Comprehensive Guide to Rounding Numbers and Significant Figures

Mastering the Art of Precision in Mathematical Calculations

Key Takeaways

Understanding significant figures is crucial for accurate scientific measurements.
Rounding numbers requires careful consideration of each digit's value.
Determining the range of possible values after rounding ensures precision in data interpretation.

Introduction to Significant Figures

In scientific and mathematical contexts, significant figures (s.f.) are essential for conveying the precision of measurements. Rounding numbers to a specific number of significant figures allows for consistency and accuracy in data representation. This guide provides a step-by-step approach to rounding various numbers to their designated significant figures and determining the range of possible values based on the rounding precision.

Q4. Rounding the Top Speed of a Racing Snail

Original Measurement and Rounding Process

Given:

The speed of a racing snail is 28 mm/s.

Rounding to One Significant Figure

To round 28 mm/s to one significant figure:

Identify the first significant digit: 2.
Examine the next digit: 8.
Since 8 ≥ 5, round the first digit up.

Therefore, 28 mm/s rounded to one significant figure is 30 mm/s.

Part 3: Rounding Numbers to Specified Significant Figures

Q1. Rounding Specific Numbers

a) 0.0829 – to 1 s.f.

Procedure:

Identify the first non-zero digit: 8.
Examine the next digit: 2.
Since 2 < 5, do not round up.

Result: 0.0829 rounded to 1 s.f. is 0.08.

b) 37.62 – to 2 s.f.

Procedure:

Identify the first two significant digits: 3 and 7.
Examine the next digit: 6.
Since 6 ≥ 5, round the second digit up.

Result: 37.62 rounded to 2 s.f. is 38.

c) 85.729 – to 3 s.f.

Procedure:

Identify the first three significant digits: 8, 5, and 7.
Examine the next digit: 2.
Since 2 < 5, do not round up.

Result: 85.729 rounded to 3 s.f. is 85.7.

d) 4782 – to 2 s.f.

Procedure:

Identify the first two significant digits: 4 and 7.
Examine the next digit: 8.
Since 8 ≥ 5, round the second digit up.
Replace subsequent digits with zeros.

Result: 4782 rounded to 2 s.f. is 4800.

e) 3929 – to 1 s.f.

Procedure:

Identify the first significant digit: 3.
Examine the next digit: 9.
Since 9 ≥ 5, round the first digit up.
Replace subsequent digits with zeros.

Result: 3929 rounded to 1 s.f. is 4000.

f) 274.65 – to 3 s.f.

Procedure:

Identify the first three significant digits: 2, 7, and 4.
Examine the next digit: 6.
Since 6 ≥ 5, round the third digit up.

Result: 274.65 rounded to 3 s.f. is 275.

Q2. Rounding the Mass of a Rock

Given:

The mass of a rock is 22.784 kg.

Rounding to 3 Significant Figures

Procedure:

Identify the first three significant digits: 2, 2, and 7.
Examine the next digit: 8.
Since 8 ≥ 5, round the third digit up.

Result: 22.784 kg rounded to 3 s.f. is 22.8 kg.

Q3. Rounding the Area of a Pond

Given:

The area of a pond is 0.0004999 km².

Rounding to 2 Significant Figures

Procedure:

Identify the first two non-zero significant digits: 4 and 9.
Examine the next digit: 9.
Since 9 ≥ 5, round the second digit up.

The number 0.0004999 km² in scientific notation is 4.999 × 10⁻⁴ km². Rounding the first two significant figures:

4.999 rounded to 2 s.f. is 5.0.
Converting back gives 0.00050 km².

Result: 0.0004999 km² rounded to 2 s.f. is 0.00050 km².

Q4. Determining the Range of Possible Values Based on Rounding

Given:

The number of people at a concert has been rounded to 20,000.

Understanding the Rounding Precision

The range of possible original numbers depends on the number of significant figures to which 20,000 was rounded:

Significant Figures	Smallest Possible Number	Largest Possible Number
1 s.f.	15,000	24,999
2 s.f.	19,500	20,499
3 s.f.	19,950	20,049

a) Rounded to 1 Significant Figure

Procedure:

With 1 s.f., only the first digit is significant: 2 × 10⁴.
The rounding unit is 10,000.
Numbers from 15,000 up to but not including 25,000 round to 20,000.

Result: Smallest number = 15,000, Largest number = 24,999.

b) Rounded to 2 Significant Figures

Procedure:

With 2 s.f., the digits are 20,000 represented as 2.0 × 10⁴.
The rounding unit is 1,000.
Numbers from 19,500 up to but not including 20,500 round to 20,000..

Result: Smallest number = 19,500, Largest number = 20,499.

c) Rounded to 3 Significant Figures

Procedure:

With 3 s.f., the digits are 20,000 represented as 20.0 × 10³.
The rounding unit is 100.
Numbers from 19,950 up to but not including 20,050 round to 20,000.

Result: Smallest number = 19,950, Largest number = 20,049.

Conclusion

Accurately rounding numbers to a specific number of significant figures is fundamental in scientific measurements and data analysis. This guide has systematically addressed the rounding process for various numbers, ensuring precision and consistency. Understanding the principles of significant figures not only enhances the clarity of data representation but also facilitates effective communication of quantitative information.