Significant Figures

What are Significant Figures

• Significant figures are the essential digits that contribute meaning to a number.
• They show how precise measurements are.
• They include all digits measured correctly plus one estimated digit.
• An estimate is an approximation based on reasonable assumptions (indirect measurements, calculations, models).
• Example note: the value of pi (π) shown in a calculator is often given with an assumed estimated digit.

Rules for Determining Significant Figures

• Rule 1: All nonzero digits are significant.
  • Examples: $1234$ has 4 significant figures; $45932$ has 5.
• Rule 2: Zeros between nonzero digits are significant.
  • Examples: $100001$ has 6 sig figs; $2009$ has 4.
• Rule 3: Leading zeros (to the left of the first nonzero) are not significant.
  • Examples: $0.0001$ has 1 sig fig; $0.0000231$ has 3.
• Rule 4: Trailing zeros to the right of a decimal point are significant.
  • Examples: $0.000123900$ has 6; $194.00$ has 5.
• Rule 5: Trailing zeros to the right of a nonzero number but not to the right of a decimal point are not significant unless specified by a symbol (e.g., overline or underline).
  • Examples: $120000000$ may be read as 2 sig figs (without notation) or 6 sig figs (with notation like overline/underline).
• Note on zeros: Leading zeros are never significant; trailing zeros may be significant if decimal point is present or if notation indicates significance.

Rounding Off Numbers

• Rounding is choosing a value close to the original by adjusting the last reported digit.
• General approach: determine the last reported digit and look at the next digit.
• Case rules (the digit to the right of the last reported digit):
  • If < 5: last reported digit stays the same; discard the rest.
  • If > 5: last reported digit increases by 1; discard the rest.
  • If = 5: round to make the last reported digit even (round half to even) or use the odd/even rule as taught:
  • Examples: $5.386\to 5.39$ (3 sig figs) and $1.879\to 1.88$
  • Tie cases: $1.875\to 1.88$ (since 8 is even) and $1.865\to 1.86$ (since 6 is even)
  • If the digits beyond 5 are more than just 5 (i.e., 5 followed by nonzero digits), treat as greater than 5.
• Practical note: if the digit to the right is not exactly 5, you follow the >5 or <5 rule rather than the tie rule.
• Summary: rounding rules depend on the digit to the right and, for exact 5, on the subsequent digits and the parity preference.

How Significant Figures Are Used in Calculations

• Addition/Subtraction:
  • The result can have as many decimal places as the measurement with the fewest decimal places.
  • The overall accuracy is limited by the least precise measurement.
  • Formula intuition: the decimal places of the result = minimum decimal places among operands.
• Multiplication/Division:
  • The result should have as many significant figures as the measurement with the fewest sig figs among operands.
  • Formula intuition: the number of sig figs in the result = minimum sig figs among operands.
• Key idea: precision is governed by the least precise input.

Example: Subtraction (Addition/Subtraction rule)

• Given: $21.94 ext{ cm}$ (2 decimal places) and $5.3 ext{ cm}$ (1 decimal place)
• Result before rounding: $27.24 ext{ cm}$
• Rounding to the fewest decimal places (1): $27.2 ext{ cm}$

Example: Mass of iron filings (Subtraction with measurement)

• Data: Mass of evaporating dish with iron filings = $18.9023\text{ g}$, mass of dish = $18.35\text{ g}$
• Actual mass of iron filings = $18.9023 - 18.35 = 0.5523\text{ g}$
• Rounding to the appropriate decimal places (2 decimals in this setup) → $0.55\text{ g}$
• Takeaway: use decimal-place rule for subtraction; round accordingly.

Example: Multiplication/Division (Significant Figures rule)

• Given: $2.25\text{ m}$ (3 sig figs) and $3\text{ m}$ (1 sig fig)
• Product: $2.25 \times 3 = 6.75\text{ m}^2$
• Round to least sig figs: $7\text{ m}^2$

Example: Terrarium volume (significant figures in a product)

• Dimensions: $L=23.54\text{ cm}$ (4 sig figs), $W=12.8\text{ cm}$ (3 sig figs), $H=10\text{ cm}$ (1 sig fig)
• Volume: $V = L\times W\times H$
• Raw product: roughly $23.54\times 12.8 \times 10 = 3024.64\text{ cm}^3$
• With least sig figs (1 sig fig from H): final volume ≈ $3000\text{ cm}^3$

Example: Density calculation (volume and sig figs)

• Dimensions: $32.13\text{ cm} \times 29.154\text{ cm} \times 19.2\text{ cm}$ (sig figs: 4, 5, 3) → least = 3 sig figs
• Mass: $23\,560\text{ g}$
• Density: $\rho = \frac{m}{V}$ (value rounded to 3 sig figs as dictated by the least precise measurement)
• Result given: approximately $1.31\text{ g/cm}^3$

Why It Is Important to Use Correct Significance

• Ensures reported values reflect measurement precision.
• Prevents over- or under-reporting of precision in data interpretation.
• Maintains consistency across calculations and communications.

Quick Exercise: Determine the number of significant figures

• Indicate the number of significant figures in each measurement:
  • 1) $3,684\text{ kg}$ → $4$ sig figs
  • 2) $5.3\text{ cm}$ → $2$ sig figs
  • 3) $0.045\text{ ft}$ → $2$ sig figs
  • 4) $150000\text{ mg}$ → ambiguous without notation; typically $2$ sig figs (1 and 5) unless notation indicates otherwise
  • 5) $0.0340\text{ km}$ → $3$ sig figs

Common Takeaways

• Significant figures convey precision, not just scale.
• Rounding rules depend on context: decimal places (for addition/subtraction) vs. sig figs (for multiplication/division).
• Always align your result with the least precise measurement involved in the calculation.

Additional Notes

• Rounding off relates to preserving value while simplifying the figure set.
• The least accurate measurement governs the precision of the final result.
• Visual cues (decimal places, trailing zeros with decimal points, and explicit notation) help determine sig figs.