Significant figures Knowledge-Based Study

Study notes

1. Measurement and Uncertainty

   In chemistry, every experimental measurement carries an inherent degree of uncertainty. This uncertainty arises from the limitations of measuring instruments and the human judgment involved in reading them.

   A recorded measurement consists of all certain digits plus one final estimated (uncertain) digit. Together, these digits are known as significant figures (or significant digits). Significant figures communicate the precision of an experimental value so that downstream calculations do not imply greater accuracy than the data genuinely support.

   For example, if the liquid level in a graduated cylinder is clearly between 21.4 mL and 21.6 mL, and appears right at the midpoint, it is reported as 21.5 mL. Here, the digits 2 and 1 are certain, whereas the digit 5 is estimated.

   Counting Significant Digits

   To determine the number of significant figures in any written measurement, apply the following systematic rules:

   • Non-zero digits: All non-zero digits are always significant. For instance, 148.2 g contains four significant figures.
   • Captive (interior) zeros: Zeros located between two non-zero digits are always significant. For instance, 4005 mL contains four significant figures, and 1.08 g contains three significant figures.
   • Leading zeros: Zeros that appear before the first non-zero digit serve solely as placeholders to locate the decimal point; they are never significant. For instance, 0.0032 kg contains only two significant figures.
   • Trailing zeros with a decimal point: Zeros at the end of a number that contains an explicit decimal point are significant. For instance, 45.00 mL contains four significant figures, and 0.050 s contains two significant figures.
   • Trailing zeros without a decimal point: Zeros at the end of a whole number without a decimal point are ambiguous. To avoid ambiguity, scientific notation is used. For example, 300 g written without a decimal point is traditionally treated as having one significant figure, whereas $3.00 \\times 10^2\\text{ g}$ explicitly shows three significant figures.

   The Multiplication and Division Rule

   When multiplying or dividing measured quantities, the final result can contain no more significant figures than the measurement with the fewest significant figures.

   Consider calculating the density of an object with a mass of 45.82 g and a volume of 12.3 mL:

   $$\\text{Density} = \\frac{45.82\\text{ g}}{12.3\\text{ mL}} = 3.725203...\\text{ g/mL}$$

   Evaluate the given data:

   • 45.82 g has four significant figures.
   • 12.3 mL has three significant figures.

   Because the least precise measurement contains three significant figures, the final result must be rounded to three significant figures: 3.73 g/mL.

   The Addition and Subtraction Rule

   When adding or subtracting measured values, the limiting factor is not the total count of significant figures, but the position of the decimal place. The result must be rounded to the same number of decimal places as the measurement with the fewest decimal places (the least decimal precision).

   Consider combining three measured masses:

   • 125.1 g (one decimal place)
   • 1.246 g (three decimal places)
   • 8.05 g (two decimal places)

   Perform the addition directly:

   $$125.1\\text{ g} + 1.246\\text{ g} + 8.05\\text{ g} = 134.396\\text{ g}$$

   Because 125.1 g has only one decimal place, the final answer must be rounded to one decimal place, yielding 134.4 g.

   Exact Numbers and Rounding Standards

   It is essential to distinguish between measured numbers and exact numbers:

   • Counted items: Exactly 12 test tubes or 4 beakers have no uncertainty.
   • Defined relationships: Conversion factors defined within a system, such as $1\\text{ m} = 100\\text{ cm}$ or $1\\text{ min} = 60\\text{ s}$, are defined exactly.

   Exact numbers possess an infinite number of significant figures and never limit the number of significant figures in a calculation.

   Rounding Rules

   When discarding unwanted digits at the end of a calculation:

   • If the first digit to be dropped is less than 5, leave the preceding digit unchanged.
   • If the first digit to be dropped is 5 or greater, increase the preceding digit by 1.
   • In multi-step calculations, retain intermediate digits in your calculator and round only the final reported answer to avoid cumulative rounding errors.