Concise Summary of Significant Figures
(use in conjunction with your text book)
As stated in your text book, the concept of significant figures relates to the inherent uncertainty
associated with measured quantities. Any measured quantity, such as the mass of an object determined
using an electronic balance or the volume of liquid measured using a buret, has some degree of
uncertainty associated with it. The so called number of significant figures associated with a
measurement includes all certain digits plus one digit that is uncertain. In general, having more
significant figures implies that the number is of higher quality. This relates to the relative uncertainty
associated with measurements (see below). Four-significant figure accuracy is better than twosignificant figure accuracy and so on.
As an example, consider a measurement made on a digital top-loading balance similar to the one
you will use in lab. Those balances can measure quantities to the nearest 0.001 g (or nearest milligram).
The uncertainty is associated with the last decimal place and so, a quantity such as 1.234 g would be
understood to have an uncertainty of 0.001 g. This means that the number could actually be between
1.233 g and 1.235 g. The numbers 1.23 would have no uncertainty associated with them.
For measurements made using a ruler, buret or thermometer, the last decimal place is determined
by reading 'between the lines' as best as you can. A measuring device is more precise if it has more
divisions or markings. Having more divisions means that the user can pinpoint the measurement more
easily and, so, there is less guesswork in arriving at the measurement. To read a buret for example, you
will notice that the buret is divided into units of milliliters (mL). Between each mL marking, there are
ten divisions. This means that the nearest tenth of a milliliter can be identified unambiguously. The
user reads between the lines and mentally divides the area so that the measurement can be recorded to
the nearest hundredth of a milliliter.
The quality of a number is sometimes assessed by noting the relative uncertainty associated with
it. As with many examples in life, an absolute number does not always convey the magnitude of the
problem. If a rich man loses 100 dollars, it is not as big a problem as when a poor man loses 100
dollars. The difference is that for the rich man, 100 dollars is a smaller proportion of his total wealth. If
you measure an object and you were "off" by a gram, would this be a big error or a small error? One
way to assess this situation is to look at the relative error. For example, if the object weighed 5 grams,
then the relative error would be:
1
1
0.2 or
100 20%
5
5
If the object weighed 500 grams, then the relative error would be:
1
1
0.002 or
100 .2%
500
500
Clearly the relative error for the second measurement is less than that for the first measurement. If the
proportional error is less, then it is a higher quality measurement. For measured quantities, relative
uncertainties are typically assessed by considering the inherent precision associated with the
measurements. A balance that reads to the nearest 0.1 mg is more precise than a balance that reads to
the nearest 1 mg. A mass of 0.234 g would have an uncertainty associated with the last decimal place.
The relative uncertainty would be:
0.001
1
100 4% or
100 4%
0.234
234
Concise list of rules for identifying the number of significant figures for a quantity:
1. All nonzero digits are significant.
e.g. 5.623g (has 4),
24.6224 (has 6)
2. Zeros between nonzero digits are always significant.
e.g. 4.033 (has 4),
20.556 (has 5)
3. Zeros at the beginning of a number are never significant. They only serve to locate the position of the
decimal point.
e.g. 0.056 (has 2),
0.0006 (has 1)
Some people are uncomfortable with this last rule and think that it makes no sense. You can see
why it must be true by considering these facts. Suppose you weighed an object on a balance and
determined that the mass was 0.024 g. This is a relatively small quantity and has only 2 significant
figures associated with it. The relative error would be 0.01 out of 0.024 or 1 out of 24 (4 %). The
number of significant figures and the relative error associated with a measurement do not change and
would always be the same regardless of how you express the number. So, if the quantity mentioned was
converted to milligrams, the mass would be 24 mg. If it was converted to kilograms, the mass would be
0.000024 kg. Clearly, if the zeros were significant, the number of significant figures would depend
upon the units used to express the measurement. This cannot be true and it is understood that the zeros
are simply place holders. Compare these two quantities: 0.244 g and 0.024 g. The first one has 3
significant figures and the second has 2. The relative error associated with the first is only 1 out of 244
(0.4 %). Clearly, it is a higher quality measurement.
4. Zeros that fall at the end of a number and to the right of the decimal point are always significant.
e.g. 0.020 (has 2),
5.00 (has 3)
Remember, trailing zeros such as in the examples above indicate the exactness of the
measurement and, therefore, provide information regarding the uncertainty. If you compare the
quantities 0.200 g and 0.2000 g, the first number is known to the nearest milligram and the second
number is know to the nearest tenth of a milligram. Stripping off the zeros would result in the loss of
information. The relative error associated with the first measurement is 1 out of 200 (0.5%) whereas the
relative error for the second measurement is 1 out of 2000 (0.05%). Be sure you do not strip off trailing
zeros during calculations since they influence the interpretation of the final answer.
5. When the number ends with zeros but has no decimal point, zeroes may or not be significant.
e.g. 500 (1, 2 or 3)
Using scientific notation will help to clarify the number of significant figures.
e.g. 5 x 102 (has 1),
5.0 x 102 (has 2),
5.00 x 102 (has 3)
Some authors indicate if the last zero is significant by using a decimal point.
e.g. 500. (has 3)
In practice, it may be understood whether or not the trailing zero is significant or not based on
the context of the procedure. If you are using a device that measures to the nearest gram, then a number
such as 100 would be understood to have 3 significant figures (relative error of 1 out of 100).
6. Numbers or conversion factors that have no uncertainty associated with them (i.e. are exact) are
understood to have an infinite number of significant figures.
e.g.
something that is counted such as 1 person,
a conversion factor that involves a defined relationship (1 L per 1000 mL)
When using values in calculations, the least certain measurement will limit how the answer is
expressed. The rules for calculations are based on the idea that the final answer should be reported with
only one uncertain digit.
Concise list of rules for using significant figures in calculations:
1. For addition and subtraction, the result can have no more decimal places than the measurement with
least number of decimal places.
e.g. 4.6442
+ 0.211
4.8552
4.855 (round to the third decimal place)
The rational is based on the following. The second number (in the example above) is uncertain in the
third decimal place. When added to the other number, the result must also be uncertain in the third
decimal place. The number 0.211 could be either 0.210 or 0.212. This would mean that the sum shown
could be either 4.8542, 4.8552 or 4.8562. Clearly, the third decimal place is uncertain. Note that for
addition and subtraction, the total number of significant figures is not the criterion used. In the example
above, 4.6442 (5 significant figures) was added to 0.211 (3 significant figures) to give a result that has 4
significant figures. Similarly, if you subtract 4.6432 from 4.6442, you will get an answer (0.10) that has
2 significant figures.
2. For multiplication and division, the result must be expressed with the same number of significant
figures as the number with the fewest number of significant figures.
e.g. 33.4 0.50 = 17
224.0/2.00 = 112
From a mathematical standpoint, multiplication and division are fundamentally different than
addition and subtraction. In addition, operations involving multiplication and division often combine
unlike units and, so, numbers cannot be compared based on decimal places. For example, density
involves dividing a mass measurement by a volume measurement. For those kinds of operations, the
relative uncertainty must be used to judge the answer. For example, if an object weighs 3.2 g and
occupies a volume of 2.00 mL, the density would be:
3.2 g
2.00 mL
1.6 g/mL
The relative uncertainty for the mass (1 out of 32 or 3%) is similar in magnitude to the uncertainty for
the density (1 out of 16 or 6%). If the density was expressed as 1.60 g/mL, then the relative uncertainty
would appear to be 1 out of 160 or 0.6%. This would be too low and would imply that the measurement
was better than it actually was.