 - 4 = 5
3 \times 6 = 18
20 \div 4 = 5
Notice that each statement gives us a different kind of relationship: combining,
removing, scaling, and distributing. Arithmetic is not merely about computing,
but about understanding how operations transform quantities.
Number Theory Foundations
Beyond basic calculation, arithmetic leads into Number Theory.
Two fundamental concepts are:
Prime Factorization: Every integer greater than 1 is either a prime
number or can be represented as a unique product of prime numbers.
For example, 60 = 2^2 \times 3 \times 5 .
GCD and LCM: The Greatest Common Divisor (GCD) is the largest
positive integer that divides each of the integers. The Least Common Multiple (LCM)
is the smallest positive integer that is divisible by both.
Modular Arithmetic
Often called "clock math," modular arithmetic deals with integers that "wrap around"
upon reaching a certain value, called the modulus.
For example, in modulo 12 (like a clock), 10 + 5 \equiv 3 \pmod{12} .
This system is the cornerstone of modern cryptography and computer science.
Why arithmetic matters
Arithmetic is often the first structured way a person learns to reason with
numbers. It teaches consistency, precision, and symbolic thinking. Without it,
topics such as algebra, geometry, probability, and calculus would not have
the same foundation.
Types of arithmetic
a) Integer arithmetic
b) Rational number arithmetic
c) Real number arithmetic
d) Approximation and estimation
Different number systems lead to different rules and subtleties. For example,
operations on whole numbers are not always identical to those on fractions or
irrational numbers, even though the basic ideas remain similar.
Properties of arithmetic
Arithmetic is governed by a few basic properties that make calculations
consistent. These include the commutative property, which says
that order does not matter in addition or multiplication, and the
associative property, which says that grouping does not matter.
There is also the distributive property, which connects
multiplication with addition:
a(b + c) = ab + ac
These properties are essential because they explain why arithmetic behaves the
way it does and why algebraic manipulation is possible.
Worked example
Suppose we want to simplify 6 × (4 + 3). Using the distributive property,
we can write:
6 × (4 + 3) = 6 × 4 + 6 × 3 = 24 + 18 = 42
This shows how arithmetic can be reorganized without changing the value.
Such transformations are central to algebra and problem solving.
Key idea
Everything else in mathematics often grows from the simple operations of
arithmetic. Even when notation becomes more advanced, the underlying idea is the
same: numbers can be transformed in consistent, meaningful ways. This is why
arithmetic remains one of the most foundational parts of mathematical thinking.
Summary
Arithmetic is the study of the basic operations on numbers and the rules that
govern them. It gives us a language for calculation, a foundation for algebra,
and the structure needed for deeper mathematical reasoning.
Web preview - Number System (Ch 1 + Ch 10 material)
foundations
The number system gives structure to all calculation.
The Number Hierarchy
Mathematics organizes numbers into sets, where each set typically expands upon
the previous one. This hierarchy is represented as:
\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R} \subset \mathbb{C}
Natural Numbers \mathbb{N}
The natural numbers are the counting numbers: \{1, 2, 3, 4, \dots\} .
They are the simplest numbers and are used for counting discrete objects.
Integers \mathbb{Z}
Integers include positive natural numbers, their negatives, and zero:
\{\dots, -2, -1, 0, 1, 2, \dots\} . They allow for the representation of
direction and magnitude (e.g., debt or temperature).
Rational Numbers \mathbb{Q}
Rational numbers can be expressed as a fraction \frac{p}{q} where p, q \in \mathbb{Z}
and q \neq 0 . Examples include \frac{1}{2} , -\frac{5}{7} , and 0.75
(which is \frac{3}{4} ).
Irrational Numbers
Irrational numbers cannot be expressed as a fraction of two integers. Their
decimal expansions are non-repeating and infinite.
Famous examples include \sqrt{2} and \pi \approx 3.14159 .
Real Numbers \mathbb{R}
Real numbers are the union of rational and irrational numbers. They correspond
to every single point on a continuous number line.
Complex Numbers \mathbb{C}
Complex numbers extend the real numbers by introducing the imaginary unit i ,
defined by the property i^2 = -1 . A complex number is written in the
form a + bi , where a and b are real numbers.
Key idea
The number system is not just a list of categories. It shows how mathematical
objects can be organized by structure, and how different kinds of numbers respond
to the operations we perform on them.
Web preview - Limits and Continuity (Ch 12 material)
calculus foundations
Limits ask what happens as a value approaches a point.
Definition
The limit of a function describes the value a function approaches
as its input gets closer and closer to some point. It is a central idea in
calculus because it captures behavior near a point without requiring exact value
at that point.
Example
Consider the function f(x) = \frac{x^2 - 1}{x - 1} . For x \neq 1 , this simplifies to
x + 1 . As x \to 1 , the function approaches 2 , even though the original
expression is undefined at x = 1 .
Continuity
A function is continuous at a point if the limit exists, the
function value exists, and they are equal. Intuitively, a continuous function
can be drawn without lifting the pen.
Why it matters
Limits and continuity are the foundation of derivatives and integrals. They allow
us to study motion, rates of change, and accumulation in a precise mathematical way.
Key idea
A limit is about approach, not necessarily equality at the point itself. This idea
is what gives calculus its power: it allows us to understand behavior near values
that may not be directly defined there.
Web preview - Differential Equations (Ch 14 material)
applied calculus
Differential equations express how quantities change.
Definition
A differential equation is an equation involving a function and
one or more of its derivatives. Such equations describe how a quantity changes
in relation to itself or to other variables.
Why they matter
Differential equations appear in physics, biology, economics, and engineering.
They can be used tomodel processes such as population growth, cooling, motion, and electrical
circuits.
Simple example
The equation dy/dx = ky describes exponential growth or decay. If k > 0, the
quantity grows; if k < 0, it decays. This is one of the simplest and most useful
models in applied mathematics.
Types
There are many kinds of differential equations, including first-order equations,
second-order equations, linear equations, and separable equations. Different
types often require different techniques for solving them.
Key idea
Differential equations are not just symbolic puzzles. They express change in a
language that can model real systems. The goal is often not only to solve them,
but to interpret what the solution means in context.
Afterword - Why Mathematics?
(missing content)
Colophon
Foundational Mathematics v0.1.0 (2026-09-20) by rokurooooo. Source chapters live at rokurooooo01.github.io. License CC BY-SA 4.0. Generated from the same HTML chapters - web and print never drift apart.