Presentation: significant figures

Before we get to the physics in this course, we have to talk about mathematics. And before we can even get to the mathematics in this course, let's talk about the numbers.

More specifically, how to properly determine the number of significant figures in measurements and calculations.

First, let's find out how to determine how many significant figures there are for a given number.

There are certain types of numbers for which significant figures are irrelevant. These include mathematical constants and irrational numbers (such as π, e, √2), or ratios (such as 1/2, 3/4, or 1/3), which are considered exact, and effectively have an infinite number of significant figures.

The next type of numbers that should be considered exact, and thus effectively have an infinite number of significant figures are defined quantities, such as 1 minute = 60 seconds. In this case one minute is defined to be exactly 60 seconds--no more, no less, no matter what.

The last type of numbers that should be considered exact, and thus effectively have an infinite number of significant figures are discrete/countable quantities, such as these plastic water bottles. If we are not concerned with the existence of fractional water bottles, then in this case we have exactly 12 water bottles--no more, no less, no matter what. (This is in contrast to very large numbers of quantities where the items would be estimated or extrapolated rather than actually counted one-by-one.)

Now let's look at some examples of (presumably) measured quantities that do need to have their significant figures counted:

0.0179 in (25-gauge wire diameter),
0.4600 in (4/0-gauge wire diameter),
4,186.01 Hz (highest piano key frequency),
27.500 Hz (lowest piano key frequency),
4,350 m (Mount Evans elevation),
200 m (Hamwolsan Mountain elevation).

Given that the measurements above are neither exact mathematical quantities, defined quantities, nor discrete/countable quantities, we can discuss the number of significant figures these values have.

We will be using this significant figures "flowchart of pain." Not so painful, really, once we get used to following the logic flow here--since the quantities listed above are not "exact," nor expressed in scientific notation, then we proceed to the first question: "less than 1?"

The first two measurements (0.0179 in and 0.4600 in) are both quantities that are less than 1, so the flowchart rule is to count all digits from left-to-right, starting from the first non-zero digit on the left.

So for 0.0179 in (where the digits in green denote significant figures), we start counting all digits starting from the "1" (the first non-zero digit on the left), then continue counting from left-to-right the "7" and "9," and thus this measurement has three significant digits. If we were to rewrite this in scientific notation, then it would be expressed as 1.79×10^-2 in. Note that scientific notation always shows the correct number of significant digits.

For 0.4600 in, we start counting all digits starting from the "4" (the first non-zero digit on the left), and then continue counting from left-to-right the "6," "0," and "0," and thus this measurement has four significant digits. If we were to rewrite this in scientific notation, then it would be expressed as 4.600×10^-1 in, which again shows the correct number of significant digits.

Now the next two measurements of 4,186.01 Hz and 27.500 Hz are both greater than 1, and both include the decimal point. The flowchart rule is to simply count all digits. Thus 4,186.01 Hz has six significant figures, and 27.500 Hz has five significant figures, and the respectively would be expressed in scientific notation as 4.18601×10³ Hz and 2.7500×10¹ Hz, each of which display their correct number of significant digits.

The next-to-last measurement of 4,350 m is greater than 1, has no decimal point, but has more than one non-zero digit, so the flowchart rule is to count all digits from right-to-left, starting from the first non-zero digit on the right. Thus for 4,350 m, we start counting all digits starting from the "5" (the first non-zero digit on the right), then continue counting the "3" and "4," and thus this measurement has three significant digits. If we were to rewrite this in scientific notation, then it would be expressed as 4.35×10³ m, as it only has three significant figures.

The last measurement of 200 m is greater than 1, has no decimal point, and does not have more than one non-zero digit (in fact, it only has one non-zero digit). The flowchart rule is then to apply the "rule of two"--where we would arbitrarily assume that such a measurement would have two significant digits, and thus express this in scientific notation as 2.0×10² m.

Second, how to handle significant figures in arithmetic operations.

The multiplication/division rule is to express the final calculation with the same number of significant figures as the quantity with the fewest number of significant figures.

Let's consider two measurements, 464.0 in and 1.40 in. Verify for yourself that they have four significant figures and three significant figures, respectively. If we were to multiply these two numbers together (to find an area expressed in units of square inches, or in²), then:

464.0 in × 1.40 in = 649.60 in²,

but this result should only have three significant figures, as this was determined by the 1.40 in quantity, which has the fewest number of significant figures (in this case, three). In order to unambiguously express this result with the proper number of significant figures, we will round this result in scientific notation to 6.50×10² in². (Note that expressing this result instead as "650 in²" would be ambiguous, as from the flowchart this would seem to have only two significant figures, from applying the "rule of two.")

Now if we were to instead divide the two measurements 464.0 in and 1.40 in (to obtain a unitless ratio), then:

(464.0 in)/(1.40 in) = 331.4285714,

but this result should again only have three significant figures, as determined by the 1.40 in quantity, which has the fewest number of significant figures (in this case, three). In order to unambiguously express this result with the proper number of significant figures, we will truncate this result in scientific notation to 3.31×10².

The addition/subtraction rule is to limit your answer to the right-most digit that is the result from all significant digits. It's better to see how this works, rather than memorizing the statement of the rule itself.

Let's add together the two measurements 464.0 in and 1.40 in. We'll write them as a "stack," aligning them properly for addition:

	464.0	in
+	001.40	in
=	?	in

Since only the hundreds, tens, ones, and one-tenths columns all have non-blank digits in the "stack," they will all be significant in the final answer. However, the last "0" in 1.40 in, in the one-hundredths column is by itself in the "stack," and so the result in that column (which would mathematically be "0") is not significant.

	464.0	in
+	001.40	in
=	465.40	in

Thus the result has only one decimal place, which will then overall have three significant figures, expressed as 465.4, and not 465.40.

The subtraction operation would be similar, again first writing them as a "stack":

	464.0	in
–	001.40	in
=	?	in

Since only the hundreds, tens, ones, and one-tenths columns all have non-blank digits in the "stack," they will all be significant in the final answer. However, the last "0" in 1.40 in, in the one-hundredths column is by itself in the "stack," and so the result in that column (which would mathematically be "0") is not significant.

	464.0	in
–	001.40	in
=	462.60	in

Thus the result has only one decimal place, which will then overall have three significant figures, expressed as 462.6, and not 462.60.


Now consider this unusual example. How many significant figures does "3" and "4" each have? How many significant figures should the product of these two quantities have? This will depend on whether the "3" and "4" given here are countable quantities (such as "3 people" and "4 apples/person"), and thus each are exact, such that the final answer of "12 apples" would be exact as well. On the other hand, if the "3" and "4" are each considered measurements that only have one significant figure each (say, from approximating lengths only to the nearest foot: "3 ft" and "4 ft"), then strictly applying the multiplication/division rule means the result of multiplying them together can only have one significant figure ("1×10¹ ft²").

Thus before performing the multiplication/division rule or the addition/subtraction rule, it will be important to know whether the quantities involved are exact counts of discrete quantities, or are actual measurements with "significant" significant figures and/or decimal places.

Note that we can multiply together or divide by quantities of different units, it is physically meaningless to add or subtract quantities of different units (as shown above), as even though the calculation on this sign arguably follows the addition/subtraction rule correctly...