Transcript Significant Digits (or Significant Figures)

Jim Pulickeel
SPH 3U1
September 2009
What’s the point of Significant
Digits?
 The radius of a circle is 6.2 cm. What is the Area?
 But pi is....
3.1415926535897932384626433832795028841971693993751058209749445
9230781640628620899862803482534211706798214808651328230664709
3844609550582231725359408128481117450284102701938521105559644622
94895493038196442881097566593344612847564823378678316527120190
91456485669234603486104543266482133936072602491412737245870066
063155881748815209209628292540917153643678925903600113305305488
20466521384146951941511609
 Which answer is more accurate?
 The original measurement of the radius was far too
crude to produce a result of that precision. The real
area of the circle could easily be as high as 128, or as
low as 113
 Sig Figs are a way of describing the precision of a
measurement. 6.2 cm has two digits of precision - two
significant digits. Anything beyond that is in the range
of round off errors - further digits are artefacts of the
calculation, which shouldn't be treated as meaningful.
What time is it?
 Someone might say “1:30” or “1:28” or “1:27:55”
 Each is appropriate for a different situation
 In science we describe a value as having a certain number
of “significant digits”
 The # of significant digits in a value includes all digits that
are certain and one that is uncertain
 “1:30” likely has 2, 1:28 has 3, 1:27:55 has 5
Significant Errors with Significant
Digits
 The Hubble Space Telescope has a lens, one of the
most precisely figured mirror ever made, but it was too
flat at the edges by about 2.2 microns (10-6m) because
of rounding and conversion errors during the grinding
process.
 During the Iraq War I (1991), 24% of American friendly
fire deaths were attributed to rounding errors in
missile launching software.
What are Significant Digits?
 Significant Digits indicate the precision of a measurement
 When recording Sig. Figs. we include the known (certain)
digits plus a final estimated digit
 What can we say about the following paper clip?
What is the length of this thing?
What About When Numbers are Given to You?
RULES FOR SIGNIFICANT DIGITS
Rounding...
Significant Digits
 It is better to represent 100 as 1.00 x 102
 Alternatively you can underline the position of the last
significant digit. E.g. 100.
 This is especially useful when doing a long calculation or
for recording experimental results
 Don’t round your answer until the last step in a calculation.
Adding with Significant Digits
 How far is it from Toronto to room 229? To 225?
 Adding a value that is much smaller than the last sig. digit
of another value is irrelevant
 When adding or subtracting, the # of sig. digits is
determined by the sig. digit furthest to the left when #s are
aligned according to their decimal.
Adding with Significant Digits
 How far is it from Toronto to room 229? To 225?
 Adding a value that is much smaller than the last sig. digit
of another value is irrelevant
 When adding or subtracting, the # of sig. digits is
determined by the sig. digit furthest to the left when #s are
aligned according to their decimal.
 E.g. a) 13.64 + 0.075 + 67 b) 267.8 – 9.36
13.64
+ 0.075
+ 67.
81
80.715
267.8
– 9.36
258.44
Sample Questions
i)
83.25
– 0.1075
83.14
ii)
4.02
+ 0.001
4.02
iii)
0.2983
+ 1.52
1.82
Multiplication and Division
 Determining sig. digits for questions involving
multiplication and division is slightly different
 For these problems, your answer will have the same
number of significant digits as the value with the fewest
number of significant digits.
 E.g. a) 608.3 x 3.45 b) 4.8  392
a) 3.45 has 3 sig. digits, so the answer will as well
608.3 x 3.45 = 2098.635 = 2.10 x 103
b) 4.8 has 2 sig. digits, so the answer will as well
4.8  392 = 0.012245 = 0.012 or 1.2 x 10 – 2
Sample Problems
i) 7.255  81.334 = 0.08920
ii) 1.142 x 0.002 = 0.002
iii) 31.22 x 9.8 = 3.1 x 102
iv) 6.12 x 3.734 + 16.1  2.3
22.85208
+
7.0
= 29.9
vii) 1700
v) 0.0030 + 0.02 = 0.02
+ 134000
vi) 33.4
Note: 146.1  6.487
+ 112.7
= 22.522 = 22.52
135700
+
0.032
=1.36 x105
146.132  6.487 = 22.5268
= 22.53
Unit conversions
 Sometimes it is more convenient to express a value in
different units.
 When units change, basically the number of significant
digits does not.
E.g. 1.23 m = 123 cm = 1230 mm = 0.00123 km
 Notice that these all have 3 significant digits
 This should make sense mathematically since you are
multiplying or dividing by a term that has an infinite number
of significant digits
E.g. 123 cm x 10 mm / cm = 1230 mm
i) 1.0 cm = 0.010 m
ii) 0.0390 kg = 39.0 g
iii) 1.7 m = 1700 mm or 1.7 x 103 mm