Significant Figures in Calculations

Introduction to Significant Figures (Sig Figs)

• Sig Figs Usage: Only required when specified, such as "report your answer with the correct number of significant figures."
• General Rule: Aim for 3-5 digits if not explicitly tracking sig figs.
• Scientific Notation: Use for very large or small numbers.

Propagating Sig Figs in Calculations

Rules for Multiplication and Division

• Answer has the same number of sig figs as the factor with the fewest sig figs.

• Example:

  • Calculation: 1.052 × 12.504 × 0.53
  • Result: 6.7208
  • Sig Figs Analysis:
    • 1.052 has 4 sig figs.
    • 12.504 has 5 sig figs.
    • 0.53 has 2 sig figs (leading zero not significant).
  • Final Answer: 6.7 (2 sig figs).

• Division Example:

  • Calculation: 2.0035 ÷ 3.20
  • Results in 0.626094
  • Sig Fig Analysis:
    • 2.0035 has 5 sig figs.
    • 3.20 has 3 sig figs.
  • Final Answer: 0.626 (3 sig figs).

Rules for Addition and Subtraction

• Answer must have the same number of decimal places as the quantity with the fewest decimal places.

• Example:

  • Calculation: 2.345 + 0.07 + 2.9975
  • Result: 5.4125
  • Decimal Places Analysis:
    • 2.345 goes to 3 decimal places.
    • 0.07 goes to 2 decimal places.
    • 2.9975 goes to 4 decimal places.
  • Final Answer: 5.41 (2 decimal places).

• Subtraction Example:

  • Calculation: 5.9 - 0.221
  • Result: 5.679
  • Cut-off: 1 decimal place due to 5.9
  • Final Answer: 5.7 (rounded up from the digit after the cut-off).

Rounding Rules

• General Rounding:
  • [0-4]: Round down.
  • [6-9]: Round up.
  • [5]: Make the last retained digit even.
• Example:
  • 92.85 becomes 92.8
  • 92.75 becomes 92.8

Multi-step Calculations

• Do Not Round Intermediate Values: Rounding should be done at the end to avoid drift from the correct answer.
• Track sig figs throughout calculations, but round only at the conclusion.