Why Significant Figures Matter
Every measurement has inherent uncertainty. When you measure something with a ruler marked in centimeters, you can't claim to know the length to the nearest micrometer. Significant figures communicate the precision of a measurement—they tell others how confident you are in your numbers.
Using too many digits implies false precision, while using too few loses valuable information. Scientists and engineers must express results with appropriate significant figures to maintain credibility and communicate measurement quality accurately.
Precision vs. Accuracy
Before diving into significant figures, it's important to understand two related but distinct concepts:
- Precision: How close repeated measurements are to each other. High precision means consistent results.
- Accuracy: How close a measurement is to the true value. High accuracy means correct results.
The Dartboard Analogy
High precision, low accuracy: Darts clustered together but far from the bullseye.
High accuracy, low precision: Darts scattered around the bullseye.
Both high: Darts tightly grouped at the bullseye—the goal of scientific measurement.
Rules for Identifying Significant Figures
Not all digits in a number are significant. Here are the rules to identify which digits count:
Rule 1: Non-Zero Digits
All non-zero digits are significant.
- 123 has 3 significant figures
- 45.67 has 4 significant figures
- 8.92 has 3 significant figures
Rule 2: Zeros Between Non-Zero Digits
Zeros between non-zero digits are significant (captive zeros).
- 105 has 3 significant figures
- 4.002 has 4 significant figures
- 20.05 has 4 significant figures
Rule 3: Leading Zeros
Leading zeros (before non-zero digits) are NOT significant. They only indicate the position of the decimal point.
- 0.0045 has 2 significant figures (4 and 5)
- 0.123 has 3 significant figures
- 0.007 has 1 significant figure
Rule 4: Trailing Zeros
Trailing zeros after a decimal point are significant. They indicate measured precision.
- 2.00 has 3 significant figures
- 45.10 has 4 significant figures
- 3.000 has 4 significant figures
Tricky Case: Trailing Zeros Without Decimals
Numbers like 100, 2500, or 700 are ambiguous. Are the zeros significant or just placeholders? Use scientific notation to clarify: 7.00 × 10² has 3 sig figs, while 7 × 10² has only 1.
Rule 5: Exact Numbers
Exact numbers have infinite significant figures. These include counted items, defined conversions, and mathematical constants.
- 12 eggs = exactly 12 (infinite sig figs)
- 1 meter = 100 centimeters (exact by definition)
- π = 3.14159... (infinite sig figs)
Practice: Count the Significant Figures
0.00340 → 3 sig figs (3, 4, and trailing 0)
4050 → 3 or 4 sig figs (ambiguous—use scientific notation)
6.022 × 10²³ → 4 sig figs
100.0 → 4 sig figs
Significant Figures in Calculations
The rules differ depending on the type of calculation:
Addition and Subtraction
Round to the least number of decimal places.
Example
12.52 + 3.1 + 0.087 = 15.707
12.52 has 2 decimal places
3.1 has 1 decimal place ← least precise
0.087 has 3 decimal places
Answer: 15.7 (1 decimal place)
Multiplication and Division
Round to the least number of significant figures.
Example
4.56 × 2.1 = 9.576
4.56 has 3 sig figs
2.1 has 2 sig figs ← least
Answer: 9.6 (2 sig figs)
Rounding Rules
When rounding to the correct number of significant figures:
- If the dropped digit is less than 5: Round down (keep the last digit unchanged)
- If the dropped digit is greater than 5: Round up (increase the last digit by 1)
- If the dropped digit is exactly 5: Round to the nearest even number (banker's rounding)
Rounding Examples
2.364 → 2.36 (round down, 4 < 5)
2.367 → 2.37 (round up, 7 > 5)
2.365 → 2.36 (round to even, 6 is even)
2.375 → 2.38 (round to even, 8 is even)
Pro Tip
Keep extra digits during intermediate calculations and only round the final answer. Rounding at each step introduces cumulative errors.
Common Mistakes to Avoid
- Counting leading zeros as significant: 0.0034 has only 2 sig figs, not 4
- Adding false precision: If your calculator shows 3.1415926, that doesn't mean your answer has 8 sig figs
- Forgetting about exact numbers: "5 apples" doesn't limit your precision
- Using the wrong rule: Remember: decimal places for +/−, sig figs for ×/÷
- Rounding intermediate steps: This compounds errors; round only the final answer
Scientific Notation and Significant Figures
Scientific notation eliminates ambiguity about significant figures. Only write the significant digits in the coefficient:
Clarifying Ambiguous Numbers
5000 with 2 sig figs → 5.0 × 10³
5000 with 4 sig figs → 5.000 × 10³
0.00340 → 3.40 × 10⁻³ (3 sig figs clearly shown)
Practical Applications
- Laboratory reports: Results must reflect measurement precision
- Engineering specifications: Tolerance depends on significant figures
- Medical dosing: Precision can be life-critical
- Financial calculations: Different rules apply (always round to cents)