Chapter 1.7: Significant Figures Notes

Chapter 1.7: Significant Figures

• There are two general types of numbers used in making quantitative measurements: exact numbers and inexact numbers. Exact numbers have no uncertainty associated with them while inexact numbers do have a degree of uncertainty.

  • Exact numbers vs Inexact Numbers (conceptual distinction)

• Exact numbers include the following:

  • Numbers obtained by counting
  • Examples: There are 5 students in the lab.
  • There are 2 hydrogen atoms in every molecule of H$_2$O.
  • Definitions
  • Examples: Conversions between units within the metric system: $1\ \mathrm{m} = 100\ \mathrm{cm}$
  • Conversions between units within the English system: $1\ \mathrm{ft} = 12\ \mathrm{in}$
  • Mathematical constants, unless they are rounded
  • Example: Pi (\$\pi\$) is the ratio of a circle’s circumference to its diameter and is an exact number: $\pi = \frac{C}{D}$ where $C$ is circumference and $D$ is diameter.
  • Exact Numbers

• Inexact Numbers

  • Numbers obtained by a measurement are inexact because there is always uncertainty associated with the device used to obtain the value.

• Rules for Determining the Number of Significant Figures (sig figs):

  • 1) Any non-zero digit is significant
  • Examples: 352 has 3 sig figs; 3.514 has 4 sig figs.
  • 2) Zeros that occur between non-zero digits are significant
  • Examples: 402 has 3 sig figs; 1.202 has 4 sig figs.
  • 3) Zeros to the left of the first non-zero digit are not significant
  • Examples: 0.0258 has 3 sig figs; 0.000659 has 3 sig figs.
  • 4) Zeros to the right of the last non-zero digit after a decimal point are significant
  • Examples: 0.120 has 3 sig figs; 0.2500 has 4 sig figs.
  • 5) The significance of zeroes to the right of the last non-zero digit in a number that does not contain a decimal point is ambiguous. In these cases, scientific notation is used for clarity.
  • Examples: 100 has 1 sig fig; $1.0\times 10^{2}$ has 2 sig figs; $1.00\times 10^{2}$ has 3 sig figs.

• Adding or Subtracting (sig figs rule for decimal places)

  • The final answer cannot have more decimal places than the original number with the least number of decimal places.
  • Example: $42.5 + 25.12 + 450.22 + 450.2 = 968.0$ (rounded to 1 decimal place, the least number of decimals among the addends is 1).

• Multiplying or Dividing (sig figs rule for significant figures)

  • The final answer must have the same number of significant figures as the original number with the fewest significant figures.
  • Example (illustrative): $42.5 \times 5.1 \times 2.0 \times 850.2 \times 8508.5 \times 10^{2}$, with the result retaining the same number of sig figs as the factor with the fewest sig figs among those multiplied.

• Exact numbers do not limit the number of significant figures in a calculation.

  • Example: $7.55\ \text{cm} \times \left[\dfrac{10\ \text{mm}}{1\ \text{cm}}\right] = 75.5\ \text{mm}$
  • This demonstrates that the conversion factor (the bracketed ratio) is exact, so it does not impose additional sig fig limits beyond the non-exact input.
  • Example 1: Exact Numbers — two significant figures
  • Example 2: $\pi (3.5\ \text{cm})^{2} = 38\ \text{cm}^{2}$ — two significant figures

• In multistep calculations, round only at the last step to avoid an accumulation of rounding errors. However, it is important to keep track of the number of significant figures that would be obtained after each individual step.

  • Example 1: $(5.19 + 0.012) \times 2.1 = 2.47714286 \Rightarrow 2.5$ (multistep slight rounding shown; final should reflect the rule)
  • Example 2: $452.3 \times 10^{5} - 2.1 = 2.20761905 \Rightarrow 2.2$
  • Example 3: $(22.3145 \times 0.9045) + (21.2146 \times 0.0915) = 22.12460115 \Rightarrow 22.12$

• Note: The transcript ends with "Example 3:" indicating there may be additional material not provided here, but the principle of rounding in multistep calculations remains as stated above.