Arithmetic Operations with Significant Figures and Rounding Off

Units And Measurements • Physics

Concept Summary

The result of a calculation involving approximate measured values must reflect the uncertainties of the original data; it cannot be more accurate than the measurements it is based on. Therefore the final answer should not have more significant figures than the least precise input. In multiplication or division, the result keeps as many significant figures as the number with the fewest significant figures. For example, 4.237 g divided by 2.51 cm³ gives 1.69 g cm⁻³, not 1.68804780876. In addition or subtraction, the rule is about decimal places: the result keeps as many decimal places as the number with the fewest decimal places, so 436.32 + 227.2 + 0.301 = 663.821 is reported as 663.8 g. Rounding off follows a convention. If the digit to be dropped is greater than 5, the preceding digit is raised by 1; if it is less than 5, the preceding digit is unchanged. If the dropped digit is exactly 5, the preceding digit is left unchanged when it is even and raised by 1 when it is odd, so 2.745 becomes 2.74 and 2.735 becomes 2.74. In a multi-step calculation, retain one extra digit in the intermediate steps and round off only at the end, otherwise rounding errors build up. Exact numbers and constants like 2π do not limit the significant figures.

Key Notes
  • The final result can never be more precise than the least precise measurement used.
  • Multiplication/division: keep as many significant figures as the factor with the fewest significant figures.
  • Addition/subtraction: keep as many decimal places as the term with the fewest decimal places.
  • Do not apply the multiplication rule to addition or subtraction (e.g. 663.8 g, not 664 g).
  • Rounding: dropped digit > 5 raises the preceding digit; < 5 leaves it unchanged.
  • If the dropped digit is exactly 5: preceding digit even stays, preceding digit odd is raised by 1.
  • In multi-step calculations keep one extra digit in intermediate steps; round at the end.
  • Rounding too early causes errors: reciprocal of 9.58 is 0.104, but reciprocal of 0.104 is 9.62, not 9.58.
  • Exact numbers like 2 in \(2\pi\) have infinite significant figures and do not restrict the answer.
Important Equations
\[ \text{Density} = \frac{4.237\ \mathrm{g}}{2.51\ \mathrm{cm^3}} = 1.69\ \mathrm{g\,cm^{-3}}\ \text{(3 s.f.)} \] \[ 436.32 + 227.2 + 0.301 = 663.821 \approx 663.8\ \mathrm{g} \] \[ 0.307\ \mathrm{m} - 0.304\ \mathrm{m} = 0.003\ \mathrm{m} = 3\times10^{-3}\ \mathrm{m} \] \[ \text{Light year} = (3.00\times10^{8})(3.1557\times10^{7}) = 9.47\times10^{15}\ \mathrm{m}\ \text{(3 s.f.)} \]

Rounding to three s.f.: \(2.746 \to 2.75\); \(1.743 \to 1.74\); \(2.745 \to 2.74\); \(2.735 \to 2.74\).

Solved: cube of side 7.203 m (4 s.f.): area \(= 6(7.203)^2 = 311.3\ \mathrm{m^2}\), volume \(= (7.203)^3 = 373.7\ \mathrm{m^3}\).

Flow Chart
Identify the operation
Mind Map
  • Arithmetic with Significant Figures
    • Multiplication / division
      • Least number of significant figures
    • Addition / subtraction
      • Least number of decimal places
    • Rounding off
      • > 5: raise
      • < 5: unchanged
      • = 5: make preceding digit even
    • Multi-step calculations
      • Keep one extra digit; round at the end
    • Exact numbers: infinite s.f.
Graphic Organiser
AspectMultiplication / DivisionAddition / Subtraction
Governing ruleLeast significant figuresLeast decimal places
Example4.237 g / 2.51 cm3 = 1.69 g cm-3436.32 + 227.2 + 0.301 = 663.8 g
Common mistakeKeeping all calculator digitsUsing the s.f. rule (writing 664 g)
Subtraction note12.9 g − 7.06 g = 5.8 g (not 5.84 g); significant figures can reduce