In scientific measurements, significant figures are the digits that carry meaningful information about the precision of the measurement. They include all the digits that are known with certainty, plus the first digit that is estimated or uncertain. Using the correct number of significant figures is crucial for honestly representing the precision of data.
Determining which digits in a number are significant follows a set of established rules:
| Rule | Explanation | Example | Significant Figures |
|---|---|---|---|
| 1. Non-Zero Digits | All non-zero digits are always significant. | 12.34 | 4 |
| 2. Captive Zeros | Zeros located between two non-zero digits are significant. | 506 | 3 |
| 3. Leading Zeros | Zeros that come before all non-zero digits are not significant. They are simply placeholders. | 0.00578 | 3 |
| 4. Trailing Zeros (with decimal) | Trailing zeros after a decimal point are significant — they indicate a specific level of precision. | 9.100 | 4 |
| 4. Trailing Zeros (no decimal) | Trailing zeros in a whole number without a decimal point are ambiguous. Use scientific notation to remove ambiguity. | 3500 | Ambiguous (2, 3, or 4) |
| 5. Scientific Notation | All digits in the coefficient are significant. | 3 |
Tip: To remove ambiguity in trailing zeros, always write numbers in scientific notation. For example, has 2 sig figs, while has 4 sig figs.
The precision of a calculated result is limited by the least precise measurement used. Different rules apply for different mathematical operations. See also Uncertainties In Final Result→ for how uncertainty propagates in calculations.
Rule: The result should be rounded to the same number of significant figures as the measurement with the least number of significant figures.
Example:
Calculate the area of a rectangle with a length of 21.3 cm and a width of 9.8 cm.
Since the least precise measurement (9.8 cm) has only two significant figures, the answer must be rounded to two significant figures.
Final Answer: (or 210 cm²)
Rule: The result should be rounded to the same number of decimal places as the measurement with the least number of decimal places.
Example:
Add the following masses: 12.11 g, 3.6 g, and 0.254 g.
12.11 (2 decimal places)
+ 3.6 (1 decimal place)
+ 0.254 (3 decimal places)
-------
15.964 g
The least precise measurement (3.6 g) has only one decimal place. Therefore, the answer must be rounded to one decimal place.
Final Answer: 16.0 g
Exact numbers (counting numbers or defined constants, e.g., 1 dozen = 12, or the 2 in ) are considered to have an infinite number of significant figures. They do not limit the significant figures in a calculated result.
Leading zeros are placeholders that indicate the magnitude of the number (i.e., where the decimal point is). They do not add to the precision of the measurement. For example, m is the same as mm — both have one significant figure.
| Operation | Rule |
|---|---|
| Multiplication / Division | Result is limited by the measurement with the least number of significant figures. |
| Addition / Subtraction | Result is limited by the measurement with the least number of decimal places. |
Proper use of significant figures is a fundamental aspect of scientific integrity, as it reflects a commitment to accurate and honest reporting of data. See also Precision And Accuracy→ for related concepts on measurement quality.