A. Romero 2006
Significant Figures
CHEM 1A
Significant figures convey the precision to which a number is known. They are a part of
any numerical answer, and are as important as the number itself and the units.
Understanding what significant figures tell us about a number:
As we go from left to right across a number, the information given by each column
indicates a higher level of precision in our measurement of this number. For example, how many
cars are there in the parking lot at the mall?
23,000 cars?
23,400 cars?
23,430 cars? or 23,432 cars?
23,432 cars seems like a better answer than 23,000 cars, but do we really know that there are
23,432 cars in the parking lot? Which of these answers should we report? This depends on how
well we measured the number of cars. Did we estimate from the size of the parking lot, or did
we actually count every last car?
If we estimated that there are about 23,000 cars, but someone else could reasonably
estimate that there may be 24,000 cars or 22,000 cars, then we should only report 23,000 cars.
This tells any person who sees our number that we are certain that there are more than 20,000
cars, and we think that there are around 23,000 cars, give or take about a 1,000 cars. In other
words, our precision only goes to the thousands column.
If we counted every car to the best of our ability, then we should report 23,432 cars. This
way, anyone who sees our number knows that our confidence in our number goes all the way
down to the units column. If someone else were to count all the cars, they might count 23,431
cars or 23,433 cars, but the difference in the values would be in the units column.
23,000 cars has two significant figures (the 2 and the 3), while 23,432 cars has five
significant figures (the 2, 3, 4, 3, and 2). The more significant figures we report with a number,
the greater the precision we are claiming the measurement has. So if we report more significant
figures than we should, we are technically lying about how well we know, or how carefully we
measured, our number.
Rules for significant figures in our own measurements:

If we are certain that a digit in a number is correct, then it is a significant figure. As we
go from left to right in a number, the first digit we come to that has any uncertainty at all
is the last significant figure that we report. This first uncertain digit has some value
because it is our best estimate, and therefore it is included as a significant figure.
However, since we are not 100% sure about this uncertain digit, we can not know (and
should not report) any numbers further to the right.

You can estimate (with some uncertainty) between the lines on glassware, a ruler, or a
scale, so you can report one more decimal place than the lines themselves have.
Example: reading a graduated cylinder
5.0 mL
4.9 mL
4.8 mL
4.7 mL
4.6 mL
3 mL
2.6 mL
2 mL
A less precise graduated cylinder
with lines in the units column can
be estimated to the tenths column.
We are sure that the number is
between 2 mL and 3 mL. The first
number we are unsure of (the six
tenths) is the last number that we
report for this measurement.
4.85 mL
A more precise graduated cylinder
with lines in the tenths column can be
estimated to the hundredths column.
The first number that we are unsure
of (the five one-hundredths) is the last
number that we report for this
measurement.
Counting significant figures in data:

Non-zero numbers (1 – 9) are always
significant

Leading zeros are never significant

Sandwiched zeros are always significant

Trailing zeros are significant only if there is a decimal point in the number
leading
sandwiched
trailing
0 .0 0 1 2 0 5 6 0 2 1 0 0 0
 
11 significant figures
Numbers that have infinite significant figures:

Quantities that are measured (mass, volume, pressure, etc.) can only be measured with as
much precision as the measuring device allows. Therefore, these values should have the
appropriate number of significant figures associated with them as described above.

Some numbers, however, are “exact.” Meaning that they are exactly that number by
definition with no uncertainty. For example, there are exactly 60 seconds in one minute
by definition. Because there is no uncertainty, exact numbers have an infinite number of
significant figures, and they do not affect the significant figures of the final answer when
they are used in a calculation.

Often in conversion factors, one number is fixed as being exact, while the other is
measured relative to this exact number. For example,
82,500 cars
1 day
if we use the information that approximately 82,500
or
cars cross a bridge each day in a calculation, we could
82,500 cars
1
day
use either of the conversion factors shown to the right.
We can define the time period as being exactly one day
(the 1 has ∞ sig.figs.), therefore it is the measured quantity, the 82,500 cars, which
would determine the significant figures of the conversion factor (3 sig.figs.).
Significant figures for numbers calculated from other values:
Multiplication and Division:

The answer to a multiplication or division problem has the same number of significant
figures as the number used in the calculation with the least number of significant figures.
Example: An event takes approximately 4.2 years. How many seconds is this?
4.2 yrs
365.25 days
24 hrs
60 min
60 sec
1 yr
1 day
1 hr
1 min
= 132,541,920 sec = 1.3 x 108 sec
2 sig.figs.
2 sig.figs.
least number
of sig.figs
5 sig.figs.
(leap year)
determines
sig.figs of
the answer
∞ sig.figs.
(exact by definition)
Addition and Subtraction:

In the answer, we can only keep columns on the right side if each number used in the
calculation has a significant figure in that column.

We keep all columns on the left side, because dropping these columns would change the
value of the number, not just the precision to which the value is known.
Examples:
125.2 3
+ 12.3 56
+ 1.3
113.37
+ 1110.
+ 6.6
138.8 86
1219.97
= 138.9
The trailing zero in the units
column of 110 is not
significant because there is
no decimal point
= 130
= 1.3 x 102
In calculations, we use all of the available digits to calculate the answer, and only
round the final answer to the appropriate number of significant figures.
Logarithms:

The answer to the logarithm of a number has the same number of places after the decimal
as the original number had significant figures.
Examples:
log ( 100 ) = 2.0
1 sig.fig.

1 place
after the
decimal
log ( 1 0 0 . 0 ) = 2.0000
4 places
after the
decimal
4 sig.figs.
The reverse is also true. The answer to a number raised to an exponent has the same
number of significant figures as the exponent had places after the decimal.
Examples:
10-7.85 = 1.4 x 10-8
the exponent has
2 places after the
decimal
2 sig.figs.
10-7.850 = 1.43 x 10-8
the exponent has
3 places after the
decimal
3 sig.figs.
Calculations including combinations of the above types of calculations:

The answer should be calculated using all available digits to avoid compounding
rounding errors. Once the calculation is complete, the final answer should then be
rounded to the appropriate number of significant figures. To determine the appropriate
number of significant figures, retrace your steps through the calculation in order, keeping
track of how many significant figures the answer to each step should have, and
determining how that affects the significant figures in the next step, and so on.