## 1. SIGNIFICANT FIGURES
Physics is called a quantitative science because it describes nature using quantities which are measurable, and seeks to discover mathematical laws relating these quantities. You may be already familiar with some of these quantities: length, time, mass, force, etc. To measure such quantities, we use measuring instruments in a prescribed way in order to assign a number (value) to that quantity. For example, to measure a length we use a measuring instrument (a marked meter stick) in a specially prescribed manner (laying the stick off on the object to be measured) to determine the numerical value we call the "length." Other measurements may require more complicated instruments, or more complex techniques, but the principle is the same. The meter stick example also illustrates the uncertainties common to all measurements. The smallest division marks on the stick represent millimeters. These are quite small, but with a little practice, most persons can estimate fractions of millimeter. Estimates smaller than l/4 millimeter are less reliable. This limits the accuracy of measurements made with a meter stick.
Suppose a small dust particle lies between two millimeter markings of the stick. We would like to specify the position of the particle. [This scale has its centimeter markings labeled 1, 2, 3, ... etc. Each centimeter is divided into 10 millimeters, which have marks, but are not labeled. Figure one shows a magnified view of a one centimeter portion of the stick.] Figure 1 illustrates the particle's position as seen by the observer. We might specify the position by saying. "The particle lies between 6 and 7 mm on the scale." This statement specifies a range in which the particle is known to lie. The position might also be specified by stating it "to the nearest millimeter." Since the particle is closer to 7 mm than it is to 6 mm, we would say, "The position is 7 millimeters, to the nearest millimeter." This statement also specifies a range in which the particle is found, but in this case the range is from 6.5 mm to 7.5 mm. This method of stating scale readings is very common in physical science. Usually, we simply state the reading as "0.7 cm" without explicitly specifying that the position was read to the nearest millimeter. Writing the value in this way implies the precision of the measurement, for if we had attempted to read the position to the nearest tenth of a millimeter we might have expressed the value as 0.78 cm. [Bear in mind that on the real meter stick (not this magnified one) the digit "8" might not be completely certain.] This convention may be formulated as a rule:
scale limited, and will be treated by rule 1.
When repeated measurements of a quantity do
Whatever the cause, indeterminate errors reveal themselves when repeated measurements give different values. A typical set of values for a measurement might look like this: 3.69 3.68 3.67 3.69 3.68 3.69 3.66 3.67 We assume that the measured quantity has just one precise value, independent of the
measuring process, and that the variability of the recorded values is caused by imperfections
of the instruments or procedure. We want to represent our knowledge, obtained from these
measurements, as We might represent this measurement by the The value 3.68 is still somewhat uncertain. Nothing to the right of the 8 was certain
enough to keep, but the 8 itself is a borderline case; it is 3.68 ± 0.02 Estimates or measures of uncertainty are called Unfortunately data errors propagate through calculations, usually producing even worse error in the results. In the following discussion we review the "rules for significant figures", a crude method for ensuring that calculated results are stated to a precision consistent with the precision of the data. The student may already be familiar with this method, but this organized review of the rules may be instructive.
To illustrate the meanings of the technical terms, consider the number
It has seven Each digit occupies a particular This number is said to be "expressed to three decimal places" since three digits are explicitly shown to the right of the decimal point. The numbers which appear in your calculator display may have an many more digits, but we know that many of those are of no significance, and are unreliable. The fifth decimal place to the left is zero, as are all other places still further left. They are not shown. When we say a number is expressed "to three decimal places" we mean that three digits are shown to the right of the decimal point. When we say a number is expressed to "seven digit accuracy" we mean that a total of seven digits are certain enough to be explicitly shown, as in the number we are considering here. Now suppose that the number 3586.297 cm represents an experimental measurement, and
we experimentally determined that its uncertainty was ñ 0.2 cm. The size of the uncertainty
tells us that the digits 9 and 7 are superfluous, and carry no significant information. They
express an amount smaller than the uncertainty. Such digits are called In example (1) the digits 3, 5, 8, and 6 are called "certain" digits, because the uncertainty is too small to affect their value. The 9 and the 7 are completely uncertain. What about the digit 2? It is not certain, because the size of the uncertainty (ñ 0.2) tells
us that it might have a value ranging anywhere from 0 to 4. Yet it is not This terminology may be summarized with a diagram. These names always relate to the first uncertain digit,
We customarily drop insignificant digits when recording data and stating results, according to the following rules.
So the proper way to write the measurement discussed above are rounded to four significant digits: 358.3 cm. This is straightforward ecept in the case where the leading insignificant digit happens to be a 5. Some books advocate a rule to handle this in an unbiased way:
This rule ensures that over many calculations you won't be introducing systematic error in one direction (up or down).
Rule 3 is a bit of "overkill", considering that significant figures rules are themselves only an approximate indication of quality of a result. Little is lost by simply discarding all insignificant digits. If you are taking a course which expects you to use this rule, you may find your results sometimes differ from the "book values" by 1. That's no "big deal".
An important feature of experimental data is that the errors combine and propagate through calculations to produce errors in the calculated results. The concept of significant figures is a crude and inadequate tool for dealing with this (we'll introduce better ways later). But it is instructive to consider how insignificant figures are generated in simple calculations. The operations of these examples are not ones you would normally do longhand. Consider this multiplication example, in which uncertain digits are shown in red bold italics. 395 Multiplying 3954 by the uncertain digit 6 gives a number in which every digit is uncertain. In the other sub multiplications, digits resulting from multiplication by the uncertain digit 4 must be counted as uncertain. Any column addition containing uncertain digits gives an uncertain result. Therefore only the first two digits of the answer are certain. The three is uncertain, and the remaining digits are completely uncertain. Therefore this result should be rounded to 114. Notice that even though the multiplicand had four significant digits, the result has only three. This illustrates a rule:
The rule for addition is different because the decimal location of the first uncertain digit determines the location of the first uncertain digit in the sum.
Example.
0.5286
The seven and the zero in the answer are certain because their value does not depend on the uncertain digit 3. The 2 is the first uncertain digit. We emphasize that the above examples are illustrative only. In practice no one would do a calculation in this manner. One would use the rules to predict the number of significant figures required in the answer, do the entire actual calculation with a digital calculator, then round off the answer to include the first uncertain digit. It is A practical common-sense method to determine the number of certain
figures is to use the calculator directly. The digit 3 in the number 9.4263 in the above example
is uncertain, but These examples demonstrate the inadequacy of Electronic calculators are so inexpensive these days that every student has one, and no one would likely do a square root longhand, as we have done here. The square root displayed on the calculator screen is 3.070228, which gives not the slightest clue how many of the digits are significant. Obviously we need to examine this matter of uncertainties in more detail, with the goal of developing a relatively simple and reliable mathematical process for estimating the precision of results of mathematical calculations. This will be done in subsequent chapters. © 2004 by Donald E. Simanek |