Chasing a number to its source, and finding where the precision stops

You see a number somewhere. "This model is 94.3% accurate." "Pi is 3.14159265358979323846264338327950288." "The atmosphere is predictable out to two weeks."

We tend to treat digits as one thing: more is better. It isn't. Past some point, the extra digits stop describing anything real and start describing the machine that printed them. I've spent the last few weeks chasing numbers back to where they were first computed, and the interesting part is never the number. It's the last digit that still means something.

Three cases, briefly.

Pi runs out at 37 digits, not 39

The popular claim is that NASA needs 39 digits of pi. It's folk knowledge that gets copied forward. The real figure comes from Marc Rayman at JPL, who worked it out for the Voyager program: 15 digits gets you to about the width of a hair, and 37 digits gets you the circumference of the observable universe. The extra two aren't a safety margin. 39 is simply what a 64-bit double hands you for free. Precision stopped being a decision and became a side effect of the float type.

Weather runs out at about two weeks

Forecasts keep getting better, but the wall isn't computing power. The atmosphere has its own predictability horizon, first put at roughly two weeks (Lorenz, 1965; Charney and colleagues, 1966). More sensors and bigger computers move it a little, not a lot, because the limit lives in the system, not the instrument. Asking for next month's weather isn't a precision request. It's a different kind of question.

Avogadro's number has no meaningful digits at all

When the mole was redefined in 2019 by fixing Avogadro's constant, 6.02214076 × 10²³ stopped being a measurement and became a definition. The digits are exact because we chose them. There is nothing left to measure. Zero of those digits carry information about the world.

The rule under all three

What sets the point where more precision stops being worth its cost?

Not the instrument's best. Not cost alone. It's the coarsest downstream link whose outcome the extra digit can still change.

If the number feeds a decision, ask where downstream an extra digit could flip the answer. For a yes/no call, one digit may be plenty. For a tolerance on a machined part, maybe three. Everything past that is decoration. Worse, it's confidence you didn't earn.

The failure mode isn't too few digits. It's too many on the wrong thing. A 2026 audit of EEG foundation models (arXiv:2606.06647) found that the dominant axis of the frozen representation is subject identity, not the task: more precision along that axis doesn't improve decoding, it amplifies the wrong quantity. Subject-disjoint cross-validation, their verdict goes, is necessary but not sufficient. A high accuracy number can be precise and useless in the same breath.

The same thing happens to accuracy scores. "94.3%" means nothing without the split, without the overlap between train and test, without a baseline. The floating-point digit is real. The meaning behind it usually isn't.

Three questions for any number you meet

1. Where was it first computed, not where it's quoted?
2. What's the last downstream decision an extra digit could still change?
3. What breaks if that digit moves?

If you can't answer the first, the number is a rumor. If you can't answer the second, the precision isn't for you. It's for whoever wanted to look careful.

---

I chase numbers like this now, as work. If you have a number you half-trust, one you've quoted in a slide or a README but never verified where it came from, send it to me. I'll trace it to its first computation and tell you exactly where its meaning stops. The method is the thing I offer, priced small: reply here or email me at [e-mail]. First serious one is free.