Significant Figures

Measured quantities are reported so that only the last digit is uncertain.

All digits, including the uncertain one, are called significant figures (or significant digits, or sigfigs, or sf).

Exact numbers can be treated as if they have an infinite number of sigfigs.

  • Defined numbers (e.g., 100 cm = 1 m)
  • Counted numbers (e.g., 24 atoms)
  • Exponents (e.g., 103)

Significant Figures

  • All nonzero digits are significant. (453 cm has 3 sf)
  • Zeros between nonzero digits are significant. (1.03 ml has 3 sf, 3902 miles has 4 sf)
  • Zeros to the left of the first nonzero digit are not significant. (0.003 g has 1 sf)
  • Zeros that are both at the end of a number and to the right of the decimal place are significant. (23.0 K has 3 sf, 10.300 g has 5 sf)
  • When a number ends in zero(s) in the units or larger places, but contain no decimal point, the zeros may or may not be significant. [Most scientists, including me, say these zeros are not significant.] Use scientific notation to remove ambiguity. (100 miles has 1 sf; 1.0 x 102 miles has 2 sf; 1.00 x 102 miles has 3 sf)
  • When a number ends in zeros, and contains a decimal point, the zeros are significant. (100. miles has 3 sf).
Rule for Multiplication and Division
Sig Figs in Calculations

In multiplication and division, the result must be reported as having no more significant figures than the measurement with the fewest significant figures.
Rule for Addition and Subtraction
In addition and subtraction, the result must be reported to the same number of decimal places as that of the term with the least number of decimal places.
Rounding Rules

Look at the leftmost digit to be removed.

  • If less than 5: truncate
    (preceding number is unchanged)
  • If 5 or greater: increase by 1
    (round up)

When a calculation involves two or more steps, retain at least one additional digit - beyond the correct number of significant figures - for intermediate calculations. Round off to the correct number of significant figures only for the answer you want to report (which is usually the final answer).

This helps to avoid acculmulated round-off error.