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S14c. Is the result of a measurement a real number?
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A single measurement (reading from a scale) always gives a rational
number, at least if the scale is in terms of rastional units.
(If the scale gives an angle in degrees which is then converted into
arc length, the measuremnt gives rational multiples of pi instead).
However, this is by convention only, since a pointer position is
just a position in 3-space which must be translated into a number
by a subjective reading or by a digital reading device of limited
resolution. Thus the true position is not determined accurately
enough to associate it with a single number.
Infinitely many rationals (and uncountably many reals) are
compatible with any observable state of the voltmeter.
That's why the error bars are intrinsic to measurement results, even to
single readings. Deleting them and claiming exact measurement results is
just laziness, acceptable when the resolution of an instrument is known.
Therefore, according to the standards of NIST (National Institute
of Standards and Technology), a measurement gives an interval
consisting of a rational number together with an error bar; see
http://physics.nist.gov/cuu/Uncertainty/
Of course, the error bar is also somewhat uncertain, but one generally
accounts for this uncertainty by rounding it upwards, to make the
whole estimate conservative.
The NIST definition has the advantage that it also applies to indirect
measurements obtained from raw measurements by some computations.
Indeed, most high quality measurements are of this kind.
Nevertheless there is no contradiction if one assumes that reality is
governed by equations in terms of exact real (or complex) numbers,
and only the measurement abilities are limited.