# Significant Figures: Rounding destroys information

Sunday December 12, 2021

Significant Figures often requires rounding so that the correct level of uncertainty is conveyed. Rounding is only supposed to happen at the “end” of calculations, and results can vary depending on what is considered an intermediate vs. a final result. In all cases, rounding is harmful and only necessary because of the limitations of Significant Figures.

It should be clear that 4.3 with an uncertainty \( \sigma \times 0.1 \) is not the same as 4.34 with the same uncertainty. It's because Significant Digits can't say 4.34 without meaning the uncertainty is \( \sigma \times 0.01 \) that we round to 4.3, if the uncertainty is \( \sigma \times 0.1 \). But this has a cost: good information is dropped.

Consider adding 1 + 1.4 + 1.4. If we do it in one go, we get 3.8 and then round to 4 for significance. But what if this is done in two steps? Maybe one team does 1 + 1.4, correctly reports their result as 2, and then a second team builds on that, adding 1.4 to get 3. The rounding that Significant Figures requires degrades the quality of results.

At some point it isn't worth tracking every digit in a result, but Significant Figures often encourages dropping too many. It may even give people the incorrect idea that if our uncertainty suggests two significant figures, we can't have three figures in our best guess at what the value is. We absolutely can, but this requires a system more expressive than Significant Figures to report.