Significant digits
a.k.a. Significant figures
Objectives
At the end of this lesson the students will be
able to:
•State why significant digits are important.
•List the rules for significant digits.
•Calculate problems and record answers with
significant digits.
Why are significant digits or
significant figures important?
The science teachers at a Baltimore County middle school wished to acquire a steel cube, one
cubic centimeter in size to use as a visual aid to teach the metric system. The machine shop
they contacted sent them a work order with instructions to draw the cube and specify its
dimensions. On the work order, the science supervisor drew a cube and specified each side to
be 1.000 cm. When the machine shop received this job request, they contacted the supervisor to
double check that each side was to be one centimeter to four significant figures. The science
supervisor, not thinking about the "logistics", verified four significant figures. When the finished
cube arrived approximately one month later, it appeared to be a work of art. The sides were
mirror smooth and the edges razor sharp. When they looked at the "bottom line", they were
shocked to see the cost of the cube to be $500! Thinking an error was made in billing, they
contacted the machine shop to ask if the bill was really $5.00, and not $500. At this time, the
machine shop verified that the cube was to be made to four significant figure specifications. It
was explained to the school, that in order to make a cube of such a high degree of certainty,
many man hours were used. The cube had to be ground down, and measured with calipers to
within a certain tolerance. This process was repeated until three sides of the cube were
successfully completed. The number of man hours to prepare the cube cost $500. The science
budget for the school was wiped out for the entire year. This school now has a steel cube worth
its weight in gold, because it is a very certain cubic centimeter in size.
Remember:
• Significant digits, which are also called
significant figures, are very important in Physics.
• Each recorded measurement has a certain
number of significant digits.
• Calculations done on these measurements must
follow the rules for significant digits.
• The significance of a digit has to do with whether
it represents a true measurement or not.
• Any digit that is actually measured or estimated
will be considered significant.
• Placeholders, or digits that have not been
measured or estimated, are not considered
significant.
It is very important,
however, to know and
understand the precision of
measurement that we use
in our daily lives.
Rules for Significant Digits
1. Digits from 1-9 are always significant.
2. Zeros between two other significant digits
are always significant
3. One or more additional zeros to the right of
both the decimal place and another
significant digit are significant.
4. Zeros used solely for spacing the decimal
point (placeholders) are not significant.
Examples of Significant Digits
EXAMPLES
453 kg
5057 L
5.00
0.007
# OF SIG. DIG.
COMMENT
All non-zero digits are always
significant.
Zeros between 2 sig. dig. are
significant.
Additional zeros to the right
of decimal and a sig. dig.
are significant.
Placeholders are not sig.
Each number that we record as a
measurement contains a certain number
of significant digits, which show accurate
or estimated digits. When we do
calculations our answers cannot be more
accurate than the measurements that they
are based on. We must be careful to follow
the following rules whenever we perform
calculations in Physics class.
Multiplying and Dividing
• RULE: When multiplying or dividing, your answer may only
show as many significant digits as the measurement showing
the least number of significant digits.
Example:
When multiplying 22.37 cm x 3.10 cm x 85.75 cm = 5946.50525 cm3 we
look to the original problem and check the number of significant
digits in each of the original measurements:
• 22.37 shows 4 significant digits.
• 3.10 shows 3 significant digits.
• 85.75 shows 4 significant digits.
Our answer can only show 3 significant digits because that is the
least number of significant digits in the original problem.
Our calculators show the answer as 5946.50525 with 9 significant
digits, we must round to the tens place in order to show only 3
significant digits. Our final answer becomes 5950 cm3.
Adding and Subtracting
RULE: When adding or subtracting your
answer can only show as many decimal
places as the measurement having the
fewest number of decimal places.
Example:
When we add
3.76 g + 14.83 g + 2.1 g = 20.69 g
We look to the original problem to see the
number of decimal places shown in each of the
original measurements. 2.1 shows the least
number of decimal places. We must round our
answer, 20.69, to one decimal place (the tenth
place). Our final answer is 20.7 g
Calculators and Significant Digits
• Many calculators display several additional,
meaningless digits, some always display only
two. Be sure to record your answer with the
correct number of significant digits. Calculator
answers are not rounded to significant digits.
You will have to round-off the answer to the
correct number of digits.
• Note that significant digits are only associated
with measurements; there is no uncertainty
associated with counting. If you counted four
laps for a runner and measured the time to be
2.34 minutes. The number of laps does not
have an uncertainty, but the measured time
does.
• When doing multi-step calculations, do not
clear your calculator.
• Round your final answer to the smallest number
of significant digits of any of the values in the
original question.
The Two Greatest Sins Regarding
Significant Digits :
• Writing more digits in an answer
(intermediate or final) than justified by the
number of digits in the data.
• Rounding-off, say, to two digits in an
intermediate answer of a multi-step
calculation, and then writing three digits in
the final answer.
Sample problems
Instructions: work the problems. When you are ready to check your
answers, go to the next page.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
37.76 + 3.907 + 226.4 = ...
319.15 - 32.614 = ...
104.630 + 27.08362 + 0.61 = ...
125 - 0.23 + 4.109 = ...
2.02 × 2.5 = ...
600.0 / 5.2302 = ...
0.0032 × 273 = ...
3
(5.5) = ...
0.556 × (40 - 32.5) = ...
45 × 3.00 = ...
Answers to Sample Problems
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
37.76 + 3.907 + 226.4 = 268.1
319.15 - 32.614 = 286.54
104.630 + 27.08362 + 0.61 = 132.32
125 - 0.23 + 4.109 = 129
2.02 × 2.5 = 5.1
600.0 / 5.2302 = 114.7
0.0032 × 273 = 0.87
3
2
(5.5) = 1.7 x 10
0.556 × (40 - 32.5) = 4.2
2
45 × 3.00 = 1.4 x 10