01_Precision — how much of this number is real?

Level: 101 → 301 · for anyone who took a science class

Every number you meet outside pure mathematics arrived by some process, and that process set a limit on how much of it you may believe. This chapter is about finding that limit and respecting it.

It is one argument in four steps. Each lesson answers a question the previous one raises, so they are worth reading in order.

#	Lesson	The question it answers
1	Exact vs approximate	Which numbers have significant figures at all?
2	Significant figures	What does the notation claim, and is it just rounding?
3	Uncertainty propagation	What are those rules an approximation of?
4	Catastrophic cancellation	Where do the digits go when they go all at once?

The through-line

Mathematics is exact. 1/3 is exactly one third, 3,200,000 + 2 is exactly 3,200,002, and no theorem in this chapter says otherwise. Everything here lives in the gap between mathematics and the world — the gap opened by measuring something rather than counting or defining it.

That gap has a well-developed theory, and significant figures are its introductory dialect: a rule you can apply with no calculus and no error bars, taught in one class period. Lesson 3 is the real thing the dialect was translating, and lesson 4 is the case where every version of the rule, dialect and original alike, has to concede that the digits are simply gone.

The chapter ends in numerical analysis, which is where this stops being a science-class topic and becomes mathematics again: conditioning is a property of a problem, stability a property of an algorithm, and telling them apart is a genuine skill with real consequences for code.

A note on the code

The examples are Python, stdlib-only, and they are illustrations rather than the subject. Python is used here because it happens to encode two of this chapter's distinctions in runnable syntax — :.2f versus :.2g, and decimal.getcontext().prec — which makes them things you can try instead of things you have to take on faith.