Are Significant Digits Unit Dependent?

A while back, I had a discussion with our development team about significant figures.  Specifically, the discussion centered around where they were going to go to lunch and when they were going to leave.  Someone piped in with, “Right here, right now,” while another countered, “I wonder how many significant digits you should take for that to actually be true…”  The first dev then stated, “First, you would need some units,” at which point I jumped into the conversation claiming, “Actually, I’m not sure that you do.  Whether you are using AU or attometers, the number of significant digits should be the same.”

The conversation then turned to whether significant digits were unit dependent.  Keeping in mind the absurdity of the conversation, and the origin that it was spawned from, I actually found the concept really interesting.  Are significant digits unit dependent?

The claim that they are not seemed intuitively true to me, but I didn’t have a proof ready to go.

Category(s): Pseudorandom
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2 Responses to Are Significant Digits Unit Dependent?

  1. Significant digits are roughly a rule of thumb, and occasionally you’ll get slightly different numbers of significant figures. This is true for all units that are different by a multiplicative factor.

    Units like Kelvin that are different from common units by an additive factor have different numbers of significant figures. 1 degree Celsius is a different level of significance than 300 Kelvin. To keep the same rough accuracy, which is the intention of significant figures, you’d need at least 3 significant figures in the Kelvin.

    Best thing to do is to carry your uncertainty around in original units and propogate conversions into the uncertainty properly.

    For time you’re probably right. If you measure time since 0 AD in seconds you get the same number of significant figures if you measure it in decimal years or decimal millennia for that matter. If you shift the baseline, then it all gets screwed up.

      The Plaid Mentat says:

      Adam – thanks for the comment! I suspected as much about multiplicative vs additive factors and I’m working on a proof using the rules of significant figures and hopefully I can expand it to also percolate relative errors. The problem with the additive part seems similar to subtractive cancellation, so I plan to tackle it that way.