Significant Figures: Quick Reference

Significant Figures in Measurement

• Precision vs. accuracy: precision = reproducibility; accuracy = closeness to the true value.
• Exact numbers vs measured numbers: counting and defined quantities are exact; other quantities derived from measurement are uncertain and described by significant figures.
• Significant figures communicate the precision of a measurement.

Exact numbers and definitions

• Counting yields exact numbers (e.g., eggs in a carton).
• Defined quantities are exact (e.g., $1\ ext{ft} = 12\ ext{in}$, $1\ ext{in} = 2.54\ \text{cm}$, $1\ \text{g} = 0.001\ \text{kg}$).

Reading measurements and uncertainty

• To measure volume with a graduated cylinder, read at the bottom of the meniscus and subdivide between marks to estimate to the tenths place.
• Example: the liquid between 21 mL and 22 mL with the meniscus near 21.3 mL; digits 2 and 1 are certain; 3 is the estimate.
  • In general, measurements on a scale with 1 mL divisions are typically estimated to the nearest $0.1\ \text{mL}$.
• Weighing a object on a balance:
  • Example: $6.72\ \text{g}$ with uncertainty $\pm 0.01\ \text{g}$; digits 6 and 7 are certain; the 2 indicates the mass is between $6.71\text{ g}$ and $6.73\text{ g}$.
  • A more sensitive balance might yield $6.723\ \text{g}$, with uncertainty $\pm 0.001\ \text{g}$ (i.e., between $6.722\text{ g}$ and $6.724\text{ g}$).
• All digits, including the last uncertain digit, are significant.

Zeros and significant figures

• Zeros can be placeholders or significant.
• Leading zeros are never significant; they locate the decimal point.
• Captive zeros (between nonzero digits) are always significant.
• Trailing zeros need context:
  • If there is a decimal point, trailing zeros are generally significant.
  • If there is no decimal point, trailing zeros are ambiguous and may not be significant.
• Example with placeholders:
  • A measurement like volumes between 200 and 300 mL could have an estimate in the tens place (e.g., 220 mL). The 0 in the ones place is a placeholder and not a significant digit.

Using exponential notation to show significant figures

• To resolve ambiguity, use scientific notation:
  • Example: 8.32407 × 10^{-3} (all digits shown are significant in the mantissa).
• Ambiguity in trailing zeros can be clarified as:
  • $1.3 \times 10^{3}$ (2 sig figs)
  • $1.30 \times 10^{3}$ (3 sig figs, if the tens place was measured)
  • $1.300 \times 10^{3}$ (4 sig figs, if the ones place was measured)
• If only decimal notation is available, assume trailing zeros are not significant.

Practical guidelines and rules

• Nonzero digits are always significant.
• Leading zeros are never significant.
• Captive zeros are always significant.
• Trailing zeros:
  • Significant if the decimal point is present; otherwise ambiguous.
  • Use exponential notation to clearly state the number of sig figs.

Quick recall rules

• Determine the first nonzero digit from the left and count all digits to the right; this is the number of significant figures, with caveat about trailing zeros before the decimal.
• When reporting measurements, include the uncertainty when possible (e.g., $6.72 \pm 0.01\ \text{g}$ or $6.723 \pm 0.001\ \text{g}$).
• If a measurement is derived from a device with limited precision, assume the last reported digit is estimated.