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Part Two: Significant Figures
T.2
Significant Figures
The use of significant figures is a method of designating the reliability of
measured quantities. Since experimentation in physics involves very many
measured quantities, it is important to know how to express the results of the
measurements and the calculations based on those measurements. Do the SelfCheck to see if you need to refresh your knowledge of the use of significant
figures.
Self- Check
Indicate the number of many significant figures in each of the following measured
quantities.
1.
6 cm
2.
4.0 m
3.
0.02 km
4.
8000 nm
5.
0.06000
In the following problems all of the numbers can be assumed to be the result of a
measurement. Solve each of the problems and give the answer with the correct
number of significant figures.
6.
(0.101)(1.03 )
(0.025 )
7.
( 4.5 × 10 3 )(2.5 × 10 −2 )
(2.15 × 10 − 4 )
8.
6.27 + 0.2 + 0.410
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Part Two: Significant Figures
Determining the Number of Significant Figures
The number of significant figures in a recorded number is the number of
digits that are certain, plus the first digit that is uncertain. For example, a length
given as 1.2 m means that the length lies somewhere between 1.15 m and
1.25 m. The distance 1.2 m has two significant figures: the numeral 1 is certain
and the numeral 2 is the first digit that is uncertain.
The limitations of the device used to make the measurement usually
determine the decimal place of the first uncertain digit in a measurement. For
example, a graduated cylinder marked as 25 ml ± 1 ml is certain for the tens of
milliliters, but the uncertainly of estimation of the liquid level is in the units of
milliliters. Thus if this graduated cylinder were used to measure a volume
estimated to be at the 21 ml there are two significant figures in the measure: the
certain numeral, 2, and the numeral 1, the first uncertain digit.
Measurements that have zeroes to the left of an understood decimal and to
the right of a non-zero digit are not significant. If the decimal is expressed then
the zeroes are significant. In the number 4000 m there is only 1significant figure.
The measure is just to the nearest kilometer. If the measure was written as
4000 m, then the measurement was made to the nearest meter, and there are 4
significant figures. This confusion can be eliminated by using scientific notation.
The measurement reported as 4 × 10 3 m clearly has only 1 significant digit, while
4.000 × 10 3 m has 4 significant digits, since all zeros expressed to the right of the
decimal point in scientific notation are significant.
Zeroes to the right of an expressed decimal and to the right of a non-zero
digit are significant. Again, putting the measurement into scientific notation helps
with the decision as to what is significant and what is not. Suppose a measure of
the thickness of the edge of a razor blade is made and recorded as 0.02 mm.
This measurement has only 1 significant figure, which is seen clearly when the
number is written in scientific notation as 2 × 10 −2 mm. If the person who made
the measurement actually measured to the nearest one-thousandth of a
millimeter, the measurement should be recorded as 0.020 mm, with 2 significant
figures. This would be 2.0 × 10 −2 mm in scientific notation. As another example,
the thickness of a piece of machined steel is expressed as 10.002 mm. This
measurement has 5 significant figures, indicating the measurement was made to
the nearest one-thousandth of a millimeter. How would this measurement be
expressed in scientific notation?______________________
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Part Two: Significant Figures
Using Significant Figures in a Product or Quotient
In multiplication and division calculations, the number of significant figures
in a result usually cannot exceed the least number of significant figures used in
the calculation, nor should it have fewer significant figures than the least number
in the measured values. If the side of a square has been measured to be
1.2 × 10 2 cm, the number of significant figures (2) must be taken into account
when calculating the area.
A = L2 = (1.2 × 10 2 ) 2 = 1.44 × 10 4 = 1.4 × 10 4 cm 2 (2 significant figures)
As another example, suppose you wish to determine the radius of a circle
whose area is given as 46.3 m2. You will need to use the formula for the area of
a circle, A = π r 2 , and the value of the ratio π . You must use the same number
of significant figures in the value of π as in the given quantity (3). This gives
A = πr 2
46.3 = 3.14 r 2
 46.3 
r =

 3.14 
1
2
= (14.7 )
1
2
= 3.83 m (3 significan t figures )
Counted numbers and defined quantities are not measured and do not
influence the number of significant figures in a calculation. Suppose that you
need to know how much cheese to purchase for a party. You estimate that there
will be 7 people present and that each will eat 0.125 kg of cheese. Your total
cheese purchase should be
(7 people )(0.125 kg of cheese ) = 0.875 kg of cheese (3 significan t figures )
In this example, the 7 (as in 7 people) did not affect the number of
significant figures in the final answer, because it was a counted quantity.
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Part Two: Significant Figures
Using Significant Figures in Sums or Differences
In addition and subtraction, the number of significant figures in the result is
determined by the position of the first uncertainty of the measure with the largest
uncertainty. For example, four different carpet installer trainees were told to use
the instrument of their choice to determine the length of one side of a closet. The
four distances were then used to find the perimeter of the closet. The
measurements they reported were 5.06 m, 5 m, 11.1 m and 11.20 m. To find the
perimeter, the measurements were added together:
5.06 + 5 + 11.1 + 11.20 = 32.36 = 32 m (2 significan t figures )
From the measurements given, the value 5 m has the greatest uncertainty,
since it was measured only to the nearest meter. Thus the sum must be rounded
to the nearest meter, which gives the answer 2 significant figures.
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Part Two: Significant Figures
Practice Problems in Significant Figures
1.
Indicate the number of significant figures in each of the following measured
quantities.
2.
3.14 s
3.
86 000
4.
3.1558 × 10 7
5.
6 × 10 7 yr
6.
0.324 × 10 1 hr
s
day
s
yr
Complete the following calculations involving measured quantities and express
the answers with the correct number of significant figures.
)(8.1× 10 3 s)
7.
(35.8
8.
(2.100 m)
(9.81 m s2 )
9.
Determine the distance of tall, d, for the situation where d = ½ at2,
a = 9.80 m/s2, and t = 4.0 s.
10.
1002.00 ml + 248.5 ml + 8.80 ml
m
s
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Part Two: Significant Figures
Answers to Self- Check
1.
1 significant figure
2.
2 significant figures
3.
1 significant figure
4.
1 significant figure
5.
3 significant figures
6.
4.2 (2 significant figures)
7.
5.2 × 10 5 (2 significant figures)
8.
6.9 (2 significant figures)
Answers to Practice Problems
1.
3 significant figures
2.
3 significant figures
3.
5 significant figures
4.
1 significant figure
5.
3 significant figures
6.
290 or 2.9 × 10 2 m (2 significant figures)
7.
0.214 or 2.14 × 10 −1 s2 (3 significant figures)
8.
d = ( 12 )(9.81)( 4.0) 2 = 78 m (2 significant figures)
9.
1259.3 ml The measure with the largest uncertainty is the 248.5 ml. The
answer must be rounded to the nearest tenth milliliter. (5 significant figures)
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