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 = ½mv2,

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

KE = ½(1.234 Kg)(2.1 m/s)2

KE = 5.44194 Kg·m·s-2

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·m·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.

X

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:

  1. Non-zero digits are always significant.
  2. Any zeros between two significant digits are significant.
  3. Any trailing zeros in the decimal portion of a number (to the right of the decimal point) are significant.
  4. 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.


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.


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.


Practice problems

How many significant digits are in each of the numbers below?

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


Significance in scientific notation


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 one significant digit in this number, because zeros after a significant digit and before a decimal are not significant.


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 four significant digits.

 


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 one significant digit in this number. The leading zero is insignificant, therefore the four zeros between the decimal and the 1 are insignificant.


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 four significant digits in this number. The two zeros sandwiched between the significant digits 1 and 6 are significant.


Practice problems

Perform the following operations on the numbers shown, and express the result using the proper number of significant digits


1. 0.00120 × 100.010
2. (1.002 × 10-3)(2.0230 × 104)
3. 1.00102 + 1.001
4. (2.3 × 102) / 2.104
5. (10.9090)2 · 9.92 x 104
6. 0.0023 / (2.1 · 3.34159)
7. 1.17 × 104 - 2.25 × 103
8. [8.909 × 1011 · 1.0 · (-2.0)] / (4.105 × 10-10)2
9. -0.0002 / (2.4125 × 104)
10. 3.14 · (2.99792458 × 108)2

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