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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. - PowerPoint PPT Presentation
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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
ExampleA 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.
ExampleA 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.
ExampleA 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
Rule Example Significant digits
# Significa
nt Figures
Nonzero digits are always significant.
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 insignifican
t)
2
Zeroes written between significant figures are significant.
30.058 3, 5, 8 and zeroes
5
Rules for Identifying Significant Figures
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.
ExampleConversion factors : 1 foot = 12 inches Counts of objects: 23 students in a classCourtesy: http://www.chem.sc.edu/faculty/morgan/resources/sigfigs/sigfigs4.html
Addition and Subtraction 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 (2 decimal places - 4 significant fig.) 0.1 J (1 decimal place - 1 significant fig.) + 0.9378 J (4 decimal places - 4 significant fig.) 51.7078 J (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.)
Courtesy: http://www.physics.uoguelph.ca/tutorials/sig_fig/SIG_dig.htm
Combination of OperationsIF 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 ProblemsPLEASE CHECK THE FOLLOWING WEBSITES TO PRACTISE:
• http://www.chem.sc.edu/faculty/morgan/resources/sigfigs/sigfigs8.html
• http://science.widener.edu/svb/tutorial/sigfigures.html
• http://www.lon-capa.org/~mmp/applist/sigfig/sig.htm