### Powerpoint

```Significant Figures
Significant Figures
• Engineers often are doing calculations with numbers
based on measurements. Depending on the technique
used, the precision of the measurements can vary
greatly.
• It is very important that engineers properly signify the
precision of the numbers being used and calculated.
Significant figures is the method used for this purpose.
Accuracy vs. Precision
Accuracy refers to how closely a measured value agrees
with the true value
Example
A scale to increments of 10 lbs is not very precise, but, if
it is well calibrated, it is accurate.
Courtesy: http://www.chem.tamu.edu/class/fyp/mathrev/mr-sigfg.html
Precision vs. Accuracy
Precision refers to the level of resolution of the
number.
Example
A scale to increments of tenths of a gram has good
precision, however, if it is not well calibrated, it would
not be accurate.
A scale measures to 0.1 lbs is more precise than one
that measures to 1 lbs.
Courtesy: College Physics by A. Giambattista, B. Richardson and R. Richardson
Significant Figures and Precision
In engineering and science, a number representing a measurement
must indicate the precision to which the measured value is known.
The precision of a device is limited by the finest division on the
scale.
Example
A meterstick, with millimeter divisions as the smallest divisions, can
measure a length to a precise number of millimeters and estimate a
fraction of a millimeter between two divisions.
Courtesy: College Physics by A. Giambattista, B. Richardson and R. Richardson
Significant Figures
The precision of a quantity is specified by the correct number of significant
figures.
Significant figures - All the digits that are measured or known accurately + the
one estimated digit
Example
=
d is to the nearest kilometer  2 significant figures
= .
d is to the nearest tenth of a kilometer  3 significant figures
More significant figures mean greater precision!!!
Courtesy: College Physics by A. Giambattista, B. Richardson and R. Richardson
Rules for Identifying
Significant Figures
Rule
Example
Significant
digits
#
Significant
Figures
58
5 and 8
2
Final or ending zeroes written to the
right of the decimal point are significant.
58.00
5, 8, and
zeroes
4
Zeroes written on either side of the
decimal point for the purpose of spacing
the decimal point are not significant.
0.058
5 and 8
(zeroes are
insignificant)
2
Zeroes written between significant
figures are significant.
30.058
3, 5, 8 and
zeroes
5
Nonzero digits are always significant.
Exact Numbers
Exact numbers: Numbers known with complete certainty.
Exact numbers are often found as conversion factors or as
counts of objects.
Exact numbers have an infinite number of significant
figures.
Example
Conversion factors : 1 foot = 12 inches
Counts of objects: 23 students in a class
Courtesy: http://www.chem.sc.edu/faculty/morgan/resources/sigfigs/sigfigs4.html
of Significant Figures
When quantities are added or subtracted, the number of
decimal places (not significant figures) in the answer
should be the same as the least number of decimal
places in any of the numbers being added or subtracted.
Example
50.67 J
0.1 J
+ 0.9378 J
51.7078 J
(2 decimal places - 4 significant fig.)
(1 decimal place - 1 significant fig.)
(4 decimal places - 4 significant fig.)
(4 decimal places - 6 significant fig.)
Result: 51.7 J ROUNDING !!! (1 decimal place - 3 sig. fig.)
Courtesy: http://www.physics.uoguelph.ca/tutorials/sig_fig/SIG_dig.htm
Multiplication, Division, etc.,
of Significant Figures
In a calculation involving multiplication, division,
trigonometric functions, etc., the number of significant
digits in the answer should be equal to the least
number of significant digits in any one of the
numbers being multiplied, divided etc.
Example
0.097 m-1 (3 decimal places - 2 significant fig.)
X 4.73 m
(2 decimal places - 3 significant fig. )
0.45881
(5 decimal places - 5 significant fig.)
Result: 0.46 ROUNDING !!! (2 decimal place - 2 sig. fig.)
Courtesy: http://www.physics.uoguelph.ca/tutorials/sig_fig/SIG_dig.htm
Combination of Operations
In a long calculation involving combination of operations, carry as
many digits as possible through the entire set of calculations and
then round the final result appropriately.
DO NOT ROUND THE INTERMEDIATE RESULTS.
Example
(5.01 / 1.235) + 3.000 + (6.35 / 4.0)=
4.05668... + 3.000 + 1.5875=8.64418...
The first division should result in 3 significant figures. The last division
should result in 2 significant figures. In addition of three numbers, the
answer should result in 1 decimal place.
Result: 8.6 ROUNDING !!! (1 decimal place - 2 sig. fig.)
Combination of Operations
IF YOU ROUND THE INTERMEDIATE RESULTS:
Example
(5.01 / 1.235) + 3.000 + (6.35 / 4.0)=
4.06 + 3.000 + 1.6=8.66
If first and last division are rounded individually before
obtaining the final answer, the result becomes 8.7 which is
incorrect.
Courtesy:http://www.chem.sc.edu/faculty/morgan/resources/sigfigs/sigfigs4.html
Sample Problems