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 Posted: Mon Apr 06, 2009 8:07 pm
What exactly are they? I was learning aobut them, and the day that we were supposed to learn about them, we had our science teacher's assistance teach us, and her voice was so annoying, and he inflections were so hard to understand that we didn't learn much that day. So what are they?
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Posted: Wed Apr 08, 2009 8:06 pm
Significant figures are a simple way to keep track of the precision of measurements. In mathematics, we can say, for example, that five is the same as 5.0, or 5.000..., and that's perfectly fine and sensible as far pure math is concerned, but in many scientific contexts there is an implicit difference between them--one is much more precise than the other. So if you say that a particular circle has a radius 1.5 cm, that's a bit different than saying 1.50 cm. In the first case, the implicit claim is that the measurement was taken to the nearest tenth of a cm (mm), while in the latter, to the nearest hundredth.
The idea is to not claim more precision than your measurements actually have. Again, if you measured the radius of a circle as 1.5 cm, it would be wrong to say that the area is πr² = 7.0686 cm². You measured the radius to the nearest tenth--if it was actually closer to 1.51 and your ruler (or caliper or whatever) wasn't precise enough to show it, you should have gotten something closer to 7.16 instead. The correct answer would be just 7.1 cm².
The answer to a calculation should not be more precise than the measurement of least precision involved. Just count the sig figs as the number of digits if there is an explicit decimal point, and only the digits before any trailing zeros if there is no explicit decimal point.
101 -> 3 sig figs
304500 -> 4 sig figs (trailing zeros don't count w/o decimal)
304500. -> 6 sig figs
304500.0 -> 7 sig figs (etc.)
If you've a calculation of, say, momentum, involving v = 101 m/s and m = 304500 kg, the momentum would be p = mv = 30800000 = 3.08E7, since the least precise quantity has three sig figs.
There are some exceptions, such as defined quantities--that an inch is 2.54 cm is a matter of definition, so that it actually has infinitely many significant figures (i.e., infinite precision), rather than just the three as would be suggested by the way it's written.
Significant figures aren't actually a very good method of keeping track of measurement uncertainties in general (e.g., for calculations involving highly nonlinear functions), but they're very simple, and for many things they're good enough.
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