Measurement is a process, and it produces three things: a number (the magnitude), a unit of measure, and a degree of error. This triplet of results can also be called a measurement. A measurement is not however an intrinsic property of an object or situation: a rod can't be 24 cm long, or have a temperature of 240°C.
Saying a rod is 24 cm long is shorthand: it means when it was measured, it was found to be 24 cm long; or it was produced by something designed to create 24 cm long rods. Let's assume the value of 24 is accurate, that is it isn't a mistake for something like 26. We next need to consider how precise it is. Is the rod 24.0 cm long? Knowing it's 24 cm long doesn't enable you to decide whether it's 24.0 cm long. There isn't enough information.
People who don't know enough about measurement -- and we're now including more or less everyone who's ever been let loose to produce a newspaper -- often assume the magnitudes in measurements can be manipulated as if they're pure numbers. They can't be. If you stick two 24 cm rods end to end, their total length is about 48 cm, but it could be 47 cm or 49 cm. That is,
2 x 24 = 48
is true, but
2 x 24 cm = 48 cm
isn't. A similar kind of error comes from mistaking conversion factor
s for facts about measurements. The conversion factor for centimetre
s to inch
es is (about)
1 cm = 0.3937008 in
but that's not a fact about measurements. If you measure something and finds it's 1 cm long, that doesn't make it 0.3937008 in long -- and indeed that value is almost certainly ludicrously wrong
, because it's so inaccurate
can be at opposite poles.
You need to know the number of significant figures. Or you need to know the error. In fact, the error is more scientific, because it can be plus or minus different amounts. Use of significant figures, in particular arithmetic on the last of them, makes a slightly unjustified assumption about the distribution of the error. But in normal circumstances we do this: we assume that a measurement of 24 cm is actually a measurement of 24 cm ± 0.5 cm. If so, that justifies the usual rules of rounding of the digits beyond the last significant place.
Often the nature of the thing being measured is enough to give us information about the precision of the measurement. So a running track 100 m long is almost certainly 100 to three significant digits, and in fact is probably 100.0 m long. But if it's 100 m to the post office, it is quite unlikely to be 1.00 x 102 m there. Applying a conversion factor of 100 m = 109.3613298 yd, we find that the running track is probably 109 yd or even 109.4 yd long, while the distance to the post office is 100 yd, or perhaps the best you could say (if it's on a street plan perhaps) is 110 yd, understood in this case as having only two significant figures.