An understanding of significant digits is critical to becoming a good engineer. The use of significant digits is closely related to the concept of uncertainty of engineering measurements. Virtually all measurements involve some degree of uncertainty about the results, usually due to limitiations in the measuring equipment (a common exception is counting discrete items). Engineers should understand and accept that their measurements are not perfect, but could be made to almost any desired degree of accuracy given enough time and resources. For example, if a room needs to be carpeted, it is not necessary to measure the length and width to much more than the closest 1/2 foot and this would be considered a fairly crude measurement. If the distance between flagpoles at the Tuscaloosa and
Birmingham city halls were measured to the closest 1/2 foot, a much more refined measurement would be required.
Uncertainty can be stated explicitly with given upper and lower bounds, for example the diameter of a piston might be expressed as:
3.125
±
0.002 inches where the indication is clear that the ``true" diameter of the piston is between 3.123 and 3.127 inches.
In this case the uncertainty is often called a “tolerance.” Another way to express the uncertainty is with the way a number is expressed. Significant digits are all digits that can be considered reliable due to either measurement or calculation. Common practice is to retain at most one doubtful digit in any expressed quantity (the doubtful digit is considered significant) and for the assumed uncertainty to be
± one-half of the unit in the doubtful digit's place. For example, in the value 123.4, the ``4" is the doubtful digit and the number would generally be assumed to have an uncertainty of
±
0.05, unless more explicit information is given.
If the value given is 10000 (trailing zeros), you cannot automatically determine the number of significant digits. If the value is known to be between 10500 and 9500, then we have 2 significant digits and the value could properly be written as 1.0E4 (the doubtful digit is in the thousands place). However, if the value were known to be between 10000.5 and 9999.5, then we have 5 significant digits and the value is 1.0000E4 (the doubtful digit is in the ones place). If a numerical value is given with the rightmost digit non-zero, then significant digits are easily counted, unless other information to the contrary is available. For example, 9.652 would generally be considered to have 4 significant digits (the 2 is the doubtful digit).
The concepts of uncertainty and significant digits are also important because of their effects on calculations. Given the piston diameter of 3.125 inches, a calculator might represent the value for the piston area as
Area =
π
* 3.125 * 3.125 / 4 = 7.669903939 inches2 which implies an extremely high degree of accuracy (a billionth of a square inch!). This degree of accuracy in measuring the piston's diameter would be considered ``false accuracy" since the area was not measured this accurately. Applying the same are formula to the uncertain extremes of the piston diameter (3.123 inches and 3.127 inches) gives limits of

7.660089604 inch2 < Area < 7.679724558 inch2
It seems much more reasonable (and much less misleading) to say that the piston area is
7.67
±
0.01 inch2. Note that the "4" in the original equation did not play a role in the determination of uncertainty since it is an exact value (from the definition of diameter), and the value of
π
may be calculated to any desired degree of accuracy.
If the diameter of the piston were stated as 3.125 inches without any inidication as to uncertainty, then what would be the calculated area?
"Small" Area =
π
* 3.1245* 3.1245/ 4 = 7.667449767 inch2
Nominal Area =
π
* 3.125 * 3.125 / 4 = 7.669903939 inch2
"Large" Area =
π
* 3.1255* 3.1255/ 4 = 7.672358505 inch2
:
1) In recording measured data, only one doubtful digit is retained, and it is considered to be a significant digit.
2) All non-zero digits are considered significant.
3) Zeros that occur between non-zero digits are considered significant.
4) Zeros that occur between a non-zero digit and the decimal point are not considered significant
(unless Rule #3 applies!). These zeros can be removed using scientific notation.
5) In dropping digits which are not significant, the last digit should be increased by 1 if the first digit dropped is 5 or greater.
6) In addition and subtraction, do not carry the result past the first column containing a doubtful digit
(going left to right).
1234.5 360310.2
+ 35.678 23400
-------- --------
1270.2 383710.2 --> 383700
^ ^
7) In multiplication, division, and other operations, carry the result to the same number of significant digits that are in the quantity used in the equation with the least number of significant digits.
233.5^2 = 54522.25 sin(34.85 ) = 0.571430
234^2 = 54756 ---> 54800 sin(34.9 ) = 0.572146 --> 0.572
234.5^2 = 54990.25 ^ sin(34.95 ) = 0.572861 ^
digit in digit in
doubt doubt
8) In a lengthy computation, carry extra significant digits throughout the calculation, then apply the significant digit rules at the end.

Note - as a general rule, many engineering values can be assumed to have 3 significant digits when no other information is available.