Math — Learning Library¶
A learning library for mathematics, built the same way as its siblings rust-learning-library ↗ and star-voting-library ↗: one idea per page, and every claim backed by a program that actually runs.
No page here hand-types what a program prints. Each lesson links a real .py file; a tool runs it, checks the output against a recorded answer key, and pastes that verified output into the page. CI fails if any of the three drift apart. So when a page says "this prints 3.2E+6", that is not a promise — it is a test result.
The examples are stdlib-only, on purpose. A library about how much of a number is real should not open with a dependency-resolution failure. If you have python3, you can run every page in this repo.
📖 Read it as a site: https://masiarek.github.io/math-learning-library/
Start here¶
01_Precision/ — How much of this number is real?
The first chapter is a single argument in four steps, and it starts from a question that sounds like it has an obvious answer and does not: are significant figures just rounding?
| Lesson | What it teaches |
|---|---|
| Exact vs approximate | Which numbers have significant figures at all — and why a counted thing has infinitely many |
| Significant figures | What the notation claims, and why the rule for + is a different rule from the rule for × |
| Uncertainty propagation | The rigorous version those rules approximate, and the two places they lie |
| Catastrophic cancellation | The one operation that destroys ten significant figures at once, silently |
Read them in that order; each one answers a question the previous one raises.
02_Measure_Zero/ — How can infinitely many points take up no room?
The second chapter is about sets so thin that they have zero length, area or volume, even when they hold infinitely many points — even uncountably many. It does not need chapter 1. On the way it meets a set that is full of gaps and still fills half the room, and a staircase that climbs from 0 to 1 while standing still almost everywhere.
| Lesson | What it teaches |
|---|---|
| What measure zero means | The definition, which never measures the set itself — and why a segment has no area but does have length |
| Countable sets | Why the rationals, found inside every interval, still have measure zero |
| The Cantor set | Uncountably many points, and total length zero |
| The fat Cantor set | No interval inside it, yet length 1/2 — and why that breaks the Riemann integral |
| The Cantor function | The devil's staircase: continuous, flat almost everywhere, and still climbing from 0 to 1 |
| Probability zero | Why probability zero is not impossible, and why on a computer it is |
Why a math library and not a Python one¶
The code here is the illustration, never the subject. Mathematics is exact — 1/3 is exactly one third, forever — and that is precisely why a chapter on significant figures has to explain that they are not a mathematical idea but a measurement one, living in the gap between the world and the arithmetic.
Python earns its place because it happens to encode several of these distinctions in syntax you can run: :.2f versus :.2g is decimal places versus significant figures, and decimal.getcontext().prec is the only knob in the standard library that counts significant digits. Those are good teaching instruments. They are not the lesson.
How the library works¶
01_Precision/
significant_figures/
README.md the lesson (prose + a generated output block)
examples/
significant_figures.py the program a reader can run
significant_figures.out its recorded output — the answer key
A lesson page never pastes output by hand. It marks the spot:
and tools/run_examples.py fills it from a real run. Inside the markers is generated; outside is yours.
python3 tools/run_examples.py # verify, and refill the pages
python3 tools/run_examples.py --update # accept current output as the answer key
python3 tools/run_examples.py --check # write nothing, fail on drift (what CI runs)
There is a second block kind, source:, which pastes the program itself for pages where the code is the lesson.
Conventions for anyone writing a page: CONTRIBUTING.md. What is planned and deliberately not written yet: ROADMAP.md.