Understanding Significant Figures in Measurements

Aug 23, 2024

Significant Figures Lecture Notes

Key Concepts

  • Definition: Significant figures (sig figs) indicate the precision of a measurement.
  • Counting Significant Figures:
    • Non-zero numbers are always significant.
    • Leading zeros (zeros to the left of the first non-zero digit) are not significant.
    • Captive zeros (zeros between non-zero digits) are always significant.
    • Trailing zeros (zeros to the right of a decimal point) are significant only if there is a decimal point. If there isn't, they are not significant.

Examples of Significant Figures

  • Count sig figs in various numbers:
    • 10026: 5 sig figs
    • 95: 2 sig figs
    • 4873: 4 sig figs
    • 404: 3 sig figs
    • 50006: 5 sig figs
    • 70080: 4 sig figs (trailing zero not counted)
    • 8000: 1 sig fig (no decimal point)
    • 800.0: 4 sig figs (decimal point present)
    • 5060: 4 sig figs (decimal present)
    • 0.0153: 3 sig figs (leading zeros not counted)
    • 0.0001008: 4 sig figs

Scientific Notation

  • In scientific notation, only the coefficient counts for significant figures.
    • For example, in 2.53 x 10^4, there are 3 sig figs.

Rounding Significant Figures

  1. Determine the number of sig figs required.
  2. Identify the uncertain digit. It's typically the last non-zero number.
  3. Round appropriately based on the next digit:
    • If the next digit is 5 or greater, round up.
    • If it’s less than 5, keep the digit the same.

Rounding Examples

  • 1 significant figure: Round 425716 to 400000.
  • 2 significant figures: Round 425716 to 430000.
  • 3 significant figures: Round 425716 to 426000.
  • 4 significant figures: Round 425716 to 425700.
  • 5 significant figures: Round 425716 to 42571.

Operations with Significant Figures

  • Addition/Subtraction: The final result should be rounded to the least number of decimal places of any number in the operation.
  • Multiplication/Division: The final result should have the same number of sig figs as the measurement with the least number of sig figs.

Examples with Operations

  • Addition: 2314 + 5.23 = 2319.23 should be rounded to 2319 (3 sig figs).
  • Subtraction: 4671 - 2.1 = 4668.9 should be rounded to 4669 (4 sig figs).
  • Multiplication: 9.6 * 7 = 67.2 should be rounded to 70 (1 sig fig).
  • Division: 34.7 / 3.1 = 11.1 should be rounded to 11 (2 sig figs).

Combined Operations

  • Always perform operations in the correct order (PEMDAS) and keep track of significant figures at each step.

Scientific Notation Operations

  • Adding/Subtracting: Make sure to adjust the exponents first so they match.
  • Multiplying/Dividing: Simply perform as normal and then apply the rules for sig figs.

Final Notes

  • Use scientific notation to express numbers with significant figures, especially large or small numbers, to maintain clarity.
  • Practice with various examples to enhance understanding and accuracy in applying significant figures in different scenarios.