## Significant Figures

If the error in a measurement can only be estimated to the nearest order of magnitude, it may be most convenient to include the error implicitly, by writing only the appropriate number of significant figures in the reported result. The right-most significant figure in a result should be the only one containing any uncertainty. Or, if you want to be quite conservative, it should instead be the last digit known with certainty. For example, if the raw measurement (or calculator display) is 1.70784, and the four digits on the right (0784) are not known exactly, you should either write your result to three significant figures, as 1.70, or to two significant figures, as 1.7.

The number of significant figures gives a rough approximation of the fractional error in the measurement. The unit size of the least significant digit is an indication of the absolute error. In the first example above, the unit size of the least significant digit is 0.01; this corresponds to the reported absolute error. The reported fractional error, then, is 0.01/1.70 = 0.6%. When collecting data and performing calculations, it's a good idea to use a few more digits than are actually significant, to avoid losing information unnecessarily. All reported results, however, should either show explicit error estimates, or be rounded to the proper number of significant figures.

As discussed in the lab manual introduction, when multiplying two numbers together, the general rule of thumb is to write the answer using the same number of significant figures as the multipliers. When the multipliers have different numbers of significant figures, the smallest is used. Put another way, the fractional error in the product will be of the same order of magnitude as the largest of the fractional errors in the numbers you started with. Thus 0.3526 x 1.2 = 0.42 (not 0.42312). This same method should be used for division.

Addition is different. Consider the example: 0.2056 + 14.25 + 576.1 = 593.1. An answer of 593.1276 is not appropriate because the last three digits (.0276) add nothing to the accuracy of the results, since one of the numbers being added (576.1) is accurate only to tenths. Here the absolute error in the sum will be of the same order of magnitude as the largest of the absolute errors in the original numbers. Subtraction works similarly.

Rules for counting significant figures in a number:

1. The leftmost non-zero digit is the first significant figure.
2. If there is no decimal point, the rightmost non-zero digit is the last significant figure.
3. If there is a decimal point, the rightmost digit is significant, zero or not.
4. Any digits between significant figures are also significant.

 Introduction Main Body •Experimental Error •Minimizing Systematic Error •Minimizing Random Error •Propagation of Error •Significant Figures Questions
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