Transcript Mathematical Operations with Significant Figures

BELL RINGER – Complete on a sheet of paper and TURN
IN before working on notes!
A student needed to calibrate a graduated cylinder [a
device to measure liquids]. She collected the following
data:
Trail #1 99.98 mL Trial #2 100.02 mL Trial #3 99.99 mL
The accepted value of the cylinder’s volume is 100.00 mL.
What is the PERCENT ERROR of her measurements?
• Average = 99.98 + 100.02 + 99.99 = 99.99
•
3
• Error = 99.99 – 100.00 = -.01
• Percent Error =
0.1 x 100 = 0.01 % error
•
100.00
Significant Figures
Dealing with uncertainty in
measurements.
What values are shown below?
• Why is it difficult to be certain about some
of the measurements you make?
– All measurements have SOME DEGREE OF
UNCERTAINTY due to limits associated with
the measuring device.
– Generally, uncertainty begins with the LAST
DIGIT of the measurement.
• In a measurement, ALL THE DIGITS
KNOWN FOR CERTAIN plus the first
ESTIMATED DIGIT are known as the
SIGNIFICANT FIGURES of the
measurement.
• It is generally accepted that when a
measurement is given, ALL NON-ZERO
DIGITS are considered SIGNIFICANT.
For example 175.4 grams
Digits known
for certain.
First estimated digit.
The Problem with Zero
• While all NON-ZERO DIGITS are
considered significant, ZEROS present a
particular problem.
– Zeros can be measurements
– Zeros can be place holders
• How do you decide whether or not a zero
is significant?
Rules for Significant Figures
• 1. ALL NON-ZERO digits are considered
•
significant.
Examples
125.45
5648
1.1211
• 2. Zeros IN THE MIDDLE OF NUMBERS
•
are SIGNIFICANT parts of a measurement.
Examples
5005
120301
• 3. Zeros AT THE BEGINNING OF A
NUMBER are not significant.
Examples 0.000003432 0.0021111
• 4. Zeros AT THE END OF A NUMBER
are only significant IF THE FOLLOW A
DECIMAL or a BAR is placed over a zero…
when this occurs, ALL digits up to and
including the zero with the bar are significant.
_
Example 45.23000 1.000
505.32000 4750000
• NOTE – If the number is in SCIENTIFIC
NOTATION only consider the COEFFICIENT
when determining Significant Figures.
• Example
4.965 x 1016
Practice Problems
• Determine how many figures are
significant in each of these measurements:
• 1. 375
2. 89.000
• 3. -0.00032
4. 4300
• 5. 12.0900
6. 0.00003200
• 7. 900001
8. 2.34 x_ 104
• 9. -0.000212000 10. 4002000
Mathematical Operations with
Significant Figures
• When completing math calculation, the
final answer must be reported rounded to
the appropriate number of significant
figures.
• The answer is rounded according to the
LAST mathematical operation completed.
Rules
• 1. Complete calculations following the
order of operations.
• 2. If the FINAL step is MULTIPLICATION
or DIVISION:
– A. Look at each value given in the problem
and find the one with the LEAST number of
significant figures.
– B. Round the FINAL ANSWER to the same
number of significant figures.
– DO NOT ROUND UNTIL THE FINAL STEP!
Mult/Div Examples
• 4.59 X 1.22 = 5.5998 = 5.5998 = 5.60
• 3 sf
3sf
3sf
3sf
• 3 sf
• 4 sf
•
•
45.6
0.002454
= 18581.90709
= 18587.90709 3sf
= 18600 3sf
ADD/SUBTRACT
• Complete calculations following order of
operations.
• If the FINAL step is addition or subtraction:
– A. Only consider digits to the RIGHT of the
decimal.
– B. Determine the fewest SF to the right of the
decimal.
– C. Round final answer to this number of SF.
ADD/SUBTRACT EXAMPLES
25.4 (1 sf)
63.66 (2 sf)
+ 102.44 (2 sf)
191.50
= 191.5
15.000 – 2.3791 = 12.6209
(3 sf)
(4 sf)
= 12.621