# Mathematical Operations with Significant Figures ```BELL RINGER
#1 Convert 125,458,621,000,000 to
Scientific Notation.
#2 Convert 0.000000125400000 to
Scientific Notation.
#1 125,458,621,000,000
*
1.25458621 x 1014
#2 0.000000125400000
1.254 x 10-7
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 BETWEEN NON-ZERO DIGITS
are SIGNIFICANT parts of a measurement.
• Examples
5005
120301
• 3. Zeros BOTH TO THE RIGHT OF a
non-zero digit AND a WRITTEN DECIMAL
are significant.
• Examples 124.000
5.000
• 4. Zeros that SERVE ONLY AS
PLACEHOLDERS are NOT
SIGNIFICANT.
• Examples 0.000003432
0.0021111
• 5. Zeros to THE RIGHT OF A NON-ZERO
DIGIT BUT to the LEFT OF AN
UNDERSTOOD DECIMAL are NOT
SIGNIFICANT…..they can be the RESULT
OF ROUNDING OFF!
• If a BAR is placed ABOVE A ZERO it makes ALL
digits OVER TO AND INCLUDING THE ZERO
WITH THE BAR SIGNIFICANT.
_
• Example
3400
1250000
• NOTE – If the number is in SCIENTIFIC NOTATION only
consider the COEFFICIENT when determining SFs.
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
• 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.