Counting Significant Digits
Dr. MJ Patterson
In science, the ultimate test of any idea is an experiment. An idea can look great on paper, and make sense when people discuss it, but if it contradicts the results of experiments designed to test the idea, then the idea is discarded.
Since nobody wants to give up a pet theory, scientists are obsessed with measurement. The procedures employed during an experiment must withstand the scrutiny of the scientific community before the results will be accepted. And, at a fundamental level, every experiment involves measuring some quantity.
Keep in mind that every measurement is limited by the caliber of the instrument used to make that measurement. Consider this experiment: In my bathroom, I keep a scale. The dial of the scale is marked to the nearest pound. If I step on the scale, the scale might read 125 pounds (no comments from the peanut gallery!). What will happen to the reading of the scale if I take off a ring and set it on the counter? Should it read less? Does it read less?
Of course, when I removed the ring, the total weight on the scale decreased, so it "should" read less. But, a typical bathroom scale is not sensitive enough to distinguish the difference of a fraction of an ounce, and so the reading remains unchanged even though there is less weight on the scale!
How do scientists deal with the limitations of their measuring instruments? One answer is significant digits (aka significant figures), which is a way of determining which numbers in any measurement are valid, and how valid any calculations based on those numbers are. Unfortunately, to the beginning science student, significant digits quickly become a quagmire of rules designed to confuse. So, over the next few modules we will try to find our way out of the swamp and make sense of significant digits and calculations.
Rules for significant digits: We will take the approach that
someone else has presented you with a number from a measurement, and you need
to determine which digits are significant or important from the experimental
point of view.
(Do not worry. There are several more rules that will come below!)
Examples: In each of the following numbers, underline all of the significant digits, and give the total number of significant digits.
1. 3.14
2. 3.14159
3. 111
4. 101
5. 10001
Solutions:
1. 3.14 - all nonzeroes - 3 sig figs (short
for significant figures)
2. 3.14159 - all nonzeroes - 6 sig figs
3. 111 - all nonzeroes
- 3 sig figs
4. 101 - the zero is sandwiched between sig
figs, so it is significant - 3 sig figs
5. 10001 - the zeroes are sandwiched between
sig figs making them significant - 5 sig figs
The next rule for significant figures is for exact numbers.
3. Exact numbers have an unlimited number of significant digits.
An exact number could be something easily counted, such as the number of people in a small room. It could also be a defined quantity, such as the 1 and the 2.54 in the conversion factor 1 in = 2.54 cm. Or, it could be something like the 2 in the equation diameter = 2 x radius for a circle.
Continuing the rules for significant digits:
4. A zero to
the left of a number is not significant.
5. An exact number
has an infinite number of significant figures.
You can think of zeroes to the left of a number as simply being decimal placeholders. They do not provide any information about how precise a measuring tool was used; instead, they indicate the magnitude of the number.
An exact number is one that has no uncertainty in the measurement. For instance, the room I am in at the moment has exactly 2 cats in it. Not 1.5 cats, and not 2.1 cats, but exactly 2 cats. In this case, the number 2 is considered to have an infinite number of sig figs, as if it were written 2.0000000000... and all of those infinite zeroes trailing at the end are significant.
For practical purposes, conversion factors are considered to have an infinite number of significant figures. For instance, 1 m = 1000 mm. Both the 1 and the 1000 have an infinite number of sig figs.
Examples: In each of the following numbers, underline all of the significant digits, and give the total number of significant digits.
1. 0.578
2. 0.00578
3. 0.0000578
4. 0.05708
5. 1 kg = 1000 g
Solutions:
1. 0.578 The 0 is to the left of the
number, and is NOT significant - 3 sig figs
2. 0.00578 The three 0's are to the left of the number and NOT
significant - 3 sig figs
3. 0.0000578 Again, the 0's to the left are NOT significant - 3
sig figs
4. 0.05708 Combining rules 2 and 3, the sandwiched 0 is significant, the left 0's are NOT - 4 sig figs
5. 1.00000,,, kg = 1000.0000,,, g
Both the 1 and the 1000 have an infinite number of sig figs
The remaining rules deal with zeroes to the right of the number.
6. A zero to
the right of the number, but the left of the decimal, is not significant.
7. A zero with a
bar under it (or sometimes over it) is significant.
8. A zero to the
right of the number, and the right of the decimal, is significant.
A zero to the right of the number, but the left of the decimal, is not significant.
In this case, the zero is simply keeping track of the decimal point. It does not add any information about how good the instrument was that was used to make the measurement.
Example 1:
In the following numbers underline the significant digits and give the total
number of sig figs.
a. 11,000
b. 101,000
c. 50
Solution 1:
a. 11,000 = the zeroes are right of the number and
left of the decimal, so they are not significant - 2 sig figs
b. 101,000 = the three zeroes to the right of the
number are left of the decimal, and so they are not significant. The
sandwiched zero is significant - 3 sig figs
c. 50 = the zero is right of the number and left of
the decimal, so it is not significant - 1 sig fig
A zero with a bar under it (or sometimes over it) is significant.
Sometimes a measurement will end in a zero that is significant, but just writing the number, such as 50, loses that extra sig fig. In these cases, to specify that the zero is significant, you draw a bar under the zero. So, if the number is written 50, it has two sig figs.
(Sometimes you will see the bar written over the number. With modern word processing software, both conventions are used, but underlining is simpler.)
Example 2:
In the following numbers underline the significant digits and give the total
number of sig figs.
a. 500
b. 500
Solution 2:
(Since underlining is already used to mean sig figs in this example, I am
changing the color of the sig figs.)
a. 500 the first zero is underlined making it
significant. The last zero is to the right of the number and left of the
decimal, so it is not significant - 2 sig figs
b. 500 the last zero is underlined making it
significant. The first zero is sandwiched between two sig figs, making it
significant - 3 sig figs
A zero to the right of the number, and the right of the decimal, is significant.
Each time you write a zero at the end of the number to the right of the decimal, you are writing something that is already understood. So why write one out there? The reason is that the tool you used to make a measurement can measure that precisely. You want to record the full precision from your tools.
For example, we have several different balances on campus. Some will weigh to the nearest one hundredth of a gram; others to the nearest one thousandth of a gram. If you use the first type, your measurement might read 4.00 g. But if you use the second type, the same measurement might read 4.000. The fact that those zeroes are recorded is important because it tells how expensive of a balance you used.
Example 3:
In the following numbers underline the significant digits and give the total
number of sig figs.
a. 95.50
b. 60.0
c. 0.008700
Solution 3:
a. 95.50 the zero is to the
right of the decimal and to the right of the number, making it significant - 4
sig figs
b. 60.0 the zero to the right
of the decimal and the number is significant. The other zero is
sandwiched between sig figs, making it significant - 3 sig figs
c. 0.008700 the zeroes to the
left of the number are not significant (decimal placeholders). The zeroes to
the right of the nubmer and the decimal are
significant (measuring instrument precision) - 4 sig figs
Rules for Significant Figures
This is the whole list!