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).