1.5 Expressing Numbers: Significant Figures

Learning Objectives

  1. Express measurements to the correct decimal place, based on the increments marked on the measurement tool.
  2. Identify the number of significant figures in measured values and indicate the level of uncertainty.
  3. Recognize exact numbers AND the fact that they have no uncertainty.
  4. Use significant figures correctly in arithmetic operations.

 

Scientists have established conventions for communicating the degree of precision of a measurement. Suppose that you are using the ruler shown below to measure the object placed above the ruler. Centimeters (cm) are shown as major divisions. The ruler also has tenths of cm (millimeters or mm) shown as minor divisions. The values for cm and tenths of a cm are certain. While tenths of centimeters are the smallest divisions on this ruler, you can and should estimate the next decimal place, hundredths of a cm.  This is an estimate and has uncertainty.  (Figure 1.7 “Measuring an Object to the Correct Number of Digits”).

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Figure 1.7 Measuring an Object to the Correct Number of Digits. What decimal place should you go to for this measurement?  What is the correct measurement?  How much uncertainty is in the measurement?

 

The concept of significant figures takes this limitation into account. The significant figures of a measured quantity are defined as all the digits known with certainty and the first uncertain, or estimated, digit. It makes no sense to report any digits after the first uncertain place, 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 may not be significant figures.

Note

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

For example, if a table is measured and reported as being 1,357 mm wide, the number 1,357 has four significant figures. The 1 (thousands), the 3 (hundreds), and the 5 (tens) are certain; the 7 (ones) is assumed to have been estimated. It would make no sense to report such a measurement as 1,357.0 or 1,357.00 because that would suggest the measuring instrument was able to determine the width to the nearest tenth or hundredth of a millimeter, when in fact 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 significant? The measuring instrument could have had marks indicating every 10 mm or marks indicating every 1 mm. Is the zero an estimate, or is the 5 an estimate and the zero a placeholder?  One way to communicate that the zero is significant is to put a decimal point after it. Another way to make it clear whether or not the zero is significant is to write the number in scientific notation, including the zero in the coefficient ONLY if it is significant.

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

  1. All nonzero digits are significant.
  2. Trapped zeros, which are zeros between significant digits, are significant.
  3. Leading zeros, which are zeros at the beginning of a number with absolute value less than 1, are not significant.
  4. Trailing zeros, which are zeros at the end of a number, are significant only if the number has a decimal point.
  5. All digits in the coefficient of a number written in scientific notation are significant.

Note that rules 3 and 4 are equivalent to saying that place holder zeros, zeros that merely position the rest of the digits for the correct order of magnitude, do not count as sig figs.  Also note that the rules apply only to measured numbers.  Exact numbers established by rules, definitions, or counting, do not have uncertainly and have unlimited sig figs.

Example 8

How many significant digits does each number have?

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

Solution

Skill-Building Exercise

How many significant digits does each measured number have?

  1. 2.1828

  2. 0.005505

  3. 55,050

  4. 5

  5. 500

 

Calculated Numbers Based on Measured Numbers

It is important to be aware of significant figures when performing calculations. 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 measured numbers that have only three significant figures each?  There are rules of thumb to determine a reasonable number of sig figs/level of uncertainly to report after calculations, but the rules are different for addition/subtraction versus multiplication/division.

For addition and subtraction, the rule is based on decimal  place values.  The answer is rounded to end at the last decimal place of the least precise measured number.  This can be visualized by stacking all of the numbers with their decimal points aligned, adding or subtracting,  and then rounding the answer to the rightmost column for which all the numbers have significant figures. Consider the following:

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The arrow points to the rightmost column in which all the numbers have significant figures—in this case, the tenths place. Therefore, round the final answer to the tenths place by looking at the number in the hundredths place.  Since there is an 8 in the hundredths place, the answer is actually closer to 1,459.1 than it is to 1,459.0, so round up to 1,459.1. The rules in rounding 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.

For multiplication or division, the rule is based on the number of sig figs. Count the number of significant figures in each measured number being multiplied or divided, then round the answer to the smallest number of sig figs.   For example:

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

In scientific notation, all significant figures are listed explicitly in the coefficient, so scientific notation provides a way to communicate significant figures without ambiguity. Simply include all the significant figures in the coefficient.  For example, 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.

Example 9

Write the answer for each expression using scientific notation with the appropriate number of significant figures.

  1. 23.096 × 90.300
  2. 125 × 9.000
  3. 1,027 610 363.06

Solution

Skill-Building Exercise

Write the answer for each expression using scientific notation with the appropriate number of significant figures.

  1. 217 ÷ 903

  2. 13.77 908.226 515

  3. 255.0 − 99

  4. 0.00666 × 321

 

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.

Concept Review Exercises

  1. Explain why the concept of significant figures is important in numerical calculations.

  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?

Key Takeaways

  • Significant figures properly report the number of measured and estimated digits in a measurement.
  • There are rules for applying significant figures in calculations.

Exercises

  1. Define significant figures. Why are they important?

  2. Define the different types of zeros found in a number and explain whether or not they are significant.

  3. How many significant figures are in each number?

    1. 140
    2. 0.009830
    3. 15,050
    4. 221,560,000
    5. 5.67 × 103
    6. 2.9600 × 10−5

  4. How many significant figures are in each number?

    1. 1.05
    2. 9,500
    3. 0.0004505
    4. 0.00045050
    5. 7.210 × 106
    6. 5.00 × 10−6

  5. Round each number to three significant figures.

    1. 34,705
    2. 34,750
    3. 34,570

  6. Round each number to four significant figures.

    1. 34,705
    2. 0.0054109
    3. 8.90443 × 108

  7. Perform each operation and express the answer to the correct number of significant figures.

    1. 467.88 23.0 1,306 = ?
    2. 10,075 5,822.09 − 34.0 = ?
    3. 0.00565 0.002333 0.0991 = ?

  8. Perform each operation and express the answer to the correct number of significant figures.

    1. 0.9812 1.660 8.6502 = ?
    2. 189 3,201.8 − 1,100 = ?
    3. 675.0 − 24 1,190 = ?

  9. Perform each operation and express the answer to the correct number of significant figures.

    1. 439 × 8,767 = ?
    2. 23.09 ÷ 13.009 = ?
    3. 1.009 × 876 = ?

  10. Perform each operation and express the answer to the correct number of significant figures.

    1. 3.00 ÷ 1.9979 = ?
    2. 2,300 × 185 = ?
    3. 16.00 × 4.0 = ?

  11. Use your calculator to solve each equation. Express each answer in proper scientific notation and with the proper number of significant figures. If you do not get the correct answers, you may not be entering scientific notation into your calculator properly, so ask your instructor for assistance.

    1. (5.6 × 103) × (9.04 × 10−7) = ?
    2. (8.331 × 10−2) × (2.45 × 105) = ?
    3. 983.09 ÷ (5.390 × 105) = ?
    4. 0.00432 ÷ (3.9001 × 103) = ?

  12. Use your calculator to solve each equation. Express each answer in proper scientific notation and with the proper number of significant figures. If you do not get the correct answers, you may not be entering scientific notation into your calculator properly, so ask your instructor for assistance.

    1. (5.2 × 106) × (3.33 × 10−2) = ?
    2. (7.108 × 103) × (9.994 × 10−5) = ?
    3. (6.022 × 107) ÷ (1.381 × 10−8) = ?
    4. (2.997 × 108) ÷ (1.58 × 1034) = ?

Answers