SIGNIFICANT DIGITS IN MEASUREMENTS

Significant means measured

Significant digits are used to avoid saying more about a measurement than we can possibly know. 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 in hours or milliseconds. The unwritten rule in everyday activities seems to be that we report as much information as we are reliably sure about - then we stop! 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 digits. 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 recorder would not have been justified in adding the 6. Therefore, the value has five significant digits. 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 also 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 digits, 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 digits. 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 digits 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 digits in a measurement include all the certain digits plus the one uncertain digit. Remember: zeros that act as place holders are never significant. Zeros that have been measured are significant.

The previous statements will help you when you are making a measurement and trying to decide how many digits you should record when writing down your measurement. The number of digits in your measurement depends on the calibration or the scales on the measuring instrument.

When you are calculating a measurement, there are some other rules you must consider for determining the correct number of significant digits in the answer.

Multiplication and Division

Look at your measurements in the problem and determine the number of significant digits in each. Numbers such as the value of pi and other defined values are not considered as measurements.

Retain in your answer only as many significant digits as you have in the measurement having the least number of significant digits.

Work the problem.

Carry out the division problems so that you have one (1) more digit than your answer needs.

Then round off to the correct number of significant digits. If the extra digit is a 5 the following rounding rule should be used: round to get an even final digit. Example: 4565 g --> 4560 g and 9875 m --> 9880 m

Addition and Subtraction This rule is based on the precision of the measurements and not number of significant digits

Look at your measurements in the problem and determine the number of decimal places in each.

Retain in your answer only as many decimal places as you have in the measurement having the least number of decimal places.

Work problem, rounding off if necessary.