Measurement and Uncertainty
3. Significant Figures
Learning outcomes
- I can determine the number of significant figures in measured values.
- I can apply the rules for significant figures during calculations.
- I can round answers to the correct number of significant figures.
- I can explain why significant figures reflect measurement precision.
- I can report calculated results using appropriate significant figures.
Introduction
When scientists make measurements, they can never know a value with perfect certainty. Every measuring instrument has a limit to how precisely it can measure.
To show the precision of a measurement, scientists use significant figures (often called significant digits or sig figs).
Significant figures tell us which digits in a measurement are meaningful and help ensure that calculated answers do not appear more precise than the original measurements.
For example:
- 12 m suggests a less precise measurement than
- 12.00 m
Although both have the same numerical value, the second measurement shows that the length was measured much more precisely.
What Are Significant Figures?
Significant figures are the digits in a measurement that carry meaningful information about its precision.
They include:
- All certain digits
- The first uncertain (estimated) digit
The last significant figure is always an estimate made by the observer.
Why Are Significant Figures Important?
Suppose two students measure the same object.
Student A records:
15 cm
Student B records:
15.000 cm
Although both values are numerically equal, Student B has measured the object much more precisely.
Significant figures communicate this difference.
Without significant figures, scientists could easily misinterpret the quality of measurements.
Rules for Counting Significant Figures
Rule 1: All Non-Zero Digits Are Significant
Every digit from 1 to 9 is significant.
Examples:
| Measurement | Significant Figures |
|---|---|
| 7 | 1 |
| 34 | 2 |
| 582 | 3 |
| 9146 | 4 |
Rule 2: Zeros Between Non-Zero Digits Are Significant
Examples:
| Measurement | Significant Figures |
|---|---|
| 101 | 3 |
| 5007 | 4 |
| 2.03 | 3 |
The zeros are part of the measured value.
Rule 3: Leading Zeros Are NOT Significant
Zeros before the first non-zero digit simply locate the decimal point.
Examples:
| Measurement | Significant Figures |
|---|---|
| 0.5 | 1 |
| 0.034 | 2 |
| 0.00780 | 3 |
Only the digits 7, 8, and 0 are significant in the last example.
Rule 4: Trailing Zeros After a Decimal Point ARE Significant
Examples:
| Measurement | Significant Figures |
|---|---|
| 4.0 | 2 |
| 6.20 | 3 |
| 12.500 | 5 |
These zeros show additional measurement precision.
Rule 5: Trailing Zeros Without a Decimal Point Are Usually Not Significant
Examples:
| Measurement | Significant Figures |
|---|---|
| 1500 | Usually 2 |
| 40000 | Usually 1 |
Because the precision is unclear, scientists often use scientific notation.
For example:
| Scientific Notation | Significant Figures |
|---|---|
| 1.5 × 10³ | 2 |
| 1.500 × 10³ | 4 |
| 4.00 × 10⁴ | 3 |
Counting Significant Figures
Let's practice.
| Measurement | Significant Figures |
|---|---|
| 25.4 | 3 |
| 0.00482 | 3 |
| 1003 | 4 |
| 7.80 | 3 |
| 9.000 | 4 |
| 0.0500 | 3 |
Significant Figures in Calculations
Different mathematical operations use different rules.
Multiplication and Division
The answer should contain the same number of significant figures as the measurement with the fewest significant figures.
Example
Calculate:
4.52 × 2.1
Calculator:
9.492
Since 2.1 has 2 significant figures, the answer becomes:
9.5
Addition and Subtraction
For addition and subtraction, use the smallest number of decimal places, not the smallest number of significant figures.
Example
15.42 + 2.1
Calculator:
17.52
Since 2.1 has 1 decimal place, the answer is:
17.5
Rounding to Significant Figures
When rounding:
- If the next digit is 5 or greater, round up.
- If it is less than 5, leave the digit unchanged.
Examples:
| Number | Rounded |
|---|---|
| 6.847 (3 s.f.) | 6.85 |
| 0.003426 (2 s.f.) | 0.0034 |
| 1257 (2 s.f.) | 1300 |
| 9.995 (3 s.f.) | 10.0 |
Notice that 10.0 has 3 significant figures.
Why Significant Figures Reflect Precision
The number of significant figures tells us how carefully a measurement was made.
Compare:
| Measurement | Precision |
|---|---|
| 12 m | Low |
| 12.0 m | Higher |
| 12.00 m | Even higher |
The extra digits indicate that the measuring instrument could distinguish smaller intervals.
Reporting Scientific Results
When writing answers:
- Include units.
- Use the correct number of significant figures.
- Avoid reporting unnecessary digits from a calculator.
For example:
Calculator:
3.141592654
If the data justify only three significant figures:
3.14
Reporting more digits would imply a level of precision that the measurements do not support.
Real-World Applications
Significant figures are important in:
- Scientific laboratories
- Medicine
- Engineering
- Manufacturing
- Construction
- Environmental science
- Space exploration
- Quality control
Correct use of significant figures helps scientists report measurements honestly and consistently.
Worked Examples
Example 1
How many significant figures are in 0.00560?
Answer:
3 significant figures
(5, 6, and the trailing 0)
Example 2
Round 18.476 to 4 significant figures.
Answer:
18.48
Example 3
Calculate:
5.62 × 2.0
Calculator:
11.24
The smallest number of significant figures is 2.
Answer:
11
Example 4
Calculate:
16.32 + 4.8
Calculator:
21.12
The smallest number of decimal places is 1.
Answer:
21.1
Example 5
Why are significant figures important?
Answer:
They show the precision of measurements and prevent calculated answers from implying greater accuracy than the original data support.
Did You Know?
Many modern digital laboratory instruments automatically display measurements with an appropriate number of significant figures based on their precision. However, scientists must still decide how many significant figures to report after performing calculations, ensuring that the final result reflects the precision of the original measurements rather than the extra digits produced by a calculator.
Key Terms
| Term | Definition |
|---|---|
| Significant Figures | The meaningful digits in a measurement, including all certain digits and the first estimated digit. |
| Precision | The level of detail or consistency in a measurement. |
| Leading Zeros | Zeros before the first non-zero digit; they are not significant. |
| Trailing Zeros | Zeros at the end of a number; they may or may not be significant depending on the presence of a decimal point. |
| Scientific Notation | A way of writing numbers using powers of ten to clearly show their significant figures. |
| Rounding | Adjusting a number to the required number of significant figures or decimal places. |
Key Takeaways
- Significant figures communicate the precision of a measurement.
- All non-zero digits are significant.
- Leading zeros are not significant.
- Zeros between non-zero digits are always significant.
- Trailing zeros after a decimal point are significant.
- In multiplication and division, the answer should have the same number of significant figures as the least precise measurement.
- In addition and subtraction, the answer should have the same number of decimal places as the measurement with the fewest decimal places.
- Scientific results should always be reported with an appropriate number of significant figures and the correct units.