2.4: Expressing Numbers - Significant Figures

Learning Objectives

• Understand the importance of significant figures in measured numbers.
• Identify the number of significant figures in a reported value.
• Use significant figures correctly in arithmetical operations.

Counting is the only type of measurement that is free from uncertainty, provided the number of objects being counted does not change while the counting process is underway. The result of such a counting measurement is an example of an exact number. If we count eggs in a carton, we know exactly how many eggs the carton contains. The numbers of defined quantities are also exact. By definition, 1 foot is exactly 12 inches, 1 inch is exactly 2.54 centimeters, and 1 gram is exactly 0.001 kilogram. Quantities derived from measurements other than counting, however, are uncertain to varying extents due to practical limitations of the measurement process used.

Uncertainty in Measurement

Scientists have established certain conventions for communicating the degree of precision of a measurement, which is dependent on the measuring device used. Imagine, for example, that you are using a meterstick to measure the width of a table. The centimeters (cm) marked on the meterstick, tell you how many centimeters wide the table is.

Many metersticks also have markings for millimeters (mm), so we can measure the table to the nearest millimeter. Most metersticks do not have any smaller (or more precise) markings indicated, so you cannot report the measured width of the table any more precise than to the nearest millimeter. However, you can estimate one past the smallest marking, in this case the millimeter, to the next decimal place in the measurement (Figure \(\PageIndex{1}\)).

Figure \(\PageIndex{1}\): Measuring the length of an Object to the correct number of digits as 1.35 cm. Three digits should be reported for the length of this object. Why?

The concept of significant figures addresses this limitation. The significant figures of a measured quantity are defined as all the digits known with certainty (those indicated by the markings on the measuring device) and the first uncertain, or estimated, digit (one digit past the smallest marking on the measuring device). It makes no sense to report any digits after the first uncertain one, so it is the last digit reported in a measurement. Zeros are used when needed to place the significant figures in their correct positions. Thus, zeros are sometimes counted as significant figures, but they are sometimes used only as placeholders.

The uncertain, or estimated, digit, is one digit past the smallest marking on the measuring device.

“Sig figs” is a common abbreviation for significant figures.

Only One Estimated Digit is Allowed

Consider the earlier example of measuring the width of a table with a meterstick. If the table is measured and reported as being 1,357 mm wide, the number 1,357 has four significant figures. The 1 (thousands place), the 3 (hundreds place), and the 5 (tens place) are certain; the 7 (ones place) is assumed to have been estimated. It would make no sense to report such a measurement as 1,357.0 (five sig figs) or 1,357.00 (six sig figs) because that would suggest the measuring device was able to determine the width to the nearest tenth or hundredth of a millimeter, when in fact it shows only tens of millimeters and therefore the ones place was estimated.

On the other hand, if a measurement is reported as 150 mm, the 1 (hundreds) and the 5 (tens) are known to be significant, but how do we know whether the zero is or is not significant? The measuring device could have had marks indicating every 100 mm or marks indicating every 10 mm. How can you determine if the zero is significant (the estimated digit), or if the 5 is significant and the zero a value placeholder?

The convention for a measurement is that the quantity reported should be all known values and the first estimated value.

Example \(\PageIndex{1}\)

Use each diagram to report a measurement to the proper number of significant figures.

1- 2-

Solution

1. The arrow is between 4.0 and 5.0, so the measurement is at least 4.0. The arrow is between the third and fourth small tick marks, so it's at least 0.3. We will have to estimate the last place. It looks like about one-third of the way across the space, so let us estimate the hundredths place as 3. Combining the digits, we have a measurement of 4.33 psi (psi stands for "pounds per square inch" and is a unit of pressure, like air in a tire). We say that the measurement is reported to three significant figures.
2. The rectangle is at least 1.0 cm wide but certainly not 2.0 cm wide, so the first significant digit is 1. The rectangle's width is past the second tick mark but not the third; if each tick mark represents 0.1, then the rectangle is at least 0.2 in the next significant digit. We have to estimate the next place because there are no markings to guide us. It appears to be about halfway between 0.2 and 0.3, so we will estimate the next place to be a 5. Thus, the measured width of the rectangle is 1.25 cm. Again, the measurement is reported to three significant figures.

Exercise \(\PageIndex{1}\)

What would be the reported width of this rectangle?

Answer

0.61 cm or 0.62 cm or 0.63 cm (all these options contain a valid estimated digit)

Are all Digits Significant?

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 it is exactly 306 million, or is some estimation involved?

The rules for deciding which digits in a measurement are significant are as follows:

1. All nonzero digits are significant. In 1,357 mm, all the digits are significant.
2. Sandwiched (or embedded) zeros, those between significant digits, are significant. Thus, 405 g has three significant figures.
3. Leading zeros, which are zeros at the beginning of a decimal number less than 1, are not significant. In 0.000458 mL, the first four digits are leading zeros and are not significant. The zeros serve only to put the digits 4, 5, and 8 in the correct decimal positions. This number has three significant figures.
4. Trailing zeros, which are zeros at the end of a number, are significant only if the number has a decimal point. In 1,500 m, the two trailing zeros are not significant because the number is written without a decimal point, therefore the number has two significant figures. However, in 1,500.00 m, all six digits are significant because the number has a decimal point.
5. When determining significant figures for numbers in scientific notation, the power of 10 is not included. So 7.0 X 10^-5 g is a measurement of mass in 2 significant figures,

Example \(\PageIndex{2}\)

How many significant figures does each number have?

1. 6,798,000
2. 6,000,798
3. 6,000,798.00
4. 0.0006798

Answer

a- four (by rules 1 and 4). b- seven (by rules 1 and 2)

c- nine (by rules 1, 2, and 4). d- four (by rules 1 and 3)

Exercise \(\PageIndex{2}\)

How many significant figures does each number have?

1. 2.1828
2. 0.005505
3. 55,050
4. 5
5. 500

Answer a

a. five. b. four. c. four. d. one. e. one.

Applied Example \(\PageIndex{3}\)

Which measuring apparatus would you use to deliver 9.7 mL of water as accurately as possible? To how many significant figures can you measure that volume of water with the apparatus you selected?

Answer

The 10 mL graduated cylinder measures in increments of 1 mL, ensuring that the estimated digit will align with the first decimal; therefore, the recorded measurement can be 9.7 mL.

The 100 mL graduated cylinder measures every 10 mL so that the estimated digit will be in the ones place, and the recorded measurement can be 9 mL. No additional estimation can be done.

Beakers are containers with approximate scales that are not accurate; therefore, they are not suitable for measurement purposes. Volume cannot be measured in beakers.

Rounding Off Numbers

Before addressing the specifics of the rules for determining significant figures in a calculated result, we must be able to round numbers correctly. To round a number, first decide how many significant figures the number should have. Once you know that, round to that many digits, starting from the left.

If the number immediately to the right of the last significant digit is less than 5, it is dropped, and the value of the last significant digit remains the same. If the number immediately to the right of the last significant digit is greater than or equal to 5, the last significant digit is increased by 1.

Consider a calculation, where the calculator result is 207.518 m. Right now, the number contains six significant figures. How would we successively round it to fewer and fewer significant figures? Follow the process as outlined in Table \(\PageIndex{1}\).

Table \(\PageIndex{1}\): Rounding examples

Number of Significant Figures	Rounded Value	Reasoning
6	207.518	All digits are significant
5	207.52	8 rounds the 1 up to 2
4	207.5	2 is dropped
3	208	5 rounds the 7 up to 8
2	210	8 is replaced by a 0 and rounds the 0 up to 1
1	200	1 is replaced by a 0

The more rounding that is done, the less reliable the figure is. An approximate value may be sufficient for some purposes, but scientific work requires a much higher level of detail.

It is important to be aware of significant figures when you are mathematically manipulating numbers. For example, dividing 125 by 307 on a calculator gives 0.4071661238… to an infinite number of digits. But do the digits in this answer have any practical meaning, especially when you are starting with numbers that have only three significant figures each? When performing mathematical operations, there are two rules for limiting the number of significant figures in an answer—one rule is for addition and subtraction, and one rule is for multiplication and division.

In operations involving significant figures, the answer is reported in such a way that it reflects the reliability of the least precise operation. An answer is no more precise than the least precise measument used to get the answer.

Significant Figures after Mathematical Operations

Significant figures are handled according to two different rules, depending on the type of calculation being performed.

Addition or Subtraction

For addition or subtraction, the rule is to stack all the numbers with their decimal points aligned. The answer should have the same number of decimal places as the quantity with the fewest decimal places. This can be applied for any numerical place, not necessary decimal places.

a- Consider the following: if you were to add 1.0023 g and 4.383 g, we note that the second number stops its significant figures in the thousandths column, while the first number has more decimal places. We therefore limit our answer to the thousandths column.

\[\begin{align*}
&\mathrm{1.0023\: g}\\ +\: &\underline{\mathrm{4.383\: g}\:\:}\\ &\mathrm{5.3853\: g}
\end{align*}\]

The answer is 5.385 g (round to the thousandths place; three decimal places)

b- A more complex example:

If you were to add 486 g, 421.23 g, and 64.77 g, we note that the first number does not have decimals, while the other two do. The arrow points to the rightmost column in which all the numbers have significant figures—in this case, the ones place. Therefore, we will limit our final answer to the ones place.

\[\begin{align*}
&\mathrm{486\: g}\\ -\: &\underline{\mathrm{421.23\: g}}\\ &\mathrm{\:\:64.77\: g}
\end{align*}\]

For the final number we need to round up to 65 g (round to the ones place; no decimal places). The standard rules for rounding numbers are simple: If the first dropped digit is 5 or higher, round up. If the first dropped digit is lower than 5, do not round up.

Multiplication or Division

For multiplication or division, the rule is to count the number of significant figures in each number being multiplied or divided and then limit the significant figures in the answer to the lowest count. An example is as follows:

The final answer, limited to four significant figures, is 4,094. The first digit dropped is 1, so we do not round up.

Scientific notation provides a way of communicating significant figures without ambiguity. You simply include all the significant figures in the leading number. For example, the number 4,000 has one significant figure and should be written as the number 4 × 10⁴. The number 450 has two significant figures and would be written in scientific notation as 4.5 × 10², whereas 450.0 has four significant figures and would be written as 4.500 × 10². In scientific notation, all reported digits are significant.

In calculations involving several steps, slightly different answers can be obtained depending on how rounding is handled, specifically whether rounding is performed on intermediate results or postponed until the last step. Rounding to the correct number of significant figures should always be performed at the end of a series of calculations, as rounding intermediate results can sometimes cause the final answer to be significantly in error.

Figure \(\PageIndex{2}\): Remember that calculators do not understand significant figures. You are the one who must apply the rules of significant figures to a result from your calculator. How many of the numbers in this display are actually significant? (Public Domain.)

Example \(\PageIndex{3}\)

Write the answer for each calculation with the appropriate number of significant figures.

1. 1.2 + 4.41 = ?
2. 56.789 + 102.2 + 1,300.099 = ?

Answer

1. 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.

2. The arrow points to the rightmost column in which all the numbers have significant figures—in this case, the tenths place. Therefore, we will limit our final answer to the tenth place. Is our final answer, therefore, 1,459.0?

   

   No, because when we drop digits from the end of a number, we also have to round the number. Notice that the first dropped digit, in the hundredths place, is 8. This suggests that the answer is actually closer to 1,459.1 than it is to 1,459.0, so we need to round up to 1,459.1. The standard rules for rounding numbers are simple: If the first dropped digit is 5 or higher, round up. If the first dropped digit is lower than 5, do not round up.

Exercise \(\PageIndex{3}\) Significant figures in addition or subtraction calculations

Complete the addition or subtraction calculations and report your answers using the correct number of significant figures based on the rules for addition and subtraction calculations. Then, answer each expression using scientific notation.

1. 101.2 + 18.702 = ?
2. 202.88 − 1.013 = ?
3. 1,027 + 610.0 + 363.06 = ?
4. 87.25 mL + 3.0201 mL = ?
5. 26.843 g + 12.23 g = ?
6. 13.77 + 908.226 + 515 = ?
7. 255.0 − 99 = ?

Answer

1. If we use a calculator to add these two numbers, we would get 119.902. However, most calculators do not understand significant figures, and we need to limit the final answer to the tenth place. Thus, we drop the 02 and report a final answer of 119.9 (rounding down) = 1.199 x 10².
2. A calculator would answer 201.867. However, we have to limit our final answer to the hundredths place. Because the first number being dropped is 7, which is greater than 5, we round up and report a final answer of 201.87 = 2.0187 x 10².
3. The calculator gives 2,000.06 as the answer, but because 1,027 has its farthest-right significant figure in the ones column, our answer must be limited to the ones position: 2,000. which in scientific notation is 2.000 × 10³.
4. 90.27 mL = 9.027 x 10 mL (while we can use scientific notation, in this case, it is not used as 10 has the power = 1)
5. 39.07 g
6. 1437 = 1.437 x 10³
7. 156 = 1.56 x 10²

Example \(\PageIndex{4}\) Significant figures in multiplication and division calculations

Complete the multiplications and divisions and report your answers using the correct number of significant figures based on the rules for multiplication and division calculations.

1. 23.096 × 90.300 = ?
2. 125 × 9.000 = ?
3. 217 ÷ 903 = ?
4. 6 × 12.011 = ?
5. 0.00666 × 321 = ?
6. 22.4 × 8.314 = ?
7. 1.381 ÷ 6.02 = ?

Answer

1. The calculator answer is 2,085.5688, but we need to round it to five significant figures. Because the first digit to be dropped (in the hundredths place) is greater than 5, we round up to 2,085.6, which in scientific notation is 2.0856 × 10³.
2. The calculator gives 1,125 as the answer, but we limit it to three significant figures and convert it to scientific notation: 1.13 × 10³.
3. 0.240 = 2.40 x 10^-1
4. 72.066 (See rule 5 under “Significant Figures.”)
5. 2.14
6. 186 = 1.86 x 10²
7. 0.229 = 2.29 x 10^-1

Example \(\PageIndex{5}\) Significant figures in Combined Calculations

Complete the calculations and report your answers using the correct number of significant figures.

1. 2(1.008) g + 15.99 g = ?
2. 137.3 + 2(35.45) = ?
3. \( {118.7 \over 2} g - 35.5 g \) = ?
4. \( 47.23 g - {207.2 \over 5.92 }g \) = ?
5. \({77.604 \over 6.467} −4.8\) = ?
6. \( {24.86 \over 2.0 } - 3.26 (0.98 ) \) = ?
7. \((15.9994 \times 9) + 2.0158\) = ?

Answer

1. 2(1.008) g + 15.99 g = 2.016 g + 15.99 g = 18.01 g
2. 137.3 + 2(35.45) = 137.3 + 70.90 = 208.2
3. 59.35 g − 35.5 g = 23.9 g
4. 47.23 g − 35.0 g = 12.2 g
5. 12.00 − 4.8 = 7.2
6. 12 − 3.2 = 9
7. 143.9946 + 2.0158 = 146.0104

Exercise \(\PageIndex{6}\)

1. Explain why the concept of significant figures is important in scientific measurements.
2. State the rules for determining the significant figures in a measurement.
3. When do you round a number up, and when do you not round a number up?

Answer

1. Significant figures represent all the known digits of a measurement plus the first estimated one. It gives information about how precise the measuring device and measurement is.

2. All nonzero digits are significant; zeros between nonzero digits are significant; zeros at the end of a nondecimal number or the beginning of a decimal number are not significant; zeros at the end of a decimal number are significant.

3. Round up only if the first digit dropped is 5 or higher.

Key Takeaways

• Significant figures properly report the number of measured and estimated digits in a measurement.
• The rule for multiplication and division is that the final answer should have the same number of significant figures as there are in the value with the fewest significant figures.
• The rule for addition and subtraction is that the final answer should have the same number of decimal places as the term with the fewest decimal places.