The term significant figures actually refers to particular digits in a number. These are sometimes called significant digits. In this document we will use the term significant figures to discuss the broader topic. That way we may still say “digit” to draw your attention to a particular digit under discussion. After you have become familiar with the topic you may use either term.
Introducing Significant Figures
Only those figures or digits of a numerical quantity which are the result of actual measurement are said to be significant. For example, if you measure the thickness of a coin, you can write it as
1.6 mm or 0.16 cm or 0.0016 m.
How many significant figures are there in this measurement? Clearly only the digits 1 and 6 are the actual measured values. Therefore we have only 2 significant figures. Zeros used as placeholders are not significant. This would include all of the zeros in 0.0016 m.
We can use scientific notation to avoid misunderstanding. We would report the measurement as:
1.6 mm or 1.6 x 101 cm or 1.6 x 103 m.
With the use of scientific notation every digit that appears is significant. Here are some examples.
2.736 has 4 significant figures
2.00 has 3 significant figures
4 x 103 has 1 significant figure.
But 4.0 x 103 has 2 significant figures.
Standard notation would not let us distinguish between the last two examples. They would both appear as 4000.
Other Numbers Having Significant Figures
Direct measurement is not the only way a number may contain significant digits. The number may be an Exact or Defined Number, it may be an integer, or the number could have been computed from numbers that have significant digits.
Defined numbers: The base of the natural logarithms is e = 2.781828… . This number has a mathematical definition and is exact. Every digit you choose to display from this number is significant. = 3.14159…, the square root of 2 (= 1.4142135…) and similar numbers are also exact. Defined unit conversion values are also exact. For example there are exactly 2.54 centimeters to the inch. This is how the inch is defined. So this number (2.54 cm/in) is exact.
Integers: When you count, the result is exact (assuming that you do not loose count). If you tell a friend that you have paid $2000 dollars for a computer, there is only one significant figure in this number. Most people will not trouble their friends with the price “One thousand nine hundred eighty seven dollars and thirty six cents” ($1987.36). On the other hand the year 2000 computer problem (Y2K) that received so much press is a number with four significant figures. The count of years is exact. The problem did not occur when computer date counters flipped to ’99, and is a dead issue when the flip to ’01.
Rational fractions: Any fraction made from integers is exact. So 2/5 or 1/137 are ratios of exact integers and are also exact. The student must be careful with these fractions. The quantity described must be inherently an integer to apply this rule. Thus one egg is exactly one twelfth (1/12) of a dozen, but the ratio of three inches to two inches (3 in/2 in) is not exact because we cannot measure lengths with unlimited precision.
When we convert rational numbers to decimal fractions they always produce a set of repeating digits. For example (106/33) = 3.2121212… .
Computed results: Any math operation with numbers having significant figures will result in a number having significant figures. However the number of significant figures in the result depends on all the inputs to the problem. These procedures are described in the section Rounding After Math Operations below.
Rounding off:
When computing results on a calculator we often end up with many digits displayed. Because computation itself cannot increase our measurement accuracy we must decide how many of these figures are significant and round the result back to the appropriate number of figures.
Here is the rule for rounding:
Once you have decided how many figures you will keep look at the first digit you will reject. When the first rejected digit is less than 5 you will round down (simply delete the rejected digits). When the first rejected digit is greater than 5 round up (this means that you will still delete the rejected digits but now you add one to the last digit you keep). If the first digit rejected is equal to 5 round up or down so that the last digit retained is even. If we always round 5 up we will distort the results for very large data sets. For example:
273.92 rounded to 4 digits is 273.9
1.97 rounded to 2 digits is 2.0
2.55 rounded to 2 digits is 2.6
4.45 rounded to 2 digits is 4.4
If you do not round after a computation, you imply greater accuracy than you measured.
Rounding After Math Operations:
The rule for choosing the number of digits to retain depends on the mathematical operation you perform. The simplest case is for multiplication or division. Here the number of significant figures in the result is equal to the number of significant figures in the least accurate value used in the computation. In the following examples the least accurate number is in bold face type.
(273.92) X (3.25) = 890.24 is rounded to 890 (3 digits because 3.25 has only 3 digits).
(1/3) X (5.20) = 1.73333 is rounded to 1.73 (3 digits because 5.20 has 3 digits). Note: 1/3 is exact since the 3 is an integer.
(1.97) X (2) = 3.94 is rounded to 4 (1 digit only because 2 has only 1 digit).
(2.0) X = 6.28318… is rounded to 6.3 (2 digits only because 2.0 has only 2 digits).
This last example using (3.14159…) is worth noting. Calculations involving pi are perhaps the most common cases where students write more significant figures than they should.
For addition or subtraction the rule can be stated as: “When some power of ten is uncertain in one of the inputs it cannot be reported in the output.” But this is easier to see by example. The sum, 12.8 + 11 = 24, not 23.8. This is because 11 is uncertain in the tenths place and so we do not know what if anything was added to the 0.8. It helps to set up the addition or subtraction in the standard place value format with the decimal points lined up:
There is a caution to be made here. In the third example standard notation does not tell us the number of significant figures in the number 200. If the correct representation of this measurement originally, was 2.0×10^2, then the answer to the subtraction should still be 2.0×10^2. This is because we do not know the value of the units place in the input number 2.0×10^2 thus we do not know the result of subtracting 5 from it.
To clarify your problem and avoid this difficulty you can use exponential notation and round the columns back to the decimal place that contains the least significant digit of the whole set of values before making the calculation. Our examples become:
Sequential Calculations and Intermediate Results:
You will often use the results of one calculation as one of the values in a subsequent calculation. That answer may then be used in yet another calculation and so on. Repeated rounding at each step can introduce errors that would not occur if you combined all of the steps algebraically and computed the final result all at once. One way to avoid these rounding errors is simply to do the algebra described and obtain the final answer in a single calculation. However the intermediate results may be of interest to your reader. In this case you should present these values with their proper number of significant figures, but keep a copy off to the side containing one or two extra figures. Use these more precise numbers from the scratch paper numbers to complete the subsequent calculations.
Last Updated November 19, 2014
