Measure a room as roughly 4 metres by 3 metres and a calculator will report the area as 12 square metres. Measure it as 4.1 by 3.2 and it reports 13.12. The second figure looks more authoritative, and both hundredths in it are invented, because the measurements behind them were only good to the nearest ten centimetres.
The output cannot be better than the input
A result carries the uncertainty of the least certain number that went into it. If a tape measure is reliable to a centimetre, no amount of arithmetic afterwards produces a millimetre-accurate answer. The digits appear because division produces them, not because they mean anything.
This is what significant figures are for. They are a way of recording how much of a number is actually known. Rounding 3.14159265 to three significant figures gives 3.14; to three decimal places it gives 3.142. Those are different operations and they answer different questions.
Where the two rules diverge
Decimal places count digits after the point. Significant figures count meaningful digits from the first non-zero one. For a number near 3 they nearly agree; for very large or very small numbers they do not agree at all.
A measurement of 0.00042 has two significant figures and five decimal places. Reporting it to three decimal places would round it to 0.000, destroying the measurement entirely. Very large and very small quantities are usually better handled in scientific notation, which makes the number of significant figures explicit, and the scientific notation converter does that conversion.
Rounding once, at the end
Rounding intermediate steps and then carrying the rounded figures forward compounds the error through the calculation. Carrying full precision through and rounding once at the end is the safer habit, which is also why spreadsheet results and hand calculations sometimes disagree in the last digit.
The rounding calculator shows decimal places, significant figures and rounding to a nearest step side by side, which makes the difference between them visible on the same number.
Precision in money and in measurement
Money is the exception that proves the rule. Currency has a genuine smallest unit, so two decimal places are not false precision, they are the actual resolution of the thing being counted. What causes trouble there is rounding the same amount repeatedly, as in the piece on adding and removing tax, where a round trip through a percentage does not return the original figure.
Stating the error instead of hiding it
The honest way to report a measurement is with its uncertainty attached, which is what percent error does: a measured 9.8 against a true value of 10 is a 2 per cent error. That single figure says more about the quality of the result than any number of decimal places, and the percent error calculator produces it from the two values.
The practical test is simple. Before writing down a digit, ask whether you could defend it if someone re-measured. If not, it is decoration.