Significant figures are used to avoid saying more about a measurement than we can possibly know or to properly report what we can and cannot measure. In everyday life we expect people to report as much information as is usefully known, not less - and certainly not more! For example, you don't report the time it takes you to tie your shoe laces as 5.685743 s or the length of the room as 23.43234205 feet. The unwritten rule in everyday activities seems to be that we report as much information as we are reliably sure about - then we stop! Such as 5 seconds and 23 feet in the previous examples. Yet when we turn to scientific calculations, that common sense rule is often violated or forgotten.

The concept of significant figures is important because measured values always have a limit of precision dependent upon the measuring device used to make that measurement. For example, if someone records the mass of a box as 15.3 kg, scientists assume that the person taking the measurement was certain that the box was greater than 15 kg and less than 16 kg, otherwise the 3 would be unnecessary. The value is said to have three significant figures. If someone records the mass of the box as 15.376 kg, scientists assume that the value is between 15.37 and 15.38 kg, otherwise the person would not have been justified in adding the 6. Therefore, the value has five significant figures. It is also assumed that the second reading of the box's mass was take on a device that would give a more precise measurement and probably a more accurate one.

Difficulties arise when dealing with values like 0.00703 g. It is tempting to say that this value has six significant figures, but actually the first zero (the Paul Revere zero) is just a convention used with decimal points, and the two zeros after the decimal point merely hold place value. The value actually only has three significant figures. This becomes more obvious when the value is converted to milligrams: 0.00703 g = 7.03 mg, telling scientists that the recorder was certain that the value was between 7.0 and 7.1 mg.

Another set of problem-causing numbers is 2500 g and 2500.0 g. It is a common error to say that the first of these two numbers has four significant figures; but as before the zeros are place holding zeros that could be eliminated if the value were converted to another unit of measurement: 2500 g = 2.5 kg or written in scientific notation 2.5 x 10³ g, leaving a value with two significant figures. (One might wonder how it is known that this value is not 2.500 kg and therefore more precise, that's the point - the measurer must properly record the measurement for you to know what was actually measured!). Now consider how many significant figures are in 2500.0 g. It is tempting to say there are two, since it looks something like the first number. However, this value actually has five significant figures, because a scientist would assume that if that last zero were not significant you would not have put it there, and instead would have written 2500 g.

The following are some "working rules" to use when making measurements:

Certain digits in a measurement are those digits read directly from the scale of the measuring instrument. This means giving a number value to the measured marks (calibrations) on the scale.

The uncertain digit in a measurement is estimated. You must "guess" this digit because this scale is not calibrated (marked off) on the measuring instrument. It would be the next finest scale to be marked off if you were going to make a more precise measuring instrument. All the measurements you make must have one estimated digit (0 thru 9).

Significant figures in a measurement include all the certain digits plus the one uncertain digit. Remember: zeros that simply act as place holders are never significant. Zeros that have been measured are significant. Just a note...zeros between significant figures are not place holders, they were part of the measuring process.

When you are using measurements to calculate other values, there are some rules you must consider for determining the correct number of significant figures in the answer. This is where students typically violate the unwritten rule of reliable reporting. For example: 89.5 feet divided by 4.23 seconds results in a speed of 21.15839243 ft/s (that's what my calculator shows). That is precise to the hundred millionth of a ft/s! That is not possible to report such precision when the measurements where only precice to the tenth and hundredth places. So rules have been agreed upon to limit the reported precision of calculations resulting from actual measurements.

Multiplication and Division