Prerequisites
Before studying significant figures, you should understand:
What Are Significant Figures?
Significant figures (or significant digits) are all the digits in a measurement that are known with certainty, plus one uncertain digit.
They indicate the precision of a measurement.
When a pharmacist measures 250.0 mg of medicine vs 250 mg, the extra zero matters!
- 250 mg (3 sig figs) → could be anywhere from 249.5 to 250.5 mg
- 250.0 mg (4 sig figs) → measured to nearest 0.1 mg, between 249.95 and 250.05 mg
That extra precision could be the difference between effective treatment and overdose!
Rules for Counting Significant Figures
Rule 1: All Non-Zero Digits Are Significant
Examples:
- 123 → 3 significant figures
- 56.78 → 4 significant figures
- 9 → 1 significant figure
Rule 2: Zeros Between Non-Zero Digits Are Significant (Captive Zeros)
Examples:
- 1002 → 4 significant figures
- 50.03 → 4 significant figures
- 2.004 → 4 significant figures
Rule 3: Leading Zeros Are NOT Significant (Placeholder Zeros)
Leading zeros only indicate the position of the decimal point.
Examples:
- 0.0025 → 2 significant figures (only 2 and 5)
- 0.00340 → 3 significant figures (3, 4, and trailing 0)
- 0.5 → 1 significant figure
Write in scientific notation to easily count sig figs:
- 0.0025 = 2.5 × 10⁻³ → clearly 2 sig figs
- 0.00340 = 3.40 × 10⁻³ → clearly 3 sig figs
Rule 4: Trailing Zeros Are Significant ONLY If Decimal Point Is Present
With decimal point:
- 250.0 → 4 significant figures
- 2.500 → 4 significant figures
- → 3 significant figures (note the decimal point!)
Without decimal point (ambiguous):
- 2500 → could be 2, 3, or 4 sig figs (ambiguous!)
- 100 → could be 1, 2, or 3 sig figs (ambiguous!)
Problem: Does 4500 have 2, 3, or 4 significant figures?
Solution: Use scientific notation to clarify:
- 4.5 × 10³ → 2 sig figs
- 4.50 × 10³ → 3 sig figs
- 4.500 × 10³ → 4 sig figs
Rule 5: Exact Numbers Have Infinite Significant Figures
Exact numbers come from:
- Counting (5 apples, 12 students)
- Definitions (1 m = 100 cm, 1 hour = 60 min)
- Mathematical constants when defined exactly (like “2” in diameter = 2 × radius)
Examples:
- There are exactly 100 cm in 1 m (not 99.99 or 100.01)
- A class has exactly 30 students
- Area of circle = πr² (the “2” in “r²” is exact, but π = 3.14159… is not)
Interactive Demo: Visualize Significant Figures
Practice identifying and using significant figures in calculations.
When multiplying/dividing, exact numbers don’t limit the significant figures in the result.
Example: If radius = 5.0 cm (2 sig figs), then diameter = 2 × 5.0 = 10. cm (still 2 sig figs, the “2” doesn’t limit it)
Quick Reference Table
| Number | Significant Figures | Explanation |
|---|---|---|
| 123 | 3 | All non-zero |
| 0.00123 | 3 | Leading zeros don’t count |
| 1.0023 | 5 | Captive zeros count |
| 1.200 | 4 | Trailing zeros after decimal count |
| 1200 | 2, 3, or 4 | Ambiguous! Use scientific notation |
| 1200. | 4 | Decimal point makes trailing zeros count |
| 1.2 × 10³ | 2 | Clear from scientific notation |
| 0.0500 | 3 | Leading zeros don’t count, trailing do |
Rounding to Significant Figures
Rounding Rules
When rounding to n significant figures:
- If next digit < 5: Round down (leave as is)
- If next digit > 5: Round up
- If next digit = 5:
- If followed by non-zero: Round up
- If followed by zeros or nothing: Round to nearest even number
Round to 3 significant figures:
- 12.34 → 12.3 (4 < 5, round down)
- 12.36 → 12.4 (6 > 5, round up)
- 12.45 → 12.4 (5 followed by nothing, round to even)
- 12.55 → 12.6 (5 followed by nothing, round to even)
- 12.451 → 12.5 (5 followed by non-zero, round up)
The “round to even when exactly 5” rule minimizes cumulative rounding errors in long calculations. It’s used in scientific and financial calculations.
Examples:
- 2.5 → 2 (round to even)
- 3.5 → 4 (round to even)
- 4.5 → 4 (round to even)
Operations with Significant Figures
Addition and Subtraction
Rule: Result should have the same number of decimal places as the measurement with the fewest decimal places.
- 12.3 (1 decimal place) ← limiting factor
- 0.45 (2 decimal places)
- 0.006 (3 decimal places)
Answer: 12.8 (rounded to 1 decimal place)
- 123.456 (3 decimal places)
- 12.1 (1 decimal place) ← limiting factor
Answer: 111.4 (rounded to 1 decimal place)
Multiplication and Division
Rule: Result should have the same number of significant figures as the measurement with the fewest significant figures.
- 2.5 (2 sig figs) ← limiting factor
- 3.42 (3 sig figs)
Answer: 8.6 (rounded to 2 sig figs)
- 12.5 (3 sig figs)
- 0.25 (2 sig figs) ← limiting factor
Answer: 50 or 5.0 × 10¹ (2 sig figs)
Note: Writing “50” is ambiguous. Better to write 5.0 × 10¹ to show 2 sig figs clearly.
Powers and Roots
Rule: Result has same significant figures as the original number.
- $(2.5)^2 = 6.25$ → 6.2 (2 sig figs, same as 2.5)
- $\sqrt{16.0} = 4.00$ → 4.00 (3 sig figs, same as 16.0)
- $(1.5)^3 = 3.375$ → 3.4 (2 sig figs, same as 1.5)
Mixed Operations
For complex calculations, carry extra digits during intermediate steps and round only at the end.
Given: radius r = 2.5 cm (2 sig figs), height h = 10.0 cm (3 sig figs)
Find: Volume V = πr²h
Step-by-step:
- r² = (2.5)² = 6.25 cm² (keep extra digits)
- πr² = 3.14159… × 6.25 = 19.6349… cm² (keep extra digits)
- V = 19.6349… × 10.0 = 196.349… cm³
Final answer: 2.0 × 10² cm³ (rounded to 2 sig figs, limited by r = 2.5)
Or: 200 cm³ if written without scientific notation (but this looks like 1 or 2 or 3 sig figs — ambiguous!)
Common Mistakes to Avoid
Wrong: 0.0045 has 4 significant figures
Correct: 0.0045 = 4.5 × 10⁻³ has 2 significant figures (only 4 and 5)
Wrong: 250 and 250. both have 2 significant figures
Correct:
- 250 → ambiguous (could be 2 or 3 sig figs)
- → 3 significant figures (decimal point makes trailing zero significant)
Wrong: 12.3 + 0.456 = 12.756 → round to 3 sig figs = 12.8
Correct: Use decimal places for addition, not sig figs!
- 12.3 (1 decimal place)
- Result: 12.8 (1 decimal place)
This happens to give 12.8 in both cases, but reasoning is different!
Wrong:
- r = 2.5 cm
- r² = 6.25 → round to 6.2 (2 sig figs)
- πr² = 3.14 × 6.2 = 19.468 → 19 cm²
Correct:
- Keep extra digits in intermediate steps
- r² = 6.25
- πr² = 3.14159 × 6.25 = 19.6349
- Final: 20 cm² (2 sig figs)
The wrong method gives 19 cm², correct gives 20 cm² — different answers!
Wrong: Diameter = 2 × 5.00 cm = 10 cm (limited to 1 sig fig by “2”)
Correct: The “2” is exact, doesn’t limit sig figs. Diameter = 2 × 5.00 = 10.0 cm (3 sig figs, same as 5.00)
Practice Problems
Level 1: JEE Main Basics
How many significant figures in each?
a) 0.00508 b) 3.1400 c) 2000 d) 2.4 × 10⁵
Solution:
a) 0.00508 = 5.08 × 10⁻³ → 3 sig figs
b) 3.1400 → 5 sig figs (trailing zeros after decimal)
c) 2000 → ambiguous (could be 1, 2, 3, or 4)
d) 2.4 × 10⁵ → 2 sig figs
Calculate: 25.2 + 1.34 + 0.006
Solution:
Sum = 26.546
Decimal places:
- 25.2 (1 decimal place) ← limiting
- 1.34 (2 decimal places)
- 0.006 (3 decimal places)
Answer: 26.5 (1 decimal place)
Calculate: 5.2 × 3.145
Solution:
Product = 16.354
Significant figures:
- 5.2 (2 sig figs) ← limiting
- 3.145 (4 sig figs)
Answer: 16 (2 sig figs)
Level 2: JEE Main/Advanced
Calculate the area of a rectangle with length 12.4 cm and width 5.3 cm.
Solution:
Area = 12.4 × 5.3 = 65.72 cm²
Significant figures:
- 12.4 (3 sig figs)
- 5.3 (2 sig figs) ← limiting
Answer: 66 cm² (2 sig figs)
Or better: 6.6 × 10¹ cm² to clearly show 2 sig figs
Density ρ = m/V where m = 25.3 g and V = 12.5 cm³. Find ρ.
Solution:
ρ = 25.3 / 12.5 = 2.024 g/cm³
Both have 3 sig figs, so result has 3 sig figs:
Answer: 2.02 g/cm³ (3 sig figs)
Calculate: $(2.0 \times 10^2) \times (3.5 \times 10^{-3}) / (5.0 \times 10^4)$
Solution:
Numerator: $2.0 \times 3.5 = 7.0$ (2 sig figs)
Result: $\frac{7.0 \times 10^{-1}}{5.0 \times 10^4} = 1.4 \times 10^{-5}$
All numbers have 2 sig figs, answer has 2 sig figs:
Answer: 1.4 × 10⁻⁵ (2 sig figs)
Level 3: JEE Advanced
The mass of a solid cube is measured as 150.0 g. One side is measured as 5.20 cm. Find the density in proper significant figures.
Solution:
Volume = (5.20)³ = 140.608 cm³ (keep extra digits)
Density = 150.0 / 140.608 = 1.0668… g/cm³
Significant figures:
- Mass: 150.0 → 4 sig figs
- Side: 5.20 → 3 sig figs → Volume has 3 sig figs
- Limiting: 3 sig figs
Answer: 1.07 g/cm³ (3 sig figs)
In an experiment, time period T = 2π√(l/g).
Given: l = 100.0 cm = 1.000 m, g = 9.81 m/s²
Calculate T with proper sig figs. (Take π = 3.14159…)
Solution:
T = 2π√(1.000/9.81)
l/g = 1.000/9.81 = 0.10193… (keep extra digits)
√(l/g) = 0.31927… (keep extra digits)
T = 2 × 3.14159 × 0.31927 = 2.0062… s
Significant figures:
- l = 1.000 → 4 sig figs
- g = 9.81 → 3 sig figs ← limiting
- “2” is exact (from definition)
- π is exact (mathematical constant)
Answer: 2.01 s (3 sig figs, limited by g)
Calculate: $\frac{(5.2)^2 \times 10.0}{2.0 \times 3.5}$
Solution:
Numerator: $(5.2)^2 \times 10.0 = 27.04 \times 10.0 = 270.4$
- (5.2)² → 2 sig figs
- 10.0 → 3 sig figs
- Numerator → 2 sig figs (limited by 5.2)
Denominator: $2.0 \times 3.5 = 7.0$
- Both have 2 sig figs
Result: $270.4 / 7.0 = 38.628...$
Overall: 2 sig figs (limited by multiple factors)
Answer: 39 or 3.9 × 10¹ (2 sig figs)
Quick Summary Chart
Counting Rules:
| Digit Type | Rule |
|---|---|
| Non-zero digits | Always count |
| Captive zeros (between digits) | Always count |
| Leading zeros | Never count |
| Trailing zeros | Count only if decimal present |
| Exact numbers | Infinite sig figs |
Operation Rules:
| Operation | Rule |
|---|---|
| Addition/Subtraction | Least decimal places |
| Multiplication/Division | Fewest sig figs |
| Powers/Roots | Same sig figs as original |
Strategy for JEE
For MCQs: Check answer options first. If all have different sig figs, that’s your clue!
For numerical answers: JEE typically expects 2-4 significant figures unless specified.
When in doubt: Use scientific notation to clearly show sig figs.
Intermediate steps: Keep at least one extra digit, round only at end.
Common JEE values:
- g = 10 m/s² (exact for simplification)
- g = 9.8 m/s² (2 sig figs)
- g = 9.81 m/s² (3 sig figs)
- π = 3.14 (3 sig figs) or use exact value
Related Topics
Within Units and Measurements
- SI Units — Measurement precision
- Dimensions — Dimensionless quantities
- Dimensional Analysis — Checking equation validity
- Errors — Relationship between errors and sig figs
Connected Chapters
- Experimental Skills — Recording measurements
- All numerical problems in Physics — Proper precision in answers
Math Connections
- Scientific Notation — Expressing large/small numbers
- Rounding — Approximation techniques