Significant figures Knowledge-Based Study

Study notes

1. What Are Significant Figures?

   In physics, every measured quantity carries experimental uncertainty. When you measure an object using an instrument, such as a meter stick or a digital balance, the recorded number reflects the precision of that instrument.

   Significant figures (often called sig figs) are all the digits in a measurement that are known with certainty, plus one final digit that is estimated or uncertain.

   Example

   If you measure the length of a block using a metric ruler marked in millimeters and find that it lies between 14.3 cm and 14.4 cm, you might estimate the reading as 14.36 cm. In this measurement:

   • The digits 1, 4, and 3 are certain.
   • The digit 6 is an estimate.
   • The value 14.36 cm contains a total of four significant figures.

   Determining Which Digits Are Significant

   To count the number of significant figures in any measured value, follow these standard rules:

   • Non-zero digits: All non-zero digits are always significant. For example, 38.4 m has three significant figures.
   • Captive zeros: Zeros located between non-zero digits are always significant. For example, 4002 s has four significant figures, and 5.08 kg has three.
   • Leading zeros: Zeros that appear before the first non-zero digit are not significant; they merely indicate the position of the decimal point. For example, 0.0075 m has only two significant figures.
   • Trailing zeros with a decimal point: Zeros at the end of a number that contains a decimal point are significant because they indicate measurement precision. For example, 45.00 s has four significant figures, and 0.020 A has two.
   • Trailing zeros without a decimal point: Zeros at the end of a whole number without a decimal point are considered ambiguous and generally not significant. For example, 1200 m is usually understood to have two significant figures.

   Multiplication and Division Rule

   When multiplying or dividing measured quantities, the final result cannot have more significant figures than the measurement with the fewest significant figures.

   Procedure

   1. Count the significant figures in each factor or divisor.
   2. Identify the smallest number of significant figures.
   3. Perform the calculation and round the final answer to that smallest number.

   Example

   Calculate the area of a rectangular surface with length $l = 4.25\text{ m}$ (three significant figures) and width $w = 2.1\text{ m}$ (two significant figures):

   $$\text{Area} = 4.25\text{ m} \times 2.1\text{ m} = 8.925\text{ m}^2$$

   Since the least precise factor (2.1 m) has two significant figures, round the answer to two significant figures: 8.9 m².

   Addition and Subtraction Rule

   When adding or subtracting measurements, the rule is based on the number of decimal places, not the total count of significant figures. The final result should have the same number of decimal places as the quantity with the fewest decimal places.

   Procedure

   1. Identify the decimal position (precision limit) of each term.
   2. Perform the addition or subtraction.
   3. Round the result to match the term with the fewest decimal places.

   Example

   Consider adding three measured masses: $12.52\text{ g}$, $341.2\text{ g}$, and $0.215\text{ g}$:

   $$12.52\text{ g} + 341.2\text{ g} + 0.215\text{ g} = 353.935\text{ g}$$

   The measurement with the fewest decimal places is $341.2\text{ g}$ (one decimal place). Therefore, round the result to one decimal place: 354.0 g.

   Resolving Ambiguity with Scientific Notation

   To eliminate ambiguity regarding trailing zeros in whole numbers, use scientific notation. Every digit written in the coefficient of scientific notation is considered significant.

   • $1.2 \times 10^3\text{ m}$ has two significant figures.
   • $1.20 \times 10^3\text{ m}$ has three significant figures.
   • $1.200 \times 10^3\text{ m}$ has four significant figures.

   Exact Numbers

   Some quantities in physics do not involve experimental measurement. These are called exact numbers and include:

   • Counted values: Exactly 4 trials, 12 objects, or 2 particles.
   • Defined conversion factors: For example, $1\text{ min} = 60\text{ s}$ or $1\text{ m} = 100\text{ cm}$.

   Exact numbers have an infinite number of significant figures and do not limit the precision or rounding of calculated results.