This is “Significant Figures”, section 2.3 from the book Beginning Chemistry (v. 1.0).
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If you use a calculator to evaluate the expression 337/217, you will get the following:
$$\frac{337}{217}=\mathrm{1.55299539171...}$$and so on for many more digits. Although this answer is correct, it is somewhat presumptuous. You start with two values that each have three digits, and the answer has twelve digits? That does not make much sense from a strict numerical point of view.
Consider using a ruler to measure the width of an object, as shown in Figure 2.6 "Expressing Width". The object is definitely more than 1 cm long, so we know that the first digit in our measurement is 1. We see by counting the tick marks on the ruler that the object is at least three ticks after the 1. If each tick represents 0.1 cm, then we know the object is at least 1.3 cm wide. But our ruler does not have any more ticks between the 0.3 and the 0.4 marks, so we can’t know exactly how much the next decimal place is. But with a practiced eye we can estimate it. Let us estimate it as about six-tenths of the way between the third and fourth tick marks, which estimates our hundredths place as 6, so we identify a measurement of 1.36 cm for the width of the object.
Figure 2.6 Expressing Width
What is the proper way to express the width of this object?
Does it make any sense to try to report a thousandths place for the measurement? No, it doesn’t; we are not exactly sure of the hundredths place (after all, it was an estimate only), so it would be fruitless to estimate a thousandths place. Our best measurement, then, stops at the hundredths place, and we report 1.36 cm as proper measurement.
This concept of reporting the proper number of digits in a measurement or a calculation is called significant figuresThe limit of the number of places a measurement can be properly expressed with.. Significant figures (sometimes called significant digits) represent the limits of what values of a measurement or a calculation we are sure of. The convention for a measurement is that the quantity reported should be all known values and the first estimated value. The conventions for calculations are discussed as follows.
Use each diagram to report a measurement to the proper number of significant figures.
Solution
Test Yourself
What would be the reported width of this rectangle?
Answer
0.63 cm
In many cases, you will be given a measurement. How can you tell by looking what digits are significant? For example, the reported population of the United States is 306,000,000. Does that mean that it is exactly three hundred six million or is some estimation occurring?
The following conventions dictate which numbers in a reported measurement are significant and which are not significant:
So, by these rules, the population figure of the United States has only three significant figures: the 3, the 6, and the zero between them. The remaining six zeros simply put the 306 in the millions position. (See Figure 2.7 "Significant Figures" for another example.)
Figure 2.7 Significant Figures
How many of the numbers in this display are actually significant?
© Thinkstock
Give the number of significant figures in each measurement.
Solution
Test Yourself
Give the number of significant figures in each measurement.
Answers
How are significant figures handled in calculations? It depends on what type of calculation is being performed. If the calculation is an addition or a subtraction, the rule is as follows: limit the reported answer to the rightmost column that all numbers have significant figures in common. For example, if you were to add 1.2 and 4.71, we note that the first number stops its significant figures in the tenths column, while the second number stops its significant figures in the hundredths column. We therefore limit our answer to the tenths column.
We drop the last digit—the 1—because it is not significant to the final answer.
The dropping of positions in sums and differences brings up the topic of rounding. Although there are several conventions, in this text we will adopt the following rule: the final answer should be rounded up if the first dropped digit is 5 or greater and rounded down if the first dropped digit is less than 5.
Express the final answer to the proper number of significant figures.
Solution
Test Yourself
Express the answer for 3.445 + 90.83 − 72.4 to the proper number of significant figures.
Answer
21.9
If the operations being performed are multiplication or division, the rule is as follows: limit the answer to the number of significant figures that the data value with the least number of significant figures has. So if we are dividing 23 by 448, which have two and three significant figures each, we should limit the final reported answer to two significant figures (the lesser of two and three significant figures):
$$\frac{23}{448}=\mathrm{0.051339286...}=0.051$$The same rounding rules apply in multiplication and division as they do in addition and subtraction.
Express the final answer to the proper number of significant figures.
Solution
Test Yourself
Express the final answer to the proper number of significant figures.
Answers
As you have probably realized by now, the biggest issue in determining the number of significant figures in a value is the zero. Is the zero significant or not? One way to unambiguously determine whether a zero is significant or not is to write a number in scientific notation. Scientific notation will include zeros in the coefficient of the number only if they are significant. Thus, the number 8.666 × 10^{6} has four significant figures. However, the number 8.6660 × 10^{6} has five significant figures. That last zero is significant; if it were not, it would not be written in the coefficient. So when in doubt about expressing the number of significant figures in a quantity, use scientific notation and include the number of zeros that are truly significant.
Express each measurement to the correct number of significant figures.
Express each measurement to the correct number of significant figures.
How many significant figures do these numbers have?
How many significant figures do these numbers have?
How many significant figures do these numbers have?
How many significant figures do these numbers have?
Compute and express each answer with the proper number of significant figures, rounding as necessary.
Compute and express each answer with the proper number of significant figures, rounding as necessary.
Compute and express each answer with the proper number of significant figures, rounding as necessary.
Compute and express each answer with the proper number of significant figures, rounding as necessary.
Perform each calculation and limit each answer to three significant figures.
Perform each calculation and limit each answer to four significant figures.