Significant Figures Calculator
What are the rules for counting significant figures?
| Rule | Example | Sig Figs |
|---|---|---|
| All non-zero digits are significant | 1,234 | 4 |
| Zeros between non-zero digits are significant | 1,007 | 4 |
| Leading zeros are NOT significant | 0.00470 | 3 (4, 7, 0) |
| Trailing zeros after decimal point ARE significant | 2.500 | 4 |
| Trailing zeros in whole numbers are ambiguous | 2,500 | 2, 3, or 4 — use sci notation |
| Exact numbers have infinite sig figs | 12 eggs, π definition | ∞ |
How do sig figs apply to calculations?
| Operation | Rule | Example |
|---|---|---|
| Multiplication / Division | Result has same sig figs as the LEAST precise input | 3.2 × 4.56 = 14.592 → 15 (2 sig figs) |
| Addition / Subtraction | Result has same DECIMAL PLACES as the least precise input | 15.3 + 1.76 = 17.06 → 17.1 (1 decimal place) |
| Mixed operations | Apply rules at each step; round only at the end | Keep extra digits in intermediate steps |
Scientific notation and significant figures
Scientific notation eliminates ambiguity. 2,500 with 3 sig figs: 2.50 × 10³. With 4 sig figs: 2.500 × 10³. With 2 sig figs: 2.5 × 10³. In scientific notation, all digits in the coefficient are significant — making the precision explicit. This is why scientists always express measurements in scientific notation. For related measurement tasks, try the Scientific Calculator or the Unit Converter.
Common significant figure mistakes
Most frequent errors: (1) Counting leading zeros as significant — 0.0045 has 2 sig figs, not 4. (2) Forgetting trailing zeros matter after the decimal — 3.40 has 3 sig figs, not 2. (3) Rounding intermediate steps — only round the final answer. (4) Treating counted numbers as measurements — "5 students" is exact, not 1 sig fig.
Significant figures rules with examples
| Rule | Example | Sig figs |
|---|---|---|
| All non-zero digits are significant | 3.47 | 3 |
| Zeros between non-zero digits are significant | 1,007 | 4 |
| Leading zeros (before first non-zero digit) are NOT significant | 0.0042 | 2 |
| Trailing zeros after decimal point ARE significant | 3.400 | 4 |
| Trailing zeros in a whole number are ambiguous | 1200 | 2, 3, or 4 (use scientific notation to clarify) |
| Exact numbers have infinite sig figs | 12 eggs, π = 3.14159... | Infinite (no rounding applies) |
Rounding to significant figures: step by step
To round 0.004782 to 3 significant figures:
- Find the first significant figure: 4 (the leading zeros are not significant)
- Count 3 sig figs from the first: 4, 7, 8
- Look at the next digit: 2 (less than 5, so round down)
- Result: 0.00478
To round 152,840 to 3 significant figures:
- First 3 sig figs: 1, 5, 2
- Next digit: 8 (≥5, so round up)
- Result: 153,000 (or 1.53 × 10⁵ in scientific notation to clearly show 3 sig figs)
Significant figures in calculations
Multiplication and division: The result has the same number of significant figures as the measurement with the fewest sig figs.
Example: 4.56 × 1.4 = 6.384 → rounded to 6.4 (2 sig figs, because 1.4 has only 2).
Addition and subtraction: The result has the same number of decimal places as the measurement with the fewest decimal places.
Example: 12.11 + 18.0 + 1.013 = 31.123 → rounded to 31.1 (one decimal place, because 18.0 has one).
Sources: NIST Reference on Constants, Units and Uncertainty; IUPAC Green Book, 3rd edition.
Frequently asked questions
How many significant figures does 0.00470 have?
Three. The leading zeros (0.00) are not significant — they are just place-holders. The digits 4, 7, and the trailing 0 are all significant. The trailing zero after 7 is significant because it is written explicitly after the decimal point, indicating precision to the ten-thousandths place.
How do sig figs work in multiplication?
The answer should have the same number of sig figs as the input with the fewest sig figs. 2.5 (2 sig figs) × 3.14159 (6 sig figs) = 7.85... → rounded to 7.9 (2 sig figs). The precision is limited by the least precise measurement in the calculation.
Are significant figures the same as decimal places?
No. Significant figures count all meaningful digits regardless of position. Decimal places only count digits after the decimal point. 123.4 has 4 sig figs and 1 decimal place. 0.00123 has 3 sig figs and 5 decimal places. Sig figs measure precision; decimal places measure scale.
What is the difference between accuracy and precision?
Accuracy is how close a measurement is to the true value. Precision is how consistently repeated measurements agree with each other. Sig figs relate to precision — the number of reliable digits in a measurement. A measurement can be precise (small variation) but inaccurate (biased away from the true value).
How do you express 2500 to exactly 3 significant figures?
Write in scientific notation: 2.50 × 10³. The trailing zero after the decimal is now unambiguous — it is explicitly significant. Alternatively, use an overbar notation: 2500̄ (bar over the zero means it is the last significant digit), but scientific notation is the universal standard.
Sources & References
Sources: NIST Guidelines for Evaluating and Expressing Uncertainty (NIST Technical Note 1297); Taylor, J.R. An Introduction to Error Analysis; IUPAC Green Book — Quantities, Units and Symbols in Physical Chemistry.
Significant figures in chemistry: molar mass and stoichiometry
Chemistry calculations require careful sig fig management. Molar mass of water (H₂O): H = 1.008 g/mol (4 sig figs), O = 15.999 g/mol (5 sig figs). Molar mass = 2(1.008) + 15.999 = 2.016 + 15.999 = 18.015 g/mol — reported to the precision of the addition rule (3 decimal places → 5 sig figs). How many moles in 5.0 g of water? 5.0 ÷ 18.015 = 0.278 mol → round to 2 sig figs (limited by 5.0): answer = 0.28 mol.
| Calculation Type | Sig Fig Rule | Example |
|---|---|---|
| Multiplication/Division | Fewest sig figs of any factor | 3.2 × 4.567 = 14.614 → 15 (2 sf) |
| Addition/Subtraction | Fewest decimal places | 12.3 + 1.45 = 13.75 → 13.8 |
| Logarithm (log x) | Decimal places = sig figs of x | log(2.34×10³) = 3.369 (3 dp) |
| Antilog (10ˣ) | Sig figs = decimal places of x | 10^3.47 = 2951 → 3000 (2 sf) |
Significant figures in physics: worked examples
Velocity: distance = 125.4 m (4 sf), time = 12.1 s (3 sf). Velocity = 125.4 ÷ 12.1 = 10.3636... → 10.4 m/s (3 sf, limited by 12.1).
Force (F = ma): mass = 2.5 kg (2 sf), acceleration = 9.81 m/s² (3 sf). F = 2.5 × 9.81 = 24.525 → 25 N (2 sf, limited by 2.5).
Energy: E = ½mv². m = 45.0 kg (3 sf), v = 3.2 m/s (2 sf). E = 0.5 × 45.0 × 10.24 = 230.4 → 230 J (2 sf).
Common sig fig exam questions and answers
| Question | Answer | Why |
|---|---|---|
| How many sig figs in 0.00470? | 3 (4, 7, 0) | Leading zeros: not significant. Trailing zero after decimal: significant |
| How many sig figs in 2500? | Ambiguous (2, 3, or 4) | Use scientific notation: 2.5×10³ = 2 sf; 2.50×10³ = 3 sf |
| How many sig figs in 100.0? | 4 | Trailing zeros after decimal point are always significant |
| How many sig figs in the number 12 eggs? | Infinite (exact) | Counted quantities are exact — not subject to measurement uncertainty |
| Round 3.7892 to 3 sig figs | 3.79 | 4th digit is 9 (≥5), so 8 rounds up to 9 |
Sources: IUPAC Green Book — Quantities, Units and Symbols in Physical Chemistry (3rd ed); NIST Guidelines for Evaluating Measurement Uncertainty (NIST TN 1297).