Measurements

1.3 Measurements

Goal

• Determine the correct number of digits to indicate the precision of a measurement or a calculated result.

Background Review

• Chapter 0 Math Skills: Section $0.3$—Order of Operations

Observations

• Quantitative Measurements:

  • Observations that involve numbers, making them more precise.

  • Examples: a piece of lead weighing $112.5 ext{ g}$, a patient's temperature of $38.5 ext{ °C}$ ($101.3 ext{ °F}$).

• Qualitative Descriptions:

  • Observations that do not involve numbers.

  • Describe qualities like color, shape, or subjective terms such as heavy or tall.

  • Examples: noting lead has a dull gray color, or a patient is coughing and their forehead feels warm.

Making Measurements

• Estimation Rule: When making a measurement, estimate to one digit beyond the smallest division markings, if possible.

• Ruler Example (Bolt length):

  • Smallest divisions are millimeters ($0.1 ext{ cm}$).

  • Bolt length is between $1.7 ext{ cm}$ and $1.8 ext{ cm}$.

  • A reasonable estimated measurement is $1.75 ext{ cm}$.

• Syringe Example (Volume):

  • Read volume from the top of the barrel seal.

  • Smallest scale divisions are $0.1 ext{ mL}$.

  • If the plunger is at the $1.8 ext{ mL}$ mark, a reasonable estimated volume is $1.80 ext{ mL}$.

  • Other reasonable interpretations could be $1.79 ext{ mL}$ or $1.81 ext{ mL}$.

Example 1.5

• **Determine the width of the shell:

  1. Using the bottom scale: Smallest division is $1 ext{ cm}$. The shell is slightly wider than $4 ext{ cm}$. A reasonable reading is $4.1 ext{ cm}$.

  2. Using the top scale: Smallest division is $0.1 ext{ cm}$. The shell is wider than $4.0 ext{ cm}$ but slightly less than $4.1 ext{ cm}$. A reasonable reading is $4.08 ext{ cm}$.

Practice Problem 1.5

• **Read the volume of liquid in the 5 mL syringe (plunger at 3.5 mark):

  • Smallest divisions visible appear to be $0.5 ext{ mL}$. (Assuming $0.5 ext{ mL}$ marks for $3.0, 3.5, 4.0$).

  • Estimating one digit beyond the $0.5 ext{ mL}$ mark, a reasonable reading is $3.50 ext{ mL}$.

Significant Digits

• Definition: Every digit that reflects the precision of a measurement, including all certain digits plus one estimated digit.

• Importance: Indicates how precisely a measurement was made; significant here refers to precision, not importance.

• Rules for Zeros:

  1. Leading Zeros (to the left of all nonzero digits): Not significant.

     • Example: $0.03 ext{ cm}$ (1 significant digit).

  2. Captive Zeros (between significant digits): Significant.

     • Example: $903 ext{ cm}$ (3 significant digits).

  3. Trailing Zeros (to the right of all nonzero digits) in a number with decimal-place digits: Significant.

     • Example: $70.00 ext{ cm}$ (4 significant digits).

  4. Trailing Zeros in an integer (no decimal point): Uncertain; assume they are not significant unless otherwise indicated.

     • Example: $4000 ext{ cm}$ (1 significant digit).

  5. Scientific Notation: All digits in the coefficient of a number in scientific notation are significant.

     • Example: $7.000 imes 10^3 ext{ g}$ has 4 significant digits.

• Indicating Significant Trailing Zeros in Integers:

  • If a number like $7000 ext{ g}$ is precisely measured to the nearest $1 ext{ g}$, it should be reported as $7000. ext{ g}$ (with a decimal point) to indicate all zeros are significant. Alternatively, use scientific notation (e.g., $7.000 imes 10^3 ext{ g}$).

Example 1.6

• Determine the number of significant digits in each measurement and underline them:

  1. $0.061 ext{ cm}$: 2 significant digits ($0.0\underline{61}$ - leading zeros are not significant).

  2. $5.200 ext{ cm}$: 4 significant digits ($\underline{5.200}$ - trailing zeros with a decimal are significant).

  3. $5.009 ext{ cm}$: 4 significant digits ($\underline{5.009}$ - captive zeros are significant).

  4. $8000 ext{ cm}$: 1 significant digit ($\underline{8}000$ - trailing zeros in an integer without a decimal are not significant by default).