Chemistry and Measurements: Significant Figures

Chemistry: An Introduction to General, Organic, and Biological Chemistry

Measured Numbers and Significant Figures

  • Learning Goal: Identify a number as measured or exact; determine the number of significant figures in a measured number.

  • The number of baseballs is counted, which means 2 is an exact number.

Measured Numbers

  • A measuring tool is used to determine a quantity such as the length or the mass of an object.
  • Measuring tools provide numbers for a measurement called measured numbers.

Reporting Length

  • To report the length of an object:
    • Observe the numerical values of the marked lines at the end of the object.
    • Estimate the last digit by visually dividing the space between the smallest marked lines.
  • This estimated number is the final digit that is reported for a measured number.
Example 1
  • Reporting Length: 4.5 cm
    • The end of the object is between the 4-cm and 5-cm marks.
    • Estimate that the end is halfway between the 4-cm and 5-cm marks and report the value as 4.5 cm.
Example 2
  • Reporting Length: 4.55 cm
    • This metric ruler is marked at every 0.1 cm.
    • You can now estimate that the length is halfway between the 4.5-cm and 4.6-cm marks and report the value as 4.55 cm.
Example 3
  • Reporting Length: 3.0 cm
    • The end of the object lines up with the 3-cm mark.
    • Because the divisions are marked in units of 1 cm, the estimated digit appears in the tenths place (0.1 cm).

Learning Check 1

  • What is the length of the red line?
    • 9.2 cm
    • 9.4 cm
    • 9.20 cm
    • 9.40 cm

Solution 1

  • What is the length of the red line?
    • 9.2 cm
    • 9.4 cm
    • 9.20 cm (we can estimate the last digit 0)
    • 9.40 cm

Significant Figures

  • In a measured number, the significant figures (SFs) are all the digits including the estimated digit.
  • All nonzero numbers are counted as significant figures.
  • Zeros may or may not be significant, depending on their position in the number.

Rules for Significant Figures

  • A number is a significant figure (SF) if it is or has the following:
    • Not a zero:
      • 4.5 g (2 SFs)
      • 122.35 m (5 SFs)
    • A zero between nonzero digits:
      • 205 degrees Celsius (3 SFs)
      • 5.008 kg (4 SFs)
    • A zero at the end of a decimal number:
        1. L (2 SFs)
      • 16.00 mL (4 SFs)
    • In the coefficient of a number written in scientific notation:
      • 4.8×1054.8 \times 10^{5} meters (2 SFs)
      • 5.70×10−35.70 \times 10^{-3} grams (3 SFs)
  • A zero is not significant if it is:
    • At the beginning of a decimal number:
      • 0.0004 s (1 SF)
      • 0.075 cm (2 SFs)
    • Used as a placeholder in a large number without a decimal point:
      • 850 000 m (2 SFs)
      • 250 000 g (3 SFs)

Counting Significant Figures

  • All nonzero numbers in a measured number are significant.
    • 38.15 cm (4 SFs)
    • 5.6 ft (2 SFs)
    • 65.6 lb (3 SFs)
    • 122.55 m (5 SFs)

Zeros Between Digits

  • Zeros between nonzero digits are significant.
    • 50.08 km (4 SFs)
    • 201 min (3 SFs)
    • 0.0702 lb (3 SFs)
    • 0.40505 m (5 SFs)

Zeros: Decimal Numbers (1 of 2)

  • Zeros at the end of decimal numbers are significant.
      1. min (3 SFs)
    • 40.00 g (4 SFs)

Zeros: Decimal Numbers (2 of 2)

  • Zeros at the beginning of decimal numbers are not significant.
    • 0.440 km (3 SFs)
    • 0.022 g (2 SFs)
    • 0.003 s (1 SF)

Zeros: Nondecimal Numbers

  • Zeros used as placeholders in a large number without a decimal point are not significant.
    • 44 000 km (2 SFs)
    • 810 cm (2 SFs)
    • 150 000 g (3 SFs)

Zeros: Scientific Notation (1 of 2)

  • Zeros in the coefficient of numbers written in scientific notation are significant.
    • 4.90×1034.90 \times 10^{3} meters (3 SFs)
    • 8.0×10−38.0 \times 10^{-3} kilograms (2 SFs)
    • 6.0330×10−56.0330 \times 10^{-5} Liters (5 SFs)

Zeros: Scientific Notation (2 of 2)

  • Keep only the significant zeros when writing numbers in scientific notation.
      1. g (3 SFs) = 5.00×1025.00 \times 10^{2} grams
    • 400 000 m (1 SF) = 4×1054 \times 10^{5} meters
    • 0.30 cm (2 SFs) = 3.0×10−13.0 \times 10^{-1} centimeters

Learning Check 2

  • Identify the significant and nonsignificant zeros in each of the following numbers, and write each number in the correct scientific notation:
    • A. 0.002 650 m
    • B. 43.026 g
    • C. 1,044,000 L

Solution 2 (1 of 2)

  • A. 0.002 650 m is written as four SFs
    • The zeros preceding the 2 are not significant.
    • The digits 2, 6, and 5 are significant.
    • The zero in the last decimal place is significant.
  • B. 43.026 g is written as five SFs
    • The zeros between nonzero digits or at the end of decimal numbers are significant.

Solution 2 (2 of 2)

  • C. 1 044 000 L is written as four SFs
    • The zeros between nonzero digits are significant.
    • The zeros at end of a number with no decimal point are not significant.

Exact Numbers (1 of 2)

  • Exact numbers are:
    • Those numbers obtained by counting items.
    • Definitions that compare two units in the same measuring system.
      • 8 cookies
      • 2 baseballs
      • 1 ft = 12 in.
      • 1 kg = 1000 g

Exact Numbers (2 of 2)

  • Exact numbers are not measured, do not have a limited number of significant figures, and do not affect the number of significant figures in a calculation.
    • Counted Numbers Items
      • 8 doughnuts
      • 2 baseballs
      • 5 capsules
    • Defined Equalities Metric System
      • 1 L = 1000 mL
      • 1 meter = 100 cm
      • 1 kg = 1000 g
    • Defined Equalities U.S. System
      • 1 ft = 12 in.
      • 1 qt = 4 cups
      • 1 lb = 16 oz

Learning Check 3

  • Identify the numbers below as measured or exact, and give the number of significant figures in each measured number.
    • A. 3 coins
    • B. The diameter of a circle is 7.902 cm.
    • C. 60 min = 1 h

Solution 3

  • A. The value 3 coins is obtained by counting, making it an exact number.
  • B. The diameter of a circle is obtained using a measuring device (ruler). The contained zero in this measured number is significant, so it contains four SFs.
  • C. 60 min = 1 h is exact by definition.

Learning Check 4

  • State the number of significant figures in each of the following measurements:
    • A. 0.030 m
    • B. 4.050 L
    • C. 0.0008 g
    • D. 2.80 m

Solution 4

  • A. 0.030 m 2 SFs
  • B. 4.050 L 4 SFs
  • C. 0.0008 g 1 SF
  • D. 2.80 m 3 SFs

Learning Check 5

  • A. Which answer contains three significant figures?
    1. 0.4760
    2. 0.00476
  • B. All the zeros are significant in
    1. 0.00307
    2. 25.300
  • C. The number of significant figures in
    1. one
    2. two
    3. three

Solution 5

  • A. Which answer contains three significant figures?
    1. 0.00476
  • B. All the zeros are significant in
    1. 25.300
  • C. The number of significant figures in
    1. three