Significant Figures

The Basics of Significant Figures
(C) - Copyright, 1996 F.W. Boyle, Jr.

Significant figures is sometimes referred to as significant digits. The idea behind significant figures is that any measurement has an associated significance and an associated error. When the results of measurements are combined to produce a new number (numerical value), the results can only be as good as the poorer (least significant) measurement. The quality of the value (known as accuracy) cannot improve simply by multiplying a less accurate value by another more accurate value. In other words, the poorness continues through.

Some examples of this are:

1) An estimate of the number of jelly beans in a jar is 3250. An average jelly bean weight is found to be 1.2977 g. How much do the jelly beans weigh?

In order to understand significant figures, we must look at each value separately.

First the number 3250 really only has 3 digits in it. The trailing zero (0) is called a place holder and is not accurate UNLESS the number is stated to be 3250.0. In this latter form the number is said to be exact. For this problem the number was not exact but was estimated to be 3250. In scientific notation, 3250 is written as 3.25x10³. Thus when written in scientific notation it is easier to determine the number of actual digits in a number.

The second number, 1.2977, has 5 digits and is already in scientific notation.

Since 3.25x10³ has only 3 digits the maximum number of digits acceptable for calculating the weight of all the jelly beans is 3 digits. To do the math we use ALL the digits thus:

                        3.25 X 10³
                      x 1.2977
                      --------
                        4.21752 x 10³

But our answer can only have 3 significant figures since the least number of significant figures in the numbers being multiplied is 3. So the next step is to look at the digit just to the right of the last significant figure. In this case the last significant figure is 1 and the number to the right of it is 7. So the number one will be rounded up to 2 and all the other digits will be dropped. The answer is then:

4.22 x 10³ g There is no set number for significant figures. Each problem must be judged on its own. It is really simple to determine the least number of significant figures by counted how many significant figures there are in each of the numbers being put into the calculation. The one with the fewest digits sets the maximum number of digits that can be kept in the final answer.

Now for some practice in counting significant figures.

0.0000375

This number appears to have 7 significant figures. However, the leading zeros are simply placeholders. This means the zeros are there only to keep the number values in their proper position. The actual count of significant figures is 3, for the 3, 7, & 5.

60270000

Again as written one might think this number had 8 significant figures but like the zeros in the first example, trailing zeros are placeholders and are not counted in the number of significant figures. The actual number of significant figures in this number is 4. Included in the count are the 6, the zero between the 6 and the 2, the 2 and the 7.

Try these numbers on your own:

    1) 1.000004

    2) 9240000

    3) 0.000000000011

    4) 0.010001

    5) 109267

    6) 3.2 x 10⁵