Measurements
always have some limitation on their precision. When you do calculations with
measurements (which you will undoubtedly have to do) the results will need to
reflect the limitations of the measurments upon which they are based. The
results will have their precision
limited by the
precision of the least precise measurement. The measurement with the smallest
number of significant figures is the least precise measurement. A result
calculated based on that measurement cannot have higher precision than the
measurement you started with. Put another way: a chain is only as strong as
the weakest link. A relay race team is only as fast as its slowest member.
Measurements and Calculations
Make the following measurements and record the information below. Performs the
indicated calculations and prepare to share the data with the rest of the
class.
Explanation and Calculation Rules
If a measurement of length is off by 0.01 cm (the maximum precision of a
typical ruler) then a calculation of area would be off by 0.01 cm2.
If you measure an area 2.25 cm on one side and 3.25 cm on the other then the
area is 7.31 cm2. The calculator says that the area is 7.3125
cm2. This is wrong because it means that you know the area
to a precision of ±0.0001 cm2. Every decimal place added to
a result implies an improvement in precision of a factor of ten. So a
measurement precise to ±0.0001 is 100 times more precise than a
measurement precise to ±0.01. You can’t possibly have a precision
that is 100 times better after doing the calculation than you had in the
original measurements.
Basically, the important thing to know is how much to round off the results
of calculations. Here are the rules:
- The least number of significant figures in any number in your
calculation determines the number of significant figures in the answer.
- Pure numbers (such as exact conversion factors and defined quantities) do
not affect the number of significant figures.
Examples:
2.5 × 3.42
Calculator gives: 8.55
Correct answer is: 8.6
8.6 is the correct answer because of the rule governing significant figures in
calculations. 2.5 has two significant figures and 3.42 has three. Since 2.5
has the least number of significant figures it is the number that determines
how many significant figures are in the answer.
3.10 × 5.789
Calculator gives: 17.9459
Correct answer is: 17.9
You may have thought the answer should be 18. but this is wrong because 3.10
has three significant figures, not two.
5.689 × 2.11
Calculator gives: 12.00379
Correct answer is: 12.0
The number of significant figures in the answer must be three because 2.1 has
only three significant figures. The trailing zero in 12.0 is important because
it is significant. Remember the rules for significant zeros: Trailing zeros in
the decimal portion only are significant.
Examples of pure numbers/exact figures: π (use the π button of your
calculator), 12 in. = 1 ft, constants in formulas, 100 years = 1 century, 100
cm = 1 m, numbers of things (how many crayons in a box).
One more note: in this activity please remember not only to pay attention to
the correct number of significant figures in the answer. Also, make sure to
show units at each step of each calculation!
Exercises
Perform the following calculations. Report answers with the correct number of
significant figures and enter calculations into your calculator in such a way
as to avoid rounding errors. Show all work, including units, at every step.
Write neatly and do your work on another sheet of paper before recording your
work and answers here.
- 2.33 cm × 6.085 cm × 2.1 cm
Calculator’s answer:
Correct answer:
No. of s.f.:
- 4/3 × π × 4.598 cm3
Calculator’s answer:
Correct answer:
No. of s.f.: