Significant Digits

Every measurement is uncertain to some extent. The reported value of any measurement should adhere to the correct number of significant digits, which depend on the precision of a measurement. Ultimately, this is controlled by both the limitations of the measuring device and the skill with which it is used. A reported measurement should contain all of the digits that are known with certainty, plus one more which is an estimate. Suppose, for example, that the mass of an object is measured on a platform balance (to the nearest 0.1 g) as 35.4 g. The number of significant digits in this case is three. We are sure of the first two figures (the 3 and 5) and know that the mass is greater than 35 g. The third digit (4) is somewhat uncertain; at best, it tells us that the true mass lies closer to 35.4 g than to either 35.3 g or 35.5 g.

The following rules can be used to determine the proper number of significant figures to be reported for a measurement:

1. The digits 1 through 9 are always significant.

2. Zeros are significant if:

a) they occur between two significant digits;
b) if they are the last digit, and are to the right of the decimal point.

Initial (leading) zeros are not significant.

e.g.:

0.007010 has two significant zeros.
0.008 has not significant zeros.
0.501 has one significant zero.

3. The precision of a measurement cannot be changed by changing the units.

4. Certain values, such as those that arise from the definition of terms, can be considered as exact. For example, there are exactly 1000 g in on kg. In other words, there are an infinite number of significant zeros to the right of the decimal point.

5. The result of addition and subtraction should be reported to the least significant digit of the most imprecise term used in the calculation.

e.g., 2.72043 + 6.7 + 0.435635 = 9.856065

should be reported as 9.9 (see 7 below concerning rounding off the result).

6. Multiplication or division should be rounded off to the number of significant digits in the least precise term used in the calculation.

e.g., 263.07 * 0.35 = 92.0745 should be reported as 92, since the least precise term in the calculation has two significant digits.

7. If a calculation produces a result with more significant digits than required, the following rules should be used to round off the number:

a) If the figure following the least significant digit is less than 5, the number is rounded down;
b) If the figure following the least significant digit is 5 or greater, the number is rounded up.

Scientific Notation

Scientific notation is a convenient shorthand way to write very large or very small numbers. This method also overcomes the problem of interpretation of significant zeros. A number written using scientific notation has only one nonzero digit to the left of the decimal point. For example, 6.00×10² is the same as 600 (three significant digits).