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Significant Figures
In scientific work, numbers are grouped in two categories: exact and inexact
1. Exact number: a number with no uncertainty in it.
Definitions – ex. A dozen is exactly 12, not 12.1 or 12.2
Counting – ex. There are 25 students in this class, not 24.9 or 25.1
 Exact numbers have an infinite number of significant figures and will never be the
limiting factor in significant figure considerations.
2. Inexact number: a number that has a degree of uncertainty.
 Inexact numbers result from measurements
 It is impossible to make an exact measurement; some uncertainty will always be
present.
3. How does uncertainty in measurements arise? Let’s consider the example below:
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We can say with certainty that the temperature in thermometer A is between 29
and 30 degrees. Since it is closer to 29 we can estimate it to be 29.2.
In thermometer B, we are certain that the temperature is between 29.2 and 29.3
degrees. We can estimate it to be 29.25 degrees.
Both temperature readings have some digits that are exactly known and one digit
that is estimated, the last digit. This is the digit that produces the uncertainty.
This is true of all measurements. Note that it would be wrong to record a
measurement with more than one estimated digit. Also note that the uncertainty
in thermometer B is less than in thermometer A.
In a measurement, the uncertainty is indicated by the number of significant
figures. Significant figures are the digits in a measurement that are known
with certainty plus one digit that is uncertain (estimated). The measurement
in thermometer A, 29.2, has 3 significant figures, while the measurement in
thermometer B, 29.25, has 4 significant figures.
The significant figures are determined by the instrument used. From the
measurement, we can determine the divisions on the instrument that was used. In
thermometer A, the last digit that is known with certainty is in the ones place, so
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the divisions on the instrument have a value of 1. In thermometer B, the last digit
that is known with certainty in the tenths place, so the divisions on the instrument
have a value of 0.1
The uncertainty can also be expressed using +/- notation after the measurement. For
example, the temperature in thermometer A could be expressed as 29.2 +/- 0.1 . This
indicates that the measurement is between 29.1 and 29.3. The uncertainty should be
reported to the place where the estimated digit is. The temperature in thermometer B
could be expressed as 29.25 +/- 0.01.
Practice problems- determine the numerical value in the following volume measurement
and underline the estimated digit. Record the number of significant figures.
a. ___________+/-____
sig. figures ________
b. ___________+/- _____
sig. figures _________
c. __________+/-____
sig. figures __________
Practice problem – determine the numerical value in the following length measurement,
underline the uncertain digit, and record the number of significant figures.
_______ +/- _____cm
sig. figs. ________
Practice problem - Determine the mass shown on the balance.
_______________ +/- _________ grams
sig. figures __________
Practice problem – draw the ruler used to make the measurement shown below:
3.2 cm
_________________________________________________________________
_________________________________________________________________
4. When measurements are made and recorded by others, we use a set of guidelines to
determine the number of significant figures.
Significant Figures
1. Nonzero digits are always significant; ex. 3.11 has 3 s.f., 244.6 has 4 s.f.
2. Leading zeros are never significant; ex. 0.00045 has 2 s.f., 0.0311 has 3 s.f.
3. Confined zeros are always significant; ex. 2.075 has 4 s.f., 2017 has 4 s.f.
4. Trailing zeros are significant if there is a decimal point in the number; ex.
62.00 has 4 s.f., 61400 has 3 s.f.
Practice problems- determine the number of significant figures in each of the following
measurements.
a. 28.6 g
b. 3440. cm
c. 910 m
d. 0.04604 L
e. 0.0067000 kg
5. Significant Figures and Mathematical Operations
 Mathematical operations should not increase or decrease the uncertainty of
measurements.
 To record correctly the numbers obtained through calculations:
1. determine the number of significant figures in the answer, as indicated by the
rules
2. round off number to correct number of significant figures
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Rule 1 – Multiplication and Division – in multiplication and division, the
number of significant figures in the product or quotient is the same as in the
number in the calculation that contains the fewest significant figures.
Ex. 6.038 x 2.57 = 15.51766 (calculator answer)
= 15.5
(correct answer; 3 sig. figs.)
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Rule 2 – Addition and Subtraction – in addition and subtraction, the answer
must have the same number of digits to the right of the decimal point as the
number in the calculation having the fewest digits to the right of the decimal
point.
Ex. 347.1 + 21.03 = 368.13 (calculator answer)
= 368.1 (correct answer; 1 decimal place)
Practice Problems– Perform the following calculations
1) (3.4617 x 107) x (5.61 x 10¯4)
2) (561.0) ÷ (21.888)
3) 3.461728 + 14.91 + 0.980001 + 5.2631
4) 22.101 - 0.9307
6. Exact numbers do not limit the number of significant figures in a calculation.
Ex. Since a pencil has a length of 19.13 cm, the length of 143 of such pencils will
be :
19.13 x 143 = 2735.59 cm (calculator answer)
= 2736 cm (correct answer; 4 significant figures)