You typed one digit. The answer came back with six.
Parallax 0.2 arcseconds in; 16.3078 light years out. The input carries one significant figure. The output displays six.
Type 0.2 into the parallax tool. It answers: 5 pc, 16.3078 light years, 1,031,325 AU.
The parallax you typed is one significant figure. The last of those answers is given to seven. Somewhere between the box and the readout, six digits of precision were created that were never measured.
What the extra digits are actually saying
They are not lying, exactly. They are answering a different question from the one you asked.
"16.3078 light years" is the correct answer to if the parallax were exactly 0.2000000, what would the distance be? That is a question about arithmetic, and the tool answers it perfectly. The question you meant was how far away is this star?, and that one carries your uncertainty with it.
For a reciprocal the translation is unusually simple. Since d = 1/p, a relative error in p becomes the same relative error in d, near enough, for small errors. So if that 0.2 really means 0.2 ± 0.01 — a decent measurement — the distance is 5 pc ± 0.25, and quoting 16.3078 ly claims to know the answer a thousand times better than you do. If it means 0.2 ± 0.05, which is what one significant figure honestly implies, the star is somewhere between 4 and 6.7 parsecs and the fourth decimal place is fiction.
The rule underneath is the one every lab course teaches and every spreadsheet forgets: a result cannot be more precise than the least precise thing that went into it. Multiplication and division carry significant figures through; addition carries decimal places. Nothing in a calculator enforces either.
Reciprocals are the dangerous case
The parallax example is worth sitting with because 1/p behaves badly in a specific direction.
Take p = 0.2 ± 0.05, so p runs from 0.15 to 0.25. Then d = 1/p runs from 6.67 pc down to 4.00 pc. The distance is not 5 ± 1.25: the error bar is lopsided, 1.67 up and 1.00 down, because the reciprocal stretches the low end of p and compresses the high end. Symmetric uncertainty in what you measured becomes asymmetric uncertainty in what you wanted.
That asymmetry gets worse as p shrinks, which is exactly where astronomy wants to go. At p = 0.02 ± 0.005 the distance runs from 40 to 66.7 pc: a stated value of 50 pc with a range spanning two thirds of itself. This is not a hypothetical hazard. It is the reason that catalogues of faint-star distances are quoted as parallaxes with uncertainties rather than as distances at all: the raw measurement has a well-behaved error and the useful number does not.
So why does this site print them?
Because the digits are doing a different job here, and it is worth being explicit about which.
These tools exist to be dragged. The reason a readout carries four decimals is so that nudging a slider produces a visible change. If the amplification readout rounded to 6.7, half the point of moving the damping slider would vanish, and the exchange rate between two inputs would be invisible. The digits are there to show motion, not to assert accuracy.
That is a defensible choice and a genuinely awkward one, because the same glyphs mean something else in a lab report. So: the digits on these pages are precise in the arithmetic sense and say nothing whatever about how well anything is known. When you copy a number off one of these tools into real work, the input's precision governs — not the readout's.
Two tools on the site take the opposite approach and are worth visiting for the contrast. The confidence interval tool refuses to give a single number at all, because its entire subject is that a range is the honest answer. And the sample size calculator makes you state the margin of error first: you specify the precision you are willing to pay for, and it tells you the cost. Both are built so that precision is an input rather than an accident of formatting.
Two digits is usually the answer
Ehrenberg's argument from 1977 has not aged: most numbers should be quoted to two significant digits, because that is roughly what people can hold in their heads and compare, and it is usually all the data supports anyway. Tables rounded to two digits get read; tables to six get skipped.
His test is a good one to apply to anything you produce. Cover the trailing digits with your thumb. If the sentence you were going to write is still true, they were never carrying meaning. They were carrying the impression of meaning, which is worse than carrying nothing.
16.3078 light years, with a thumb over it, is "about 16 light years", and that is the entire content of the measurement.
References (1)
- the case for two significant digits, made better than anyone has made it since Ehrenberg (1977). Rudiments of Numeracy. Journal of the Royal Statistical Society Series A 140(3).