Digits

A common error committed by students is to display too many digits in a numerical answer. Many engineers consider an excessive number of digits to be an incorrect solution The rules displaying digits can be summarized below:

1. The number of digits displayed indicates the precision of the measurements and other values used to arrive at the answer.
2. When adding or subtracting, the digits displayed are as precise as the least precise value being added or subtracted.
3. When multiplying or dividing, the answer is only as precise as the least precise value being multiplied or divided.
4. When in doubt, be reasonable.

When using scientific notation there is no ambiguity in the number of significant digits. For instance, has four significant figures. Likewise, also has four. Written without scientific notation 500.2 would be known to have four significant figures, but the number of digits in in 7,000 would be unknown. Depending on the type of work, it is common to not use scientific notation in engineering. The rules for determining the number of significant figures state that all nonzero digits are significant, all zeros between nonzero digits are significant,  and all digits written when using scientific notation are significant.

Examples of calculations using significant figures:

because the least precise number is to the nearest tenth resulting in an answer to the same precision.

because the least number of significant digits is one, so the answer is given to one digit.

When displaying a solution, the number of digits represents the precision of the solution. There are two basic kinds of numbers in calculations.  The first is a measurement, and the second is a count.  Measurements have a finite number of digits as a result of the instrument used to take the measurement and the human error in measuring the measurement. The number of significant digits is shown when the measurement is written. A count has an infinite number of digits, but the number of digits shown is a whole number with no digits in the decimal places. It is the units that identify counts. Counts are either a thing or a person. The example below demonstrates this.

  In this example each widget (a thing) weighs 2.4 N, so two widgets weigh 5.8 N. It is the widget weight (a measurement) that controls the number of digits.

Another way to think of the measurements is with the terms “accuracy” and “precision” which are shown below using the metaphor of a dartboard.

Measurements of a single variable such as weight or length are precise when they are close together and this is shown by a relatively large number of significant digits. Accuracy is determined by the closeness of the measurement to its “true” value. True is in quotations because of the fact that the true value of any measurement is always unknown and unknowable. One way to convince yourself that the true value of a measurement is unknowable is to imagine measuring the width of a room. Start by measuring with an increasing level of precision as shown in the following table.

Method	Value
Pace	33 ft
Yardstick	33.2 ft
Tape	33.18 ft
Laser	33.184 ft
More precise laser	33.1837 ft

As the table demonstrates, the width of the room can be measured to an ever increasing precision leaving its true value unknown