Significant Figures: Counting and Rounding

Key Concepts

• Significant Figures (Sig Figs): Measure of precision in numerical data.
• Rules for Counting Sig Figs:
  • Any nonzero number is significant.
  • Zeros between nonzero digits are significant.
  • Leading zeros (zeros before the first nonzero digit) are not significant.
  • Trailing zeros (zeros after the last nonzero digit) are significant if there's a decimal point.

Examples

• 126 → 3 sig figs
• 95 → 2 sig figs
• 4873 → 4 sig figs
• 404 → 3 sig figs (zeros between nonzero digits)
• 50006 → 4 sig figs
• 7080 → 4 sig figs (trailing zeros not counted without decimal)
• 8000 → 1 sig fig
• 800.0 → 4 sig figs (decimal makes all zeros significant)
• 0.153 → 3 sig figs (leading zeros not counted)
• 0.001008 → 4 sig figs (trailing zeros after decimal are significant)
• 5.070080 → 7 sig figs (all zeros after decimal are significant)

Scientific Notation

• Ignore the multiplier (e.g., 3.06 * 10^5 has 3 sig figs).
• Examples:
  • 2.53 * 10^4 → 3 sig figs
  • 1.00 * 10^8 → 4 sig figs
  • 4.20 * 10^7 → 3 sig figs

Rounding Sig Figs

• One Sig Fig: Round based on the first digit (e.g., 4257 → 4000 if next digit < 5).
• Two Sig Figs: Consider next digits (e.g., 4257 → 4300 because 5 rounds up).
• Three Sig Figs: 4257 → 4260 (6 rounds up).
• Four Sig Figs: 4257 → 4257 (no rounding needed as next digit is not > 5).

Combined Operations and Rounding Examples

Rounding Exercises

• 158.1054 to:
  • One sig fig: 200
  • Two sig figs: 160
  • Three sig figs: 158
  • Four sig figs: 158.1
  • Five sig figs: 158.11
• 0.35047 to:
  • One sig fig: 0.4
  • Two sig figs: 0.35
  • Three sig figs: 0.350
  • Four sig figs: 0.3505

Special Cases

• 100 with 2 sig figs: Use scientific notation: 1.0 * 10^2
• 50000 with:
  • One sig fig: 5 * 10^4
  • Two sig figs: 5.0 * 10^4
  • Three sig figs: 5.00 * 10^4

Addition and Subtraction

• Align numbers by decimals.
• Final answer is uncertain at the place of least significance.
• Examples:
  • 2314 + 5.23 = 2319.23 rounded to 2319.23 uncertain at 0.01 place.

Multiplication and Division

• Final answer should have the same number of sig figs as the number with the least sig figs in the operation.
• Examples:
  • 9.6 * 7 = 67.2 rounded to 70
  • 5363.172 / 13.2 = 406.3009 rounded to 40.6

Complex Operations

• Order of Operations (PEMDAS): Perform operations in parentheses first, then exponents, multiplication, etc.
• Intermediate values should be calculated exactly; round the final result according to sig figs.
• Example: (4.31 * 52.31) + 6.814 has intermediate multiplication result rounded to least number of sig figs before final addition.

Scientific Notation Operations

• Addition: Adjust exponents to match before adding coefficients.
• Rounding: Round final result to the least number of sig figs from the initial numbers.
• Example: 4.23 * 10^6 + 5.1 * 10^6 = 9.3 * 10^6
• Multiplication: Add exponents, multiply coefficients, round to least sig figs.
• Example: 1.5 * 10^2 * 2.13 * 10^3 = 3.2 * 10^5

Summary

• Use rules for counting significant figures based on position and presence of decimal point.
• Apply correct rounding rules especially in combined operations.
• Use scientific notation to manage significant figures in large or small numbers effectively.

Keep practicing with various types of problems to master the counting and rounding of significant figures effectively!