Significant Figures, Why do they Matter ?

1
Significant Figures, Why do they Matter ?
Truth in advertising When you give the result
of a calculation, you communicate two
things 1) The number itself 2) How well
that the number is known If you measure the
sinking velocity of a foram while out at sea on a
ship and you give the result 1 cm/sec But
you neglect to mention that your boat was in 20
ft seas Your proper answer would correctly be
1cm/sec /- 620 cm! How well is your results
known ???
2
A Few Rules on Significant Figures
1. When you multiply or divide round off to the
same number of figures as in the factor with the
least number of figures. 4.56 3 significant
figures x1.4 2 significant
figures 6.384
6.4
3
A Few Rules on Significant Figures
2. When you add or subtract, it is the position
of the digits that is important, NOT how many
there are. Imagine (or actually do this!) that
the various terms that are added or subtracted
together are in a vertical column with the
decimal points all lined up. Spot the term that
goes least far to the right. Round off to that
position.
12.11 2 decimal places 18.0 1
decimal place 1.013 3 decimal
places 31.123
31.1
4
Significant Figures and Decimal Places
20.0 1 decimal place 4.00 2 decimal places
24.0 1 decimal place
20.0 3 significant figures x 4.0
2 significant figures 80. 2
significant figures
5
I am a ZERO
1. Nonzero Integers Nonzero integers are ALWAYS
significant.
3798 4 significant figures 26.5 3
significant figures
2. Leading Zeros Zeros which precede nonzero
digits are NEVER significant
0.0025 2 significant figures 02 1
significant figure
6
I am a ZERO
3. Captive Zeros Zeros which fall between
nonzero digits are ALWAYS significant
1.008 4 significant figures
3079 4 significant figure
4. Trailing Zeros Zeros at the right end of a
number is significant ONLY with a decimal
point.
100 1 significant figure 100. 3
significant figures 3300 2 significant
figures 3.300 4 significant figures
7
A Few Examples
Identify the number of significant figures in
each example
12 1900 1098 20,100 2.001 x
103 .0001 7.300 1.01 x 10-5 220,400
How would you convert 35.9 m to cm using
significant figures ?
8
(No Transcript)
9
Measurement Errors
Measurements are central to science. The laws of
science are discovered by measurements. Any law
which is contradicted by a measurement must be
re-considered.
10
Measurements

• Measurements are approximate subject to
  uncertainties given by your measuring device.
• Errors are not mistakes! - Errors cannot be
  avoided when making measurements. Mistakes can
  usually be avoided...

11
Measurements
When making a measurement, your accuracy is only
good to ½ of the smallest graduation you have
available. What is the length of the gray bar
above ?
12
Propagation of Errors

• Making more than one measurement
• Combine your measurements using addition or
  multiplication.

Rule 1 When 2 measurements are combined by
addition ?????z sqrt(?x2
?y2) Rule 2 When 2 measurements are combined
by multiplication ?z/Z sqrt((?x/x)2
(?y/y)2)
13
Propagation of Errors

• Keep all significant digits to the end.
• The Factor-of-Two rule If one measurement has
  much larger error than another, the smaller one
  will have a negligible effect on the final
  answer.
  
  If A
  has 1 error and B has 2 error then

error sqrt ( 12 22 ) 2.24 2
14
Speaking Logarithmically
BIG numbers and small numbers... how to deal with
them.

• Logarithms
• Exponents
• Scientific Notation

15
Speaking Logarithmically
Increased CO2 absorbed by oceans (Feely et al.,
2004)

• Solubility product of calcite is 0.00000000447
• Can also be written in scientific notation

4.47 x 10-9
Geochemists write this as 10-8.35 What is
this ?
16
Speaking Logarithmically

• The age of the Earth is 4,600,000,000 yrs
• Can also be written in scientific notation

4.6 x 109
One of these people might write it as 109.6
What is this ?
To speak to someone like a geochemist it might
help to learn this language...?
17
Logarithms
109.6 has an exponent which is the logarithm of
4,600,000,000. So log (4,600,000,000)
9.6 But the notation is accurate 109.6
4,600,000,000 These all follow from the basic
definition of a logarithm.
Logb N x,
If bx N
So a logarithm is nothing more than an exponent.
Logb N x is asking b to what
power equals N ?
18
(No Transcript)
19
Logarithmic Notation
There are many ways to express a number. Ex.)
Distance from Earth to the Moon is 380,000 km.
Scientific Notation 3.8 x 105
Logarithmic Notation 105.6 (assumes
log with base 10)
Natural Logarithm (ln) loge N x The
transcendental number e has a value of 2.71828
(to 6 figures) The above equation asks e
to what power equals N ? If ln (380,000)
12.848, then what is this in logarithmic
notation ?
20
Moving from one Logarithmic Base to Another
You may remember another rule of logs to
transform from one base (a) to another base (b)
logb N (loga N) / (logb a)
Do an example convert 105.5798 to 164.6399
21
What Does This Tell us About Scientific Notation ?
Let's try to translate between scientific
notation and logarithmic notation 105.5798
Take the distance to the moon of 380,000 km. In
Logarithmic notation this is written as
105.579 But in scientific notation this is
3.80 x 105 We an rewrite this according to rules
for multiplying exponents.
380,000 105.5798 100.5798 x 105
3.80 x 105
22
What Does This Tell us About Scientific Notation ?
Let's try to translate between scientific
notation and logarithmic notation
We an rewrite this according to rules for
multiplying exponents.
380,000 105.5798 100.5798 x 105
3.80 x 105
This works because 100.5798 3.80 or
log (3.80) 100.5798
23
Try an Example
Write the age of the Earth in 1.)
scientific notation and 2.)
logarithmic notation 3.) Show the
translation