70 DIY Science: Illustrated Guide to Home Chemistry Experiments
That brings up another important point. Mathematical
operations cannot increase the degree of accuracy. If you add,
subtract, multiply, or divide values, the number of significant
figures remains the same. For example, if you multiply
the two values 4.03 and 1.16, the mathematical result is
4.6748, but that result ignores sig figs. Each of the two initial
values is known to three sig figs, so the result can have no
more than three sig figs and should therefore be recorded
as 4.67. Similarly, if you multiply the values 4.0 and 1.17, the
mathematical result is 4.68. But because one of the initial
values had only two sig figs, you must round the result to
two sig figs, or 4.7.vALAN oG ERSUS dIGITAL
Interpolation is possible only with measuring
instruments that use analog readouts. Instruments
that use digital readouts display only known values—
assuming they are properly designed—and therefore
provide no data for interpolation. For example, an
electronic centigram scale provides a readout to 0.01
g, but no additional information that could be used for
interpolation. Conversely, a quad-beam manual centigram
scale provides known values to 0.01 g, and may allow
interpolation down to the milligram level.
Alas, some digital instruments present values well
beyond their actual resolutions. For example, I saw one
inexpensive digital pH meter whose display specifies pH
to 0.01. Unfortunately, the manual states that the actual
resolution of the instrument is only ±0.2 pH, so that last
decimal place is entirely imaginary.SGNICANTI fI fIGURES IN THE REAL woRLd
Significant figures are an attempt to encode two
separate pieces of information—a known value and
an uncertainty—into one number, with sometimes-
ambiguous results. Chemistry texts often devote a great
deal of attention to significant figures, but most working
scientists use a more commonsense approach. They use
two separate numbers to express the two values, in the
form 123.456±0.078 or 123.456(78), where 123.456 is
the nominal value and 0.078 or (78) is the uncertainty.
For example, if a balance indicates that the mass of a
sample is 42.732 g, that mass might be recorded as
42.732±0.005 or as 42.732(5), indicating an uncertainty
of 5 units in the third decimal place.If you use a ruler with millimeter (mm) graduations, you can
obtain a better value for the length of the wire. For example, if
the wire extends past the 23 mm line but does not reach the
24 mm line, you can state with certainty that the length of the
wire is between 23 mm and 24 mm. By interpolation, you may
estimate that the actual length of the wire is 23.8 mm. That value
has three sig figs, two known and one interpolated. By using
a better instrument, you have increased the accuracy of your
measurement by a factor of ten.
If you use a caliper with 0.1 mm graduations, you can obtain an
even better value for the length of the wire. For example, you
may find that the length of the wire falls between 23.7 mm and
23.8 mm. Once again, you may add another significant figure by
interpolation and estimate the length of the wire as 23.74 mm, a
value with four sig figs. But, although you now know the length of
the wire with much greater accuracy than before, you still don’t
know the exact length of the wire. You can continue using better
and better measuring instruments, but even the best possible
instrument can still provide only an approximation of the wire’s
length. Some approximations are better than others, obviously,
and one common way to quantify the quality of an approximation
is to specify the number of significant figures.
Any calculation that uses measured data must take the
uncertainly of the measuring instruments, as quantified
by significant figures, into account. When you’re making a
calculation using measurements from more than one instrument,
the least accurate instrument determines the level of certainty of
the results. For example, you might want to determine a runner’s
average speed over a 1 km course. To get accurate results, you
might use a laser rangefinder accurate to 1 m to set the start
and finish lines and a stopwatch accurate to 0.01 second to
time the run. The accuracy of these measuring instruments is
high, and the result will have a correspondingly high number
of sig figs. Conversely, if you measure the length of the course
with your car’s odometer, you know the distance only to within
0.1 kilometer (which can be interpolated to one further sig
fig), so even using the most accurate stopwatch available
cannot increase the accuracy of your result. Similarly, if you
use a sundial to time the run, the high accuracy of the laser
rangefinder is wasted because the elapsed time for the run is
not known accurately.