0.1 A guide to significant figures

The number of significant figures is important because this indicates the precision of your data and results. Determining the number of significant figures is not a trivial task; you can't simply count the number of places to the right of the decimal as you may have been taught in grade school. The following paragraphs attempt to provide some guidance that will be of immediate use in your homeworks and exams. A failure to account for significant figures will affect your grade for each question. Even more importantly, you should apply these guidelines throughout your career and to evaluate the quality of any data you are presented with.

• Any digit that is not zero is significant (e.g., 329 has three significant figures).

• Zeros between nonzero digits are significant (e.g., 309 has three significant figures).

• Zeros to the right of a decimal point are significant as long as they are to the right of a nonzero digit (e.g., both 0.50 and 0.0050 each have two significant figures).

• Zeros to the right of the decimal but before a non-zero digit are essential but not significant. 300, 3, and 0.003 each have one significant digit. Note that the consequence of this rule is that you can freely change units without creating or deleting significant figures. Three millimeters and 0.003 m each have one significant figure. In general, it is best to report your data in as close to the original units as possible.

• Numbers with trailing zeros that do not contain decimal points are ambiguous (e.g., 12,000 means only that the real value is between 11,500 and 12,499. In contrast, 12,000. or 1.2000 x 104 indicates that the actual value is known to be between 11,999.5 and 12,000.5).

• Counting numbers (e.g., number of rain gages, number of days with rain) have an infinite number of significant figures for calculation purposes because the values are integers. The results, however, must still take into account the accuracy of the data. As one example, we are probably more certain of the number of rain gages in a given area than the number of days with rain.

• For addition and subtraction, the number of significant figures in the final answer is determined by the uncertainty associated with the numbers being added. 201 plus 56 yields 257; even though 56 only has two significant figures, the final answer has three significant figures because each number has a similar precision (plus or minus 0.5). Subtracting 236 from 12,000 yields an answer with only two significant figures (12,000) because the uncertainty in 12,000 (plus or minus 500) is greater than the number being subtracted. If you express 12,000 - 236 as 11,764, you are creating more precision than actually exists and this is not allowed! From a scientific point of view, you should ignore the 236 and try to improve the precision associated with the larger number (12,000).

• For multiplication and division, calculators and computers make it easy to carry all the figures throughout the calculations, and in such cases you should always carry at least one more significant figure than you are likely to need. It is then your responsibility to round the final answer according to the smallest number of significant figures in any of the original numbers. Until you have a lot of practice, a series of calculations can help you determine the allowable precision of your answer. If you are multiplying 257 times 0.2, for example, yields 51.4, but in reality you only know only that your final answer is somewhere between 40 and 60. The reasoning for this is that the uncertainty in 0.2 ranges from 0.15 to 0.25. So at one extreme 0.15 times 257 is 39, and this rounds to 40; at the other extreme, 0.25 times 257 is 64, which rounds to 60). Thus the best estimate of your final answer is 50, as this is the rounded value from 51.4. The important point is that your answer should have only one significant figure, and you choose 50 rather then 40 or 60 because 50 is the rounded version of your calculated answer of 51.4 and 50 is statistically more likely to be close to the "true", but unknown, value.