Not 100% sure if math SE is appropriate for this, but:

If I have a computation that involves exact numbers only, say to compute the ratio $\frac{123}{157} = 0.78343949...$, where these two are from an exact count of particular objects, how many significant figures is appropriate to include in the answer, and why?

  • $\begingroup$ It depends on the context. If you need exact numbers, leave it as a fraction. If approximate answers are satisfactory, then what is error tolerance in this application. Where are you rounding your other numbers? If you have used $\pi = 3.14$, and $g = 9.76$ then $3$ significant digits is sufficient. $\endgroup$ – Doug M Dec 12 '17 at 20:41
  • $\begingroup$ In my particular context, no other guidelines are given. That's the issue I'm having. I'd agree that usually I'll just figure it out based on context. But when it isn't given, do I just guess? $\endgroup$ – Michael Stachowsky Dec 12 '17 at 20:42
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    $\begingroup$ If there is no context it is safest to assume that you need to be exact. In which case you would leave it as a fraction. $\endgroup$ – wgrenard Dec 12 '17 at 20:43
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    $\begingroup$ I would leave it as a fraction, unless I had other context that made me think a decimal would be more friendly. And then I would use that context to guide me. $\endgroup$ – Doug M Dec 12 '17 at 20:47
  • $\begingroup$ Fair points. Thanks! $\endgroup$ – Michael Stachowsky Dec 12 '17 at 20:52

I think the answer depends a lot on the context you need to report the solution in.

If it is an assignment or test, 2 or 4 significant digits are usually enough.

If you are using this number in further calculations, the best practice would be keep it as a ratio for as many steps as possible. In the final step, compute the decimal value as required.

  • $\begingroup$ I agree on what is usually considered to be "enough", for sure. It does depend on context. My question was more so - what are the rules governing this if context is not available? Or, more fundamentally, do such rules even exist? $\endgroup$ – Michael Stachowsky Dec 12 '17 at 20:40
  • $\begingroup$ In light of the above comments, this answer is great. Thanks $\endgroup$ – Michael Stachowsky Dec 12 '17 at 20:52
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    $\begingroup$ I have not come across rules. One usually is trying to make a point when reporting a number (e.g, A/B = 3.16, which is greater than $\pi$). Devoid of even this context, in situations where you are required to write as decimals, I stand by my answer of 2 or 4 significant digits. It strikes a balance between detail and aesthetics/space taken up and I don't see an argument for greater precision. You're welcome, and thanks to you too! $\endgroup$ – Srikiran Dec 13 '17 at 1:00

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