Transcript Significant figures (download)

SIGNIFICANT figures
Two types of numbers: exact and inexact.
Exact
numbers are obtained by counting or by
definitions – a dozen of wine, hundred cents in a dollar
All
measured numbers are inexact.
Learning objectives
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Define accuracy and precision and distinguish
between them
Make measurements to correct precision
Determine number of SIGNIFICANT FIGURES in a
number
Report results of arithmetic operations to correct
number of significant figures
Round numbers to correct number of significant
figures
All analog measurements involve a
scale and a pointer
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Errors arise from:
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Quality of scale
Quality of pointer
Calibration
Ability of reader
ACCURACY and PRECISION
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ACCURACY: how closely a number agrees
with the correct value
PRECISION: how closely individual
measurements agree with one another –
repeatability
–
Can a number have high precision and low
accuracy?
Significant figures are the number of
figures believed to be correct
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In reading the number the last digit quoted is a best estimate.
Conventionally, the last figure is estimated to a tenth of the
smallest division
2.3 6
2.0
2.1
2.2
2.3
2.4
2.5
The last figure written is always an
estimate
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In this example we recorded the
measurement to be 2.36
The last figure “6” is our best estimate
It is really saying 2.36 ± .01
2.0
2.1
2.2
2.3
2.4
2.5
Precision of measurement (No. of
Significant figures) depends on scale –
last digit always estimated
Smallest Division = 1
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Estimate to 0.1 – tenth of smallest division
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3 S.F.
99.6
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97
98
99
100
Lower precision scale
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Smallest Division = 10
Estimate to 1 – tenth of smallest division
2 S.F.
96
70
80
90
100
Precision in measurement follows the
scale
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Smallest Division = 100
Estimate to 10 – tenth of smallest division
1 S.F.
90
0
100
Measuring length
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What is value of large
division?
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What is value of small
division?
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Ans: 1 cm
Ans: 1 mm
To what decimal place
is measurement
estimated?
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Ans: 0.1 mm (3.48 cm)
Scale dictates precision
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What is length in top
figure?
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What is length in middle
figure?
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Ans: 4.6 cm
Ans: 4.56 cm
What is length in lower
figure?
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Ans: 3.0 cm
Measurement of liquid volumes
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The same rules apply
for determining
precision of
measurement
When division is not a
single unit (e.g. 0.2 mL)
then situation is a little
more complex.
Estimate to nearest .02
mL – 9.36 ± .02 mL
Reading the volume in a burette
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The scale increases
downwards, in contrast
to graduated cylinder
What is large division?
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Ans: 1 mL
What is small division?
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Ans: 0.1 mL
RULES OF SIGNIFICANT FIGURES
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Nonzero digits are always significant
(four) 283 (three)
Zeroes are sometimes significant and
sometimes not
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38.57
Zeroes at the beginning: never significant 0.052 (two)
Zeroes between: always
6.08 (three)
Zeroes at the end after decimal: always
39.0 (three)
Zeroes at the end with no decimal point may or may
not: 23 400 km (three, four, five)
Scientific notation eliminates
uncertainty
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2.3400 x 104 (five S.F.)
2.340 x 104 (four S.F.)
2.34 x 104
(three S.F.)
23 400. also indicates five S.F.
23 400.0 has six S.F.
Note: significant figures and decimal
places are not the same thing
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38.57 has four significant figures but two decimal
places
283 has three significant figures but no decimal
places
0.0012 has two significant figures but four decimal
places
A balance always weighs to a fixed number of
decimal places. Always record all of them
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As the weight increases, the number of significant figures in
the measurement will increase, but the number of decimal
places is constant
0.0123 g has 3 S.F.; 10.0123 g has 6 S.F.
Significant figure rules
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Rule for addition/subtraction: The last digit
retained in the sum or difference is
determined by the position of the first
doubtful digit
37.24 + 10.3 = 47.5
1002 + 0.23675 = 1002
225.618 + 0.23 = 225.85
Position is key
Significant figure rules
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Rule for multiplication/division: The product
contains the same number of figures as the number
containing the least sig figs used to obtain it.
12.34 x 1.23 =
15.1782
= 15.2 to 3 S.F.
0.123/12.34 = 0.0099675850891
= 0.00997 to 3 S.F.
Number of S.F. is key
Rounding up or down?
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5 or above goes up
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37.45 → 37.5 (3 S.F.)
123.7089 → 123.71(5 S.F.); 124 (3 S.F.)
< 5 goes down
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37.45 → 37 (2 S.F.)
123.7089 → 123.7 (4 S.F.)
Scientific notation simplifies large and
small numbers
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1,000,000 = 1 x 106
0.000 001 = 1 x 10-6
234,000 = 2.34 x 105
0.00234 = 2.34 x 10-3
Multiplying and dividing numbers in
scientific notation
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(A x 10n)x(B x 10m) = (A x B) x 10n + m
(A x 10n)/(B x 10m) = (A/B) x 10n - m
Adding and subtracting
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(A x 10n) + (B x 10n) = (A + B) x 10n
(A x 10n) - (B x 10n) = (A - B) x 10n