Exact vs approximate¶

Level: 101 · for anyone who took a science class

One line: Significant figures are bookkeeping for what an instrument could resolve, so a number that was counted or defined has infinitely many of them — and mistaking one kind for the other is the most common error in the whole subject.

Two kinds of number wear the same clothes¶

1284 and 12.84 look like the same species. They are not, and no amount of staring at the digits will tell you which is which. The difference is in how the number got here.

Counted or defined → exact. Ballots in a box. Trials in an experiment. Inches in a foot. Sides of a triangle. These have no uncertainty, because no instrument was involved: there is no better ruler that could refine "12 inches per foot", since the 12 is what the foot means. An exact number has infinite significant figures.

Measured → approximate. A length, a mass, a voltage, a population estimate. Every one of these carries the fingerprint of the device that produced it, and the digits you write down are a statement about that device.

The test is one question:

Could a better instrument change this digit?

Yes → measured. Significant figures apply. No → exact. They do not.

What the program prints¶

Verified output of exact_vs_approximate.py — regenerated by tools/run_examples.py, never hand-typed.

1. COUNTED THINGS ARE EXACT
   Nobody 'measures' a ballot. You either have it or you don't.
     ballots cast            1,284   sig figs: infinite
     inches in a foot           12   sig figs: infinite
     trials run                  3   sig figs: infinite
   No instrument could refine any of these. There is no last
   uncertain digit, because there is no uncertain digit at all.

2. MEASURED THINGS CARRY THEIR INSTRUMENT
   The same physical length, three rulers:
         2.5 cm   2 s.f.   means 2.45 to 2.55           (a school ruler, marked in mm)
        2.54 cm   3 s.f.   means 2.535 to 2.545         (a vernier caliper)
      2.5400 cm   5 s.f.   means 2.53995 to 2.54005     (a micrometer)
   Same object. The digits describe the RULER, not the object.

3. THE TEST
   Ask: could a better instrument change this digit?
     yes -> the digit is measured, and sig figs apply
     no  -> the number is exact, and sig figs do not

4. WHY IT MATTERS IN ARITHMETIC
   Three measured lengths, averaged. The 3 is a COUNT, so it is
   exact and does not limit the answer:
     (2.51 + 2.47 + 2.54) / 3
     = 7.52 / 3
     = 2.506666666666666666666666667
     -> 2.51 cm      3 s.f., limited by the RULER, never by the 3

   Had the 3 been treated as 1 significant figure, the answer would
   collapse to 3 cm -- throwing away two digits that were really there.

5. THE TRAP
   A census figure looks counted and is not:
     '3,200,000 people'  is an ESTIMATE, 2 s.f., +/- 50,000
     '3,200,002 people'  would be a COUNT, 7 s.f., exact
   Same units, same object, same digits. Only the method differs,
   and the method is the entire question.

The digits describe the ruler¶

Section 2 above is the heart of it. One physical object, measured three ways, gives 2.5 cm, 2.54 cm, and 2.5400 cm. The object did not change. Its length did not change. What changed is who was holding what.

This is why 2.5 and 2.50 are different numbers in science and identical numbers in arithmetic. In arithmetic the trailing zero is decoration. In a lab notebook it is a claim: "I could see that place, and it was zero." Writing 2.50 when you used a ruler marked in millimetres is not tidiness; it is an overstatement of your equipment.

Why exact numbers matter in a calculation¶

Take three measured lengths and average them. You divide by 3 — and that 3 is a count, so it is exact and does not limit anything:

(2.51 + 2.47 + 2.54) / 3  =  2.50666…  →  2.51 cm

Three significant figures out, because the ruler gave three, and the 3 you divided by has infinitely many. Treat that 3 as one significant figure and the answer collapses to 3 cm, discarding two digits that were genuinely measured. The rule is not "the smallest number in the expression wins" — it is "the least precise measurement wins", and exact numbers are simply not in the contest.

Conversion factors work the same way. Multiplying by 2.54 cm/inch (exact by definition since 1959) costs you nothing; multiplying by a measured density costs you whatever that density's precision was.

The trap¶

A number can look counted and be estimated. This is the one worth remembering, because it is how a reasonable person gets it wrong:

Reported	How it was produced	Significant figures
3,200,000 people	statistical estimate	2 — ±50,000
3,200,002 people	an actual head count	7 — exact

Same units, same object, nearly the same digits. Only the method differs, and the method is the entire question. When you meet a number in the wild, the useful reflex is not "how many digits does it have?" but "who produced this, and with what?"

The reverse trap is just as live. Vote totals, inventory counts, and row counts in a database are exact — reporting a turnout of 52.3% from exact counts is a display rounding, not a measurement claim, and there is no uncertainty to propagate. But a simulated or sampled figure that looks equally crisp may carry a real confidence interval that the decimals conceal.

Run it yourself¶

From the root of your clone of this repository:

python3 01_Precision/exact_vs_approximate/examples/exact_vs_approximate.py