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Free pilot
Mathematics written around how people learn
OpenKira is an experiment in free university-level textbooks, designed for<br>anyone who wants to learn university mathematics—independently or alongside<br>formal study. The books are free to read and use. We put out a new edition<br>every month, based on what readers tell us worked<br>and what didn't.
Open the Introduction to Proof and Analysis pilot<br>Analysis Pilot
Why this approach ↓
Version 1.2 · Free PDF · No account · Updated every month · Answers below in the library
The approach
A considered approach to managing working memory and overwhelm
OpenKira isn't here to replace teachers or the books that already work for you.<br>It's another way into the subject: something dependable to use on your own or<br>alongside a course, especially when denser texts are hard to follow.
01
One coherent path
Each topic follows a dependable sequence, reducing the decisions needed before learning can begin.
02
Reasoning made visible
Models, scaffolds, worked examples, and “spot the mistake” tasks explain why a method works.
03
A textbook that listens
We read what you send in. If something isn't clear or useful, we will try to fix it in the next monthly edition.
The library
A path that we propose
More than 100 modules planned, beginning with Introduction to Proof and Analysis<br>and Linear Algebra studied side by side, then branching towards master's-level<br>pure and applied mathematics. The note under each module shows what to study first.
01
Foundations<br>Begin these together · 2 modules
Pilot in progress<br>Introduction to Proof and Analysis
Begin alongside Linear Algebra
Textbook<br>Answers
Topics covered
A first course in writing proofs, then a first course in analysis:<br>172 sheets from logic and sets through the real numbers, sequences, and integration.
01
Sets and Logic<br>Propositions, connectives, proof methods, sets, and quantifiers
02
Functions and Equivalence Relations<br>Maps, inverses, orders, equivalence classes, and quotients
03
Integers<br>Peano axioms, induction, divisibility, unique factorisation, and congruences
04
Real Numbers<br>Rationals, field and order axioms, completeness, countability, and Dedekind cuts
05
Sequences<br>Epsilon-N limits, Cauchy sequences, Bolzano–Weierstrass, limsup and liminf
06
Series<br>Convergence tests, absolute convergence, power series, and the exponential
07
Continuity<br>Epsilon-delta limits, IVT, EVT, compactness, and uniform continuity
08
Differentiation<br>Differentiability, mean value theorem, L'Hôpital, Taylor, and convexity
09
Integration<br>Darboux and Riemann integrals, FTC, improper integrals, and Lebesgue's criterion
Planned<br>Linear Algebra
Begin alongside Introduction to Proof and Analysis
02
Core toolkit<br>Build from the two foundations · 6 modules
Planned<br>Group Theory
From Introduction to Proof and Analysis + Linear Algebra
Planned<br>Probability and Statistics
From Introduction to Proof and Analysis
Planned<br>Fourier Series, Multivariable Calculus, and Differential Equations
From Introduction to Proof and Analysis + Linear Algebra
Planned<br>Numerical Analysis
From Introduction to Proof and Analysis + Linear Algebra
Planned<br>Mathematical Computing
From Introduction to Proof and Analysis + Linear Algebra
Planned<br>Mechanics and Modelling
From Introduction to Proof and Analysis + Fourier Series, Multivariable Calculus, and Differential Equations
03
Advanced undergraduate<br>Enter the main branches · 12 modules
Planned<br>Real Analysis
From Introduction to Proof and Analysis + Fourier Series, Multivariable Calculus, and Differential Equations
Planned<br>Complex Analysis
From Introduction to Proof and Analysis
Planned<br>Measure and Probability
From Real Analysis + Probability and Statistics
Planned<br>Topology
From Real Analysis
Planned<br>Abstract Algebra
From Group Theory + Linear Algebra
Planned<br>Graph Theory
From Introduction to Proof and Analysis
Planned<br>Geometry of Curves and Surfaces
From Fourier Series, Multivariable Calculus, and Differential Equations + Linear Algebra
Planned<br>Partial Differential Equations
From Fourier Series, Multivariable Calculus, and Differential Equations
Planned<br>Dynamical Systems
From Fourier Series, Multivariable Calculus, and Differential Equations + Real Analysis
Planned<br>Optimisation
From Fourier Series, Multivariable Calculus, and Differential Equations + Linear Algebra
Planned<br>Statistical Theory and Inference
From Probability and Statistics + Real Analysis
Planned<br>Stochastic Processes
From Probability and Statistics + Fourier Series, Multivariable Calculus, and Differential Equations
04
Specialist foundations<br>Prepare for master's-level pathways · 14 modules
Planned<br>Functional Analysis
From Real Analysis + Linear Algebra
Planned<br>Fourier Analysis and Distributions
From Fourier Series, Multivariable Calculus, and Differential Equations + Real Analysis
Planned<br>Galois Theory
From Group Theory +...