Definition
Significant figures (or significant digits) are the digits in a measured value that carry meaningful information about its precision.
They include:
- all certain digits obtained from a measurement,
- plus one estimated (uncertain) digit.
The number of significant figures indicates how precisely a quantity has been measured.
Why Are Significant Figures Important?
No measurement is perfectly exact.
The number of significant figures communicates the reliability of a measured value.
For example,
- 2 m is much less precise than
- 2.000 m
although they represent the same measured value with different levels of precision.
Rules for Identifying Significant Figures
Rule 1
All non-zero digits are significant.
| Number | Significant Figures |
| 35 | 2 |
| 482 | 3 |
| 7.19 | 3 |
Rule 2
Zeros between non-zero digits are significant.
| Number | Significant Figures |
| 1002 | 4 |
| 5.08 | 3 |
| 20.03 | 4 |
Rule 3
Leading zeros are not significant. They only indicate the position of the decimal point.
| Number | Significant Figures |
| 0.5 | 1 |
| 0.047 | 2 |
| 0.00360 | 3 |
Rule 4
Trailing zeros to the right of a decimal point are significant.
| Number | Significant Figures |
| 3.40 | 3 |
| 8.200 | 4 |
| 0.0500 | 3 |
Rule 5
Trailing zeros in whole numbers without a decimal point are usually not considered significant unless their significance is explicitly indicated, such as by using a decimal point or scientific notation.
For example,
| Number | Possible Interpretation |
| 1500 | significance is ambiguous |
| 1500. | 4 significant figures |
| 1.500 × 10³ | 4 significant figures |
| 1.5 × 10³ | 2 significant figures |
Scientific notation removes this ambiguity by clearly indicating the intended number of significant figures.
Significant Figures in Scientific Notation
Only the digits in the coefficient are significant.
| Scientific Notation | Significant Figures |
| 3.2 × 10⁵ | 2 |
| 4.560 × 10⁻² | 4 |
| 8.000 × 10⁶ | 4 |
Rounding to Significant Figures
When reducing the number of significant figures:
- If the first discarded digit is less than 5, leave the previous digit unchanged.
- If the first discarded digit is 5 or greater, increase the previous digit by one.
| Original | Rounded |
| 3.14159 → 3 s.f. | 3.14 |
| 6.876 → 2 s.f. | 6.9 |
| 0.004562 → 2 s.f. | 0.0046 |
| 7850 → 2 s.f. | 7.9 × 10³ |
Using scientific notation makes the intended number of significant figures clear.
Significant Figures in Calculations
Multiplication and Division
The final answer should contain the same number of significant figures as the quantity with the fewest significant figures.
Example:
Since 2.5 has 2 significant figures,
Therefore, the answer should be reported as 8.6.
Addition and Subtraction
For addition and subtraction, the result is determined by the least number of decimal places, not the least number of significant figures.
Example:
Rounded to one decimal place:
Examples
Example 1
How many significant figures are in 0.00650?
Leading zeros are not significant.
The digits 6, 5, and the final 0 are significant.
Answer: 3 significant figures
Example 2
How many significant figures are in 4020?
The zero between 4 and 2 is significant.
The final zero is ambiguous because there is no decimal point.
Answer: The number of significant figures is ambiguous. It is commonly interpreted as having 3 significant figures unless additional notation is provided.
Example 3
Express 4500 with two significant figures.
Answer:
Scientific notation clearly indicates two significant figures.
Exact Numbers
Some numbers are exact rather than measured. Examples include counted quantities (such as 12 students) and defined quantities (such as 1 m = 100 cm). Exact numbers are considered to have an unlimited number of significant figures and do not limit the number of significant figures in calculations.
Key Points
- Significant figures indicate the precision of a measured quantity.
- All non-zero digits are significant.
- Leading zeros are never significant.
- Zeros between non-zero digits are significant.
- Trailing decimal zeros are significant.
- Scientific notation clearly shows the intended number of significant figures.
- Multiplication and division use the fewest significant figures.
- Addition and subtraction use the fewest decimal places.
- Exact numbers have an unlimited number of significant figures and do not limit the precision of calculated results.