Significant Digits and Precision
The concept of significant digits comes from the act of measurement. When a
measurement is made, the final digit recorded has some uncertainty in it. When using a
marked scale, the last recorded digit should be an estimate of the fraction of the space
between marks, as when measuring a length using a meter stick.
For example, in the
picture to the right, the
edge falls between 2.2
and 2.3 cm. A
reasonable estimate is
that it is half way
between, so the
recorded value should
be 2.25 with 3
significant figures, the
last one being an
estimate.
For digital readings, the electronics sets the level of significance. The last digit displayed
is significant and has some uncertainty. For example, a reading of 3.45 Volts on a digital
voltmeter means the actual voltage is between 3.445 and 3.455 volts.
Measurements from a marked scale, such as a meter stick,
should end with the last digit representing an estimate of the
location of the position as a decimal fraction of the distance
between the finest marks.
The last digit displayed on a digital readout is uncertain to ±1.
Calculating with significant digits
Adding and subtracting:
Suppose Hope and Les are measuring the width of a room. Hope takes a long
metal rod and measures it to be 253.4 cm. long and she finds that the room is three of
these rods and a bit more. Less brings a precise caliper over and measures the extra bit as
23.042 cm. The total length of the room is then
3 x (253.4 cm.) + 23.042 cm. = 783.242 cm.
Now, let’s look at this carefully. Hope knows the length of the rod is 253.4 cm. or
between 253.3 and 253.5 cm. If we add 23.042 to 253.4, we get 276.442; but, which
digits really mean something?
Let’s see what range is covered by taking the high and low limits for 253.4 and adding
23.042.
253.3 + 23.042 = 276.342
and
253.5 + 23.042 = 276.542
These differ in the tenths place, so the smallest digit which is significant is the first place
to the right of the decimal. Another way to see this is:
253.4
+ 23.042
276.442
The .042 at the end is not significant because it is smaller than the uncertainty in 253.4.
When adding or subtracting two numbers, the number with the
fewest digits past the decimal determines the last significant
digit in the answer. If the least precise number added has its
last digit in the hundredths place, the answer will have no
significant digits past the second decimal place.
Multiplying and dividing numbers:
To see how to figure the number of significant digits in the answer when two numbers are
multiplied, let’s look at calculating the area of a rectangle. Ariel is set to calculate the
area of a sheet of paper and her partners, Len and Hyde measure the sides. Len measures
the short side as 2.31 cm. and Hyde measures the long side as 10.22 cm. Ariel multiplies
these to get 23.6072, but which figures are significant? They ask Sigurd Figgis for
advice.
Sig Figgis tells them to consider drawing the rectangle showing the uncertainty in Len’s
and Hyde’s numbers, calculate the area of the uncertain region and see how this
compares to the digits in the answer.
Area = 10.22 cm x 2.31 cm = 23.6072 cm2
Significant digits:
4
3
?
The shaded regions cover the uncertainty of ±.01 cm for each measurement. The area of
the longer shaded area is larger and will be sufficient to show the level of significance in
the answer. The calculation gives:
(.02) x 10.22 =.2044
So the uncertainty is in the tenths place and the area should be reported as 23.6 cm2. This
has three significant digits, the same as the factor with the fewest digits (2.31).
When multiplying or dividing two numbers, the one with the
fewest significant digits determines the number of significant
digits in the answer.