Honesty requires that we don't report what we don't know.
In the sciences, we often find ourselves in the position of having to combine various measurements, each with a different precision, into one result. It's important to recognize that such a result, whether it requires addition, subtraction, addition or multiplication (or some other operation) among some group of measurements, is only as good as the least-precise measurement.
The result of a calculation is only as precise as the least-precise piece of data.
Here's an example. Let's say we want to calculate the kinetic energy of a moving object. The formula is
$$KE = \frac{1}{2} mv^2$$
where m is the mass in Kilograms (Kg) and v is its velocity in meters per second (m/s or m·s-1).
Now let's suppose that the mass is known to the nearest milligram: m = 1.234 Kg, and the velocity is known to the nearest tenth of a m/s: v = 2.1 m/s. The kinetic energy (KE) is then
$$ \begin{align} KE &= \frac{1}{2} (1.234 \; Kg)\left(2.1 \; \frac{m}{s} \right) \\[5pt] &= 5.44194 \; Kg \cdot m \cdot s^{-2} \end{align}$$
Now our question is: Is it right to report this number with this much precision (i.e. to the 100,000th place?
Answer: no. In fact, we can only truthfully report it to the same precision as our weakest input, 2.1 m/s, which, we say, has two significant digits. Our answer can then only have two significant digits, so we'd round it to
$$KE = 5.4 \; Kg \cdot m \cdot s^{-2}$$
It turns out that there are agreed-upon rules for deciding how many digits of precision to report, and the rest of this section will teach you how to decide.
precision / precise
In math and science, precision doesn't necessarily imply accuracy. Precision is a measure of the spread of a series of measurements of the same thing. If many measurements of the same quantity are performed and no measurement is too far from the average, the measurement is precise. If many measurements lie farther from the average, then the measurement is less precise.
Significance rules
The rules for deciding significance of digits are:
- Non-zero digits are always significant.
- Any zeros between two significant digits are significant.
- Any trailing zeros in the decimal portion of a number (to the right of the decimal point) are significant.
- The digits of all exact numbers (like a population) are all significant.
Rule 1: Non-zero digits are always significant.
This rule is fairly self-explanatory. If a measurement yields a non-zero number, then that number has significance to the experiment. Let's say we measure the following temperatures in some experiment. The table gives the number of significant digits in each result.
| Temperature (˚C) | Signif. digits |
|---|---|
| 27 | 2 |
| 27.1 | 3 |
| 1.272 | 4 |
| 325.14 | 5 |
| 2794.2267 | 8 |
| 2274.01 | 6 |
| 2274.010 | 7 |
Let's take a look at each of those temperatures. 27˚C has two significant digits. We assume here that the reader of the thermometer intended to measure to that precision and gave us the most precise number possible. Good lab technique demands that we use all of the precision that an instrument gives us*.
The temperatures 27.1˚C, 1.272˚C, 325.14˚C and 2784.2267˚C contain no zeros, thus the number of significant digits in each is just the count of nonzero numbers (3, 4, 5 & 8, respectively).
The last two numbers are more difficult because they contain zeros. For that we'll have to move on to our other significance rules.
Both of those numbers contain a zero between nonzero numbers, and the last also has a zero that trails all of the non-zero digits after the decimal point.
Rule 2: Any zeros between two significant digits are significant.
Here are some examples of masses measured to varying degrees of precision, and a count of the significant digits of each.
| Mass (Kg) | Signif. digits |
|---|---|
| 68 Kg | 2 |
| 86.01 Kg | 4 |
| 86.009 Kg | 5 |
| 320.4 Kg | 4 |
| 320.40 Kg | 5 |
The first mass, 69 Kg, has two significant digits according to rule 1 above.
The second has a zero between the decimal and a non-zero number. This digit must be considered to be significant. We must assume that the operator of the balance read to the full precision of the instrument and that the zero in the tenths place is meaningful, thus the number has four significant digits.
For the same reason, 86.009 Kg has five significant digits. The next mass, 320.4 Kg has a zero before the decimal, but between non-zero numbers. It, too is significant.
Finally, the last number in the table has two things going on: a zero sandwiched between two non-zero numbers is significant, but we'll need our next rule for that trailing zero.
Rule 3. Any trailing zeros in the decimal portion (to the right of the decimal point) are significant.
Here are some examples (times this time) of numbers that require our third rule:
| Time (seconds) | Signif. digits |
|---|---|
| 1.220 s | 4 |
| 1.23200 s | 6 |
| 1.2020 s | 5 |
| 27.60 s | 4 |
| 270.1010 s | 7 |
The first time in the table has four significant digits. We must assume that the person who performed the measurement intended for the zero in the thousandths place to be significant, else she'd have left it off.
The second time, t = 1.23200 has two trailing zeros, thus six significant digits, for the same reason.
Notice that two of the numbers in this table contain both zeros between non-zero numbers and zeros as the last digit after the decimal point. For example, the first zero in 270.1010 s lies between 7 and 1, and is thus significant. The same goes for the second zero (sandwiched between 1's). The final zero comes after the decimal part of the number, and is thus significant.
Rule 4. All digits of an exact number are significant.
Imagine that the population of a city is reported to be 370,650 people. According to our rules, the trailing zero of this number would not be significant. But this number is an exact number. We assume that all people were counted and that the number came out as it did. The trailing zero is significant because if we just lopped it off, we'd have a different number.
The speed of light, for another example, is defined exactly as c = 2.99792458 × 108 m/s. It contains nine significant digits, as many as are known.
There are 500 sheets of paper in a ream of paper. There are three significant digits in this number (not just one) because the 500 is a count or inventory of the sheets. It is an exact number.
Operation differences
The rules are slightly different for addition/subtraction and multiplication/division.
The rules for how many significant digits to keep are slightly different for multiplication or division and for addition or subtraction.
Addition & subtraction
If we need to subtract 10.2 from 12345.6, we'd have 12345.6 - 10.2 = 12335.4. It wouldn't make any sense to round this answer to 12000 (the number of significant digits in 10.2) because we're subtracting digits one-to-one. The rule for addition and subtraction is to report the result using the least number of significant digits after the decimal. In this example, that number is one digit, so our answer 12335.4 is good.
Examples:
1094.4543 - 234.11 = 860.3443
Here we would report the result with two significant digits after the decimal, 860.34, because 234.11 has the least number of significant digits to the right of the decimal (2).
340002.1004 + 45402.28 = 385404.3804
We would report with the result with two significant digits after the decimal, 385,404.38.
Multiplication & division
For multiplication or division, our result can only have the total number of significant digits of the number having the least of them. For example:
2.111 x 2.5 = 5.2775, which we would report with two significant digits (and using good rounding rules), 5.3, because 2.5 is our weakest number in terms of significant digits, having only two.
Here are some more examples:
| 1013 - 2.1132 | = 1011 |
| 343.323 - 0.0064322 | = 343.316 |
| 43.44 × 23.712 | = 1030 |
| 689.8432 / 1.233 | = 559.484 |
| 101325 / 23.44 | = 4323 |
| 67.4909 + 873.424 | = 940.833 |
| -45.00901 × 2.3211 | = 104.47 |
| -3.244 - 3.144 | = -6.388 |
| 6.529856 + 5.7891 | = 12.3190 |
| 23.91234 / 7.0001 | = 3.4160 |
Sums and differences: The number of significant digits after the decimal in the result is the same as in the added or subtracted number with the fewest significant digits after the decimal.
Products and quotients: The number of total significant digits in the result is the same as in the number with the fewest in the product or quotient.
Practice problems
| 1. | 5.7 | 6. | 500 | 11. | 90,000 |
| 2. | 5.70 | 7. | 5.000 | 12 | 90,000.0001 |
| 3. | 5.07 | 8. | 5001 | 13. | 0.001 |
| 4. | 5.0700 | 9. | 5001.2040 | 14. | 0.000000100 |
| 5. | 50.070 | 10. | 90200 | 15. | 1.00000100 |
Solutions to problems 1-12
When numbers are expressed in scientific notation, the same rules for significance apply. We sometimes just need to just mentally reverse the scientific notation to visualize the number first. Here are some examples.
1.00 × 105
This number is 100000, a 1 followed by five zeros. There is only
1.007 × 105
This number is 100700. We get it by sliding the decimal of 1.007 five places to the right and filling in with zeros. The two trailing zeros are insignificant, but the two zeros sandwiched between the 1 and 7 are. This number has
1.00 × 10-5
This number is 0.00001, a 1.00 with the decimal point moved five places to the left, filled in with zeros. There is only
1.006 × 10-5
This number is 0.00001006, a 1.006 with the decimal point moved five places to the left, filled in with zeros. There are
Practice problems
Perform the following operations on the numbers shown, and express the result using the proper number of significant digits
| 1. |
$0.00120 \times 100.010$ Solution$$0.00120 \times 100.010 = 0.120012$$ The first number has only three significant digits, so that determines the number in our result: $$\bf = 0.120$$ |
| 2. |
$(1.002 \times 10^{-3})(2.0230 \times 10^4)$ Solution$$ \begin{align} (1.002 &\times 10^{-3})(2.0230 \times 10^4) \\ &= 20.27046 \end{align}$$ The first number has only four significant digits, so that determines the number in our result: $$\bf = 20.27$$ |
| 3. |
$1.00102 + 1.001$ Solution$$1.00102 + 1.001 = 2.00202$$ This is an addition, so we need to mindful of the rule. The second number has only three significant digits after the decimal. So must our result: $$\bf = 2.002$$ |
| 4. |
$(2.3 \times 10^2) / 2.104$ Solution$$\frac{2.3 \times 10^2}{2.104} = 109.31558$$ The numerator has only two significant digits, so our result can have only two. In order to achieve that, we'll have to round 109 to 110. $$= \bf 110$$ |
| 5. |
$(10.9090)^2 · 9.92 \times 10^4$ Solution$$ \begin{align} (10.9090)^2 &· 9.92 \times 10^4\\ &= 1.18054 \times 10^7 \end{align}$$ 9.92 has three significant digits, the least of the two numbers, so our result can have only three. $$= \bf 1.18 \times 10^7$$ |
| 6. |
$0.0023 / (2.1 · 3.34159)$ Solution$$\frac{0.0023}{2.1(3.34159)} = 3.27759 \times 10^{-4}$$ 2.1 is the weak link here, with only two significant digits. So our result can have only two: $$= \bf 3.3 \times 10^{-4}$$ |
| 7. |
$1.17 \times 10^4 - 2.25 \times 10^3$ Solution$$ \begin{align} 1.17 \times 10^4 &- 2.25 \times 10^3 \\ &= 9.450 \times 10^3 \end{align}$$ This is an addition problem, so the result should have the same number of significant digits to the right of the decimal than the fewest present in the sum, that's the .25 in 2.25 or the .17 in 1.17. The result is $$= \bf 9.4 \times 10^3$$ |
| 8. |
$\frac{8.909 \times 10^{11} · 1.0 · (-2.0)}{(4.105 \times 10^{-10})^2}$ Solution$$ \begin{align} &\frac{8.909 \times 10^{11} · 1.0 · (-2.0)}{(4.105 \times 10^{-10})^2} \\ &= 1.05738 \times 10^{31} \end{align}$$ The 1.0 and the 2.0 have only two significant digits; so must the result: $$= \bf 1.0 \times 10^{31}$$ |
| 9. |
$-0.0002 / (2.4125 \times 10^4)$ Solution$$\frac{-0.0002}{2.4125 \times 10^4} = 8.290155 \times 10^{-9}$$ The leading zero in 0.0002 doesn't count, so the least number of significant digits is four. The result should be reported as $$= \bf 8.290 \times 10^{-9}$$ |
| 10. |
$3.14 · (2.99792458 \times 10^8)^2$ Solution$$ \begin{align} 3.14 &· (2.99792458 \times 10^8)^2 \\ &= 2.822091 \times 10^{17} \end{align}$$ The 3.14 has the fewest significant digits (3). So must the result: $$= \bf 2.82 \times 10^{17}$$ |
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