Scientific notation writes very large or very small numbers compactly as a number between 1 and 10 multiplied by a power of 10 — 5,200,000 becomes 5.2 × 10⁶, and 0.00034 becomes 3.4 × 10⁻⁴. Every real measurement is an estimate, not a perfectly exact value, because measuring tools have limits. Absolute error is the size of the possible mistake (e.g. ±0.5cm on a ruler). Relative error compares that error to the measurement itself (error ÷ measurement), and percentage error expresses that as a percentage — useful because an error of 1cm matters far more on a 10cm measurement than on a 10km one.
Example
A distance of 45,000,000km (roughly the distance to Mars at closest approach) is written in scientific notation as 4.5 × 10⁷ km. If a ruler measures 20cm with a possible error of ±0.5cm, the relative error is 0.5/20 = 0.025, or a percentage error of 2.5% — small errors matter more on smaller measurements.
Key terms
Scientific notation:
Writing a number as a value between 1 and 10 multiplied by a power of 10.
Percentage error:
How large a measurement error is, expressed as a percentage of the measurement.
Questions
1. Scientific notation writes numbers as:
A value between 1 and 10 multiplied by a power of 10
Always a whole number with no decimal
A fraction only
A percentage only
2. 5,200,000 in scientific notation is:
5.2 × 10⁶
5.2 × 10⁵
52 × 10⁵
5.2 × 10⁷
3. Every real measurement is:
An estimate, not a perfectly exact value
Always perfectly exact
Impossible to make
Only ever a whole number
4. Absolute error is:
The size of the possible mistake in a measurement
Always zero
A percentage only
Unrelated to measuring tools
5. Relative error compares:
The error to the measurement itself
Two completely unrelated numbers
Only the tool's brand
Nothing meaningful
6. 0.00034 in scientific notation is:
3.4 × 10⁻⁴
3.4 × 10⁴
34 × 10⁻⁵
0.34 × 10⁻³
7. Percentage error expresses relative error as:
A percentage
Always a whole number
A colour
A fraction of 10
8. Write 0.0056 in scientific notation:
5.6 × 10⁻³
5.6 × 10³
56 × 10⁻⁴
0.56 × 10⁻²
9. Write 720,000 in scientific notation:
7.2 × 10⁵
7.2 × 10⁶
72 × 10⁴
7.2 × 10⁴
10. A ruler measures 20cm with a possible error of ±0.5cm. What is the percentage error?
2.5%
0.5%
5%
25%
11. A scale measures 2kg with a possible error of ±0.02kg. What is the percentage error?
1%
2%
0.2%
10%
12. Why does the same 1cm absolute error matter more on a 10cm measurement than on a 10km measurement?
The relative and percentage error is far larger for the smaller measurement
Absolute error always matters the same amount regardless of the total measurement
1cm is always an insignificant error regardless of context
Percentage error is not affected by the size of the original measurement
13. A measurement of 8m has an absolute error of ±0.4m. What is the percentage error?
5%
4%
8%
0.4%
14. Scientific notation is especially useful for:
Very large or very small numbers
Only numbers between 1 and 10
Numbers with no decimal places
Only negative numbers
15. Why might scientists studying the size of atoms (extremely small) or distances between galaxies (extremely large) rely heavily on scientific notation?
It keeps numbers with many zeros compact, readable and easier to compare or calculate with
Scientific notation only works for numbers close to 1
Extremely large and small numbers cannot be written using scientific notation
Standard decimal notation is always clearer for extreme values
16. A carpenter measures a plank as 2m with a possible error of ±1cm, while a jeweller measures a ring as 2cm with a possible error of ±1mm. Which measurement has the larger percentage error?
The jeweller's ring measurement (0.5% vs 0.5% — actually equal, testing careful reading)
The carpenter's plank measurement, since 1cm is a bigger absolute number
Both have identical relative error automatically regardless of context
Percentage error cannot be compared between different types of measurement
17. Why might reporting only the absolute error of a measurement (e.g. "±2cm") sometimes be less informative than reporting the percentage error?
Percentage error puts the error in context relative to the size of the measurement, making its significance clearer
Absolute error is always more informative than percentage error in every situation
Percentage error and absolute error always convey identical information
Context has no bearing on how meaningful an error measurement is
18. Why might a scientific calculation combining several measurements (each with their own small errors) end up with a larger overall uncertainty than any single measurement alone?
Errors from each individual measurement can accumulate when combined in a calculation
Combining measurements always cancels out and eliminates all error
Only the largest single error in a calculation ever matters, with no compounding effect
Measurement errors never affect the results of combined calculations
19. A pharmaceutical company measures the mass of an active ingredient in a tablet as 5mg with a possible error of ±0.1mg. Why is knowing the percentage error (2%) especially important in this context?
A small absolute error can represent a significant percentage of a very small dose, which could meaningfully affect the medicine's safety or effectiveness
Percentage error is irrelevant when the absolute error is already known
A 0.1mg error is always insignificant regardless of the total dose
Measurement error has no real-world consequence in pharmaceutical contexts
20. Why might astronomers reporting the distance to a star in scientific notation (e.g. 4.0 × 10¹³ km) include an estimated percentage error alongside the figure?
Even very precise instruments have measurement limits, and reporting the uncertainty honestly reflects the true confidence in the figure
Astronomical measurements are always exact with absolutely no possible error
Scientific notation removes the need to ever report any measurement uncertainty
Percentage error is only ever relevant for small, everyday measurements
21. Understanding scientific notation and measurement error mainly helps you to:
Work with extreme numbers compactly and understand the practical limits of any real measurement
Assume every measurement is always perfectly exact with no possible error
Avoid ever comparing errors across different measurement contexts
Treat absolute and percentage error as always meaning the same thing
Answer key (parent copy)
1. A value between 1 and 10 multiplied by a power of 10
2. 5.2 × 10⁶
3. An estimate, not a perfectly exact value
4. The size of the possible mistake in a measurement
5. The error to the measurement itself
6. 3.4 × 10⁻⁴
7. A percentage
8. 5.6 × 10⁻³
9. 7.2 × 10⁵
10. 2.5%
11. 1%
12. The relative and percentage error is far larger for the smaller measurement
13. 5%
14. Very large or very small numbers
15. It keeps numbers with many zeros compact, readable and easier to compare or calculate with
16. The jeweller's ring measurement (0.5% vs 0.5% — actually equal, testing careful reading)
17. Percentage error puts the error in context relative to the size of the measurement, making its significance clearer
18. Errors from each individual measurement can accumulate when combined in a calculation
19. A small absolute error can represent a significant percentage of a very small dose, which could meaningfully affect the medicine's safety or effectiveness
20. Even very precise instruments have measurement limits, and reporting the uncertainty honestly reflects the true confidence in the figure
21. Work with extreme numbers compactly and understand the practical limits of any real measurement