Significant Digits
In science classed you will be making many
measurements. We must realize that there is
a MAJOR difference between a defined (or
exact) number and a number representing a
measured quantity (measurements).
We know that when we make measurements
we must always estimate the last digit in the
measurement. Since the last digit is an
estimate, there is some uncertainty in all
measurements. Defined numbers, such as a
dozen eggs, have no uncertainty. There are
exactly 12 eggs in a dozen because it is
defined as such.
The accuracy of your measurements is how
close your value is to the actual or accepted
value. Accuracy depends on how carefully
you make your measurements. Precision,
however, depends on the measuring
instrument. We can determine the precision
of a measurement by the number of
significant digits.
Rules for identifying significant digits
in a written measurement:
1. All non zero digits are significant
2. All zeroes between non zero digits
are significant
3. Leading zeroes are never significant
4. Trailing zeroes:
-are significant if there is a decimal
point (12.300)
-are NOT significant if there is no
decimal point (12300)
We often use our measurements to calculate
We often use our measurements to calculate
other values. Nothing calculated from
measurements can be more precise than the
least precise measurement.
This is very different from rules you have
learned in math class. For example, if we
are asked to find the area of a box that is 6
cm by 8 cm. Any decent math student
knows that the area would be 48 cm². A
science student must look at the problem
differently.
In science class, a measurement of 6cm has
only one significant digit. It means that the
length is greater than 5cm and less than
7cm. If the person doing the measurement
had measured it more accurately, he/she
would have written it as 6.0cm or 6.00cm.
Since the lengths are “more or less” 6cm and
8cm, the science student that the area could
vary. The maximum are would be 63cm²
(7cm · 9 cm) and the minimum area would
be 35 cm² (5cm · 7cm). So, how do we
write the answer?
Since the original measurements are
precise to only one significant digit, the
answer can have only one significant digit
and the answer must be rounded to 50cm²!
Since the original measurements
were not very precise, we should not expect
calculations using imprecise measurements
to result in very precise answers.
If we measure the volume of a jar to be
exactly 900 ml how can we write this
number so that all three digits are
significant? The easiest way is to use
scientific notation and write the number as
9.00 · 10².
The rules for significant digits apply to
measured numbers. Defined numbers (12
eggs to a dozen, the average of 3
measurements) do not have significant digits
and the rules for significant digits do not
apply.
Rules for Using Significant
Digits in Your Calculations
1. When adding and subtracting answers
should be rounded to the same precision
(Accuracy) as the least accurate tool used.
EXAMPLE: 6.28 g + 51.0 g =?
Solution: 6.28 g is accurate to the hundredth
51.0 g is accurate to the tenth
The sum 57.28 g must be rounded to the
same accuracy as the least accurate tool.
Therefore the answer is 57.3 g.
4.0000____
2) Do the following calculations. Round to
the correct number of significant digits.
3.44 x 1.256 _____
2. When multiplying and dividing measured
values, answers should have the same
number of significant digits as the original
number with the fewest number of
significant digits.
0.0075 x 3.664 _____
73.45 / 105 _____
4.55 / 0.3 _____
EXAMPLE: 750 cm · 0.237 cm =?
SOLUTION: 750
2 Significant Digits
X 0.0237 3 Significant Digits
17.775 cm² which must be
rounded to two significant digits since 2 is
the lowest number of significant digits in the
original problem. Therefore the answer is
18 cm²
FINAL NOTE: If you are doing several
mathematical operations on your calculator,
leave all the digits on the calculator and do
your rounding on the final answer.
(22.45 / 1.27) x 0.12 _____
(4500 x 2.44) / 156.0 _____
(4.55 x 104) (7.04 x 105) __________
(5.606 x 109) (4.015 x 106) / (11.18 x 105)
___________
(9.66 x 1011) (3.599 x 104) / (3.91 x 10-3)
___________
(1.1618 x 1015) (4.015 x 106) / (11.18 x 105)
1) How many significant figures are in each
of the following?
3) Do the following calculations. Round to
the nearest number of significant figures.
0.189 ____
2.212 + 4.5 = ____
891.00 ____
0.88 – 0.2344 = ____
900____
5.5 + 0.23 + 3.06 + 8.09 = ____
190 ____
3.3 – 11.0 + 15 = ____
89.00 ____
(12.3 + 6.6) (8.99 – 0.44) = ____
5400____
(65.4 – 0.3) (3.66 + 18) = ____
0.0018____
(2.34 – 1.6) (12.9 – 12.7) = ____
0.0016____
.00001____
440____
10,500,000____