Significant figures
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Significant figures.

With thanks to Professor Webster.

The number of significant figures in a result is the number of figures that are reliably known. As an example, measure the diameter of the top of a coffee cup with a millimeter scale ruler. Observe the top is between 80 and 81 mm wide. That is two significant figures. By observing carefully, it might be judged to be between 80 mm and 80.5 mm. The observer could not use the ruler to write 80.23761 mm because that would mean a distinction with respect to 80.23762 mm.

Suppose that diameter is used in some calculations. The calculator displays perhaps ten digits. Please resist the temptation to write them all down. They are not meaningful since the input data are not known that well.

Set up your calculator or computer software to display an appropriate number of significant figures.

Practice Link

Example of error propagation.
The structural rigidity of a circular cylinder of radius r and length L in torsion is
M / theta = G pi r4 /2L
with M as the applied moment or torque, theta as the twist angle, G as the shear modulus, and pi = 3.141592...
The shear modulus is to be calculated from the moment, angle and specimen dimensions.
What is the effect of a one percent error in the length?
What is the effect of a one percent error in the radius?
Replace L by 1.01 L. G changes by a factor of 1.01, so a one percent error in the length gives rise to a one percent error in the shear modulus G.
Replace r by 1.01 r. G changes by a factor of 1.04, so a one percent error in the radius gives rise to a four percent error in the shear modulus G. The calculation is more sensitive to errors in the radius than errors in the length because the radius is taken to the fourth power. Moreover, for a long cylinder used in a torsion test, the radius is considerably smaller than the length. Therefore the effect of a comparable absolute uncertainty in dimension will correspond to a larger percent value in the radius than in the length, hence a larger effect on the calculated modulus.
More formal analysis is to be provided in class.