Significant Figures

I. Definition

  • Numbers believed to be correct with uncertainty only in the last digit
  • Include accurately known figures and first doubtful figure

II. Types of Values

  • ExactCounted clearly (e.g., 3 pencils)
  • MeasuredContains some uncertainty

III. Rules for Determining Significant Figures

  1. All digits from direct measurement are significant
  2. All nonzero digits (1-9) are always significant
    • 27 has 2 significant digits
    • 275 has 3 significant digits
  3. Zeros after decimal point before nonzero digits are not significant
  4. Zeros between significant digits are significant
    • 2705 has 4 significant digits
  5. Trailing zeros after decimal point are significant
    • 275.00 has 5 significant digits
  6. Zeros used for spacing the decimal point are not significant (placeholders only)
    • 0.03 has 1 significant digit
    • 0.027 has 2 significant digits
  7. For large numbers, trailing zeros may or may not be significant
    • In 84,000, the number of significant figures is ambiguous:
      • It could be 2 significant figures (8.4 × 10^4)
      • It could be 3 significant figures (8.40 × 10^4)
      • It could be 5 significant figures (8.4000 × 10^4)
    • To avoid ambiguity, use scientific notation or explicitly state the number of significant figures

IV. Scientific Notation

  • Used to clarify significant figures in large numbers
  • Example: 29,000 → 2.90 x 10^5 (3 significant figures)

V. Rounding Off Numbers

A. Whole Numbers

  • Look at next smaller place value
  • < 5: Round down
  • ≥ 5: Round up

Examples:

  • Round 1,274 to nearest hundred: 1,300
  • Round 85 to nearest ten: 90
  • Round 42 to nearest ten: 40
  • 152 rounded to the nearest ten: 150
  • 467 rounded to the nearest hundred: 500
  • 891 rounded to the nearest ten: 890
  • 2403 rounded to the nearest thousand: 2000
  • 764 rounded to the nearest hundred: 800
  • 1937 rounded to the nearest ten: 1940
  • 3829 rounded to the nearest hundred: 3800
  • 6500 rounded to the nearest thousand: 7000
  • 3245 rounded to the nearest hundred: 3200
  • 1284 rounded to the nearest ten: 1280
  • 5555 rounded to the nearest thousand: 6000
  • 729 rounded to the nearest hundred: 700
  • 1836 rounded to the nearest ten: 1840
  • 4901 rounded to the nearest hundred: 4900
  • 237 rounded to the nearest ten: 240
  • 805 rounded to the nearest hundred: 800
  • 1517 rounded to the nearest thousand: 2000

B. Decimal Numbers

  • Look at digit to the right of rounding position
  • < 5: Treat as zero
  • ≥ 5: Add 1 to previous digit, treat rest as zero

Examples:

  • Round 3.14159 to 2 decimal places: 3.14
  • Round 0.0678 to 2 decimal places: 0.07
  • Round 5.6789 to 3 decimal places: 5.679
  • 3.47 rounded to the nearest tenth: 3.5
  • 6.321 rounded to the nearest hundredth: 6.32
  • 9.856 rounded to the nearest hundredth: 9.86
  • 2.499 rounded to the nearest whole number: 2
  • 15.675 rounded to the nearest tenth: 15.7
  • 0.983 rounded to the nearest hundredth: 0.98
  • 7.045 rounded to the nearest tenth: 7.0
  • 11.777 rounded to the nearest hundredth: 11.78
  • 14.999 rounded to the nearest whole number: 15
  • 20.523 rounded to the nearest tenth: 20.5
  • 8.256 rounded to the nearest hundredth: 8.26
  • 4.319 rounded to the nearest tenth: 4.3
  • 12.001 rounded to the nearest hundredth: 12.00
  • 3.459 rounded to the nearest tenth: 3.5
  • 6.789 rounded to the nearest hundredth: 6.79
  • 1.234 rounded to the nearest whole number: 1
  • 9.555 rounded to the nearest hundredth: 9.56

VI. Importance

  • Conveys precision of measurements
  • Essential in scientific and quantitative contexts
  • Simplifies complex figures without sacrificing accuracy

VII. Examples

  • 100.8 s: 4 significant figures (1.008 × 10^2 s)
  • 0.00580 km: 3 significant figures (5.80 × 10^-3 km)
  • 210.0 g: 4 significant figures (2.100 × 10^2 g)