Transcript Lecture 3

How many significant figures will the answer to
3.10 x 4.520 have?
You may have said two. This is too few. A common
error is for the student to look at a number like
3.10 and think it has two significant figures. The
zero in the hundredth's place is not recognized
as significant when, in fact, it is. 3.10 is the key
number which has three significant figures.
Three is the correct answer. 14.0 has three
significant figures. Note that the zero in the
tenth's place is considered significant. All trailing
zeros in the decimal portion are considered
significant.
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Another common error is for the student to think
that 14 and 14.0 are the same thing. THEY ARE
NOT. 14.0 is ten times more precise than 14.
The two numbers have the same value, but they
convey different meanings about how
trustworthy they are.
Sometimes student will answer this with five. Most
likely you responded with this answer because it
says 14.012 on your calculator (the correct
answer should be reported as 14.01). This
answer would have been correct in your math
class because mathematics does not have the
significant figure concept.
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2.33 x 6.085 x 2.1. How many significant
figures in the answer?
Answer - two.
Which is the key number?
Answer - the 2.1.
Why?
It has the least number of significant figures
in the problem. It is, therefore, the least
precise measurement.
Answer = 30
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(4.52 x 10-4) ÷ (3.980 x 10-6).
How many significant figures in the answer?
Answer - three.
Which is the key number?
Answer - the 4.52 x 10-4.
Why?
It has the least number of significant figures in the
problem. It is, therefore, the least precise
measurement. Notice it is the 4.52 portion that
plays the role of determining significant figures;
the exponential portion plays no role.
Answer = 113.6
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4.20x3.52
Which is the key number?
Both have 3 significant figures. In this case,
the number with smaller value, regardless
of the decimal point, is the key number
(3.52). The answer is 14.8 (the correct
answer should be reported as 14.78 as will
be seen shortly.
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Look at the following multiplication problem:
35.63 * 0.5481 * 0.05300  88.5470578%
1.1689
he key number is 35.63 which has 4 significant
figures. Therefore, the answer should be
88.55%
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42.68 * 891
 546.57
132.6 * 0.5247
It is clear that the key number is 891 and the
answer should have 3 significant figures.
However, in cases where the answer is less than
the key number as is our case, the answer
retains an extra digit as a subscript. The answer
is 546.6
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97.7 x 100.0
32.24
 36.04
687
When multiple operations are involved, do it
in a step by step procedure. The
parenthesis above has 97.7 as the key
number.
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301.36 36.04
687
 0.4911
The second process has 687 as the key number.
Finally as the answer is less than the key number
an additional digit was added as a subscript. It is
noteworthy to observe that extra digits were
retained temporarily in all steps and rounding off
to the correct number of significant figures was
done in the final answer.
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Logarithms
The digits to the left of the decimal point are not
counted since they merely reflect the log 10x and
they are not considered significant. The zeros to
the right of the decimal point are all significant.
Examples
Log 2.0x103 = 3.30 (two significant figures in both
terms). The blue digit in the answer is not
significant.
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Log 1.18 = 0.072 (three significant figures in
both terms, the blue zero in the answer is
significant)
Antilog of 0.083 = 1.21 (three significant
figures in both terms)
Log 12.1 = 1.083 (three significant figures in
both terms, the blue digit in the answer is
not significant)
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Errors
Errors can be classified according to nature
of the error into two types, determinate
and indeterminate errors. A determinate
error (sometimes called a systematic
error) is an error which has a direction
either positive or negative. An example of
such an error is performing a weight
measurement on an uncalibrated
balance(for instance it always add a fixed
amount to the weight).
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Another important example is measuring volume
using a burette that has extra or less volume
than indicated on its surface. When using the
abovementioned balance or burette, our results
will always be higher or lower depending on
whether these tools have positive or negative
bias. This means that determinate error is
unidirectional. Sometimes a determinate error
can be significant if the analyst is careless or
inexperienced neglecting enough drying times in
a gravimetric procedure, using a too high
indicator concentration in a volumetric
procedure, etc.
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An indeterminate error is a random error
and has no direction where sometimes
higher or lower estimates than should be
observed are obtained. In most cases,
indeterminate errors are encountered by
lack of analyst experience and attention.
Indeterminate errors are always present
but can be minimized to very low levels by
good analysts and procedures.
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