Significant Figures

Measurements always carry some uncertainty. Significant figures (sig figs) are a shorthand for how precisely a number is known. Mishandling them is a classic source of bookkeeping errors in experimental physics.

Concept

A significant figure is a digit that contributes to the precision of a measurement. The general rules:

Counting Sig Figs

1. All non-zero digits are significant. 423 → 3 sig figs.
2. Zeros between non-zero digits are significant. 1002 → 4 sig figs.
3. Leading zeros are NOT significant (they only locate the decimal point). 0.0034 → 2 sig figs.
4. Trailing zeros after the decimal point ARE significant. 2.30 → 3 sig figs; 2.300 → 4 sig figs.
5. Trailing zeros in whole numbers without decimal are ambiguous. 1500 → 2, 3, or 4 sig figs. Use scientific notation: $1.5 \times 10^3$ (2 sf), $1.50 \times 10^3$ (3 sf), $1.500 \times 10^3$ (4 sf).

Examples

Arithmetic with Sig Figs

Addition / Subtraction

The result has the same number of decimal places as the term with the fewest decimal places.

$12.11 + 0.3 = 12.41 \to 12.4$ (one decimal place).

Multiplication / Division

The result has the same number of significant figures as the factor with fewest sig figs.

$2.5 \times 1.234 = 3.085 \to 3.1$ (2 sig figs).

Exact Numbers

Exact integers ($\pi$ to full precision, count of objects) have infinite sig figs and never limit the result.

Rounding

• If the digit being dropped is < 5: round down.
• If > 5: round up.
• If = 5 with nothing after: round to even (banker's rounding); commonly schools just round up.

Worked Example

Q: A rectangular plate measures $3.21\,\text{cm}$ by $2.7\,\text{cm}$. What is its area to the correct number of sig figs?

Solution: $A = 3.21 \times 2.7 = 8.667\,\text{cm}^2$

The factor with fewer sig figs is $2.7$ (2 sig figs). Therefore: $A = 8.7\,\text{cm}^2.$

Q (bonus): Add $24.686 + 2.343 + 3.21$ with correct sig figs.

Sum = $30.239$. Fewest decimal places is $3.21$ (2 dp). Answer: $30.24$ — wait, $3.21$ has 2 decimal places, so we round to 2 decimal places: $30.24$.

Common Confusions

• Trailing zeros: 100 is ambiguous; 100. (with the dot) signals 3 sig figs.
• Scientific notation removes all ambiguity — use it for clarity.
• Don't confuse sig figs (precision) with decimal places (number after the dot).
• Carry extra digits through intermediate calculations; only round the final answer.

Key Takeaways

• Sig figs encode measurement precision.
• Multiplication/division: keep the least number of sig figs.
• Addition/subtraction: keep the least number of decimal places.
• Use scientific notation to make sig figs unambiguous.
• Round only at the end of a calculation.