Significant Figures and Rounding: Comprehensive Solutions

Mastering the Art of Precision in Numerical Calculations

Key Takeaways

Understanding Significant Figures: Significant figures indicate the precision of a number and are crucial in scientific calculations.
Rounding Rules: Proper rounding ensures the accuracy and reliability of numerical data.
Application Across Various Contexts: Significant figures are applied consistently across different numerical scenarios to maintain clarity and precision.

Introduction to Significant Figures

Significant figures (often abbreviated as sig figs) are the digits in a number that contribute to its precision. This concept is fundamental in various fields, including science, engineering, and mathematics, where precise measurements and calculations are essential. Understanding significant figures helps in expressing uncertainty and ensuring that numerical results are both accurate and meaningful.

What Are Significant Figures?

Significant figures encompass all the certain digits in a measurement plus one final digit, which is somewhat uncertain or estimated. The number of significant figures in a number tells us about its precision; more significant figures indicate a higher precision.

Rules for Identifying Significant Figures

All non-zero digits are always significant.
Any zeros between significant digits are significant.
Leading zeros (zeros before the first non-zero digit) are not significant.
Trailing zeros in a number with a decimal point are significant.
Trailing zeros in a whole number with no decimal shown are not significant unless indicated otherwise.

Detailed Solutions and Explanations

Q1. Identify the First Significant Figure in 81049

To determine the first significant figure in the number 81049, we scan from left to right to find the first non-zero digit.

Solution: The digits are 8, 1, 0, 4, 9. The first non-zero digit is 8.

Q2. Identify the Second Significant Figure in 73.29

For the number 73.29, we identify significant figures by recognizing all non-zero digits and any zeros between them.

Solution: The first significant figure is 7 and the second is 3.

Q3. Identify the Second Significant Figure in 1.7328

In the number 1.7328, each digit contributes to the precision.

Solution: The first significant figure is 1, and the second is 7.

Q4. Identify the Third Significant Figure in 2.60153

The number 2.60153 contains multiple digits, including zeros.

Solution: The third significant figure is 0, as it is between significant digits.

Q5. Determine the Number of Significant Figures in Given Numbers

a) 34902

Analyzing the number 34902:

All non-zero digits (3, 4, 9, 2) are significant.
The zero between 4 and 9 is also significant.

Solution: 34902 has 5 significant figures.

b) 0.014

For the number 0.014:

Leading zeros (0.0) are not significant.
Only the digits 1 and 4 are significant.

Solution: 0.014 has 2 significant figures.

Q6. Rounding the Number 1628 to Significant Figures

The number 1628 has 4 significant figures. We will round it to 3 and 2 significant figures respectively.

a) Rounding to 3 Significant Figures

Consider the first three digits: 1, 6, and 2. The next digit is 8.

Since 8 ≥ 5, round the last kept digit (2) up to 3.

Solution: 1628 rounded to 3 significant figures is 1630.

b) Rounding to 2 Significant Figures

Consider the first two digits: 1 and 6. The next digit is 2.

Since 2 < 5, the digit remains unchanged.
Adjust the number to its correct place value.

Solution: 1628 rounded to 2 significant figures is 1600.

Q7. Rounding Numbers to 1 Significant Figure

a) 523

Expressed as 5.23 × 10², we focus on the first digit.

The next digit (2) < 5, so the first digit remains 5.

Solution: 523 rounded to 1 significant figure is 500.

b) 27

Expressed as 2.7 × 10¹, we focus on the first digit.

The next digit (7) ≥ 5, so the first digit is rounded up from 2 to 3.

Solution: 27 rounded to 1 significant figure is 30.

c) 345

Expressed as 3.45 × 10², we focus on the first digit.

The next digit (4) < 5, so the first digit remains 3.

Solution: 345 rounded to 1 significant figure is 300.

d) 0.684

Expressed as 6.84 × 10⁻¹, we focus on the first digit.

The next digit (8) ≥ 5, so the first digit is rounded up from 6 to 7.

Solution: 0.684 rounded to 1 significant figure is 0.7.

Q8. Rounding Numbers to 2 Significant Figures

a) 568

Expressed as 5.68 × 10², we focus on the first two digits.

The next digit (8) ≥ 5, so the second digit is rounded up from 6 to 7.

Solution: 568 rounded to 2 significant figures is 570.

b) 932

Expressed as 9.32 × 10², we focus on the first two digits.

The next digit (2) < 5, so the second digit remains 3.

Solution: 932 rounded to 2 significant figures is 930.

c) 0.629

Expressed as 6.29 × 10⁻¹, we focus on the first two digits.

The next digit (9) ≥ 5, so the second digit is rounded up from 2 to 3.

Solution: 0.629 rounded to 2 significant figures is 0.63.

d) 6566

Expressed as 6.566 × 10³, we focus on the first two digits.

The next digit (6) ≥ 5, so the second digit is rounded up from 5 to 6.

Solution: 6566 rounded to 2 significant figures is 6600.

Part 2: Advanced Rounding Exercises

Q1. Rounding to 1 Significant Figure

a) 32

Expressed as 3.2 × 10¹, focus on the first digit.

The next digit (2) < 5, so the first digit remains 3.

Solution: 32 rounded to 1 significant figure is 30.

b) 1024

Expressed as 1.024 × 10³, focus on the first digit.

The remaining digits determine the place value.

Solution: 1024 rounded to 1 significant figure is 1000.

c) 0.00257

Expressed as 2.57 × 10⁻³, focus on the first digit.

The next digit (5) ≥ 5, so the first digit is rounded up from 2 to 3.

Solution: 0.00257 rounded to 1 significant figure is 0.003.

d) 9399

Expressed as 9.399 × 10³, focus on the first digit.

The next digit (3) < 5, so the first digit remains 9.

Solution: 9399 rounded to 1 significant figure is 9000.

Q2. Rounding to 2 Significant Figures

a) 3478

Expressed as 3.478 × 10³, focus on the first two digits.

The next digit (7) ≥ 5, so the second digit is rounded up from 4 to 5.

Solution: 3478 rounded to 2 significant figures is 3500.

b) 0.2149

Expressed as 2.149 × 10⁻¹, focus on the first two digits.

The next digit (4) < 5, so the second digit remains 1.

Solution: 0.2149 rounded to 2 significant figures is 0.21.

c) 25693

Expressed as 2.5693 × 10⁴, focus on the first two digits.

The next digit (6) ≥ 5, so the second digit is rounded up from 5 to 6.

Solution: 25693 rounded to 2 significant figures is 26000.

d) 12371

Expressed as 1.2371 × 10⁴, focus on the first two digits.

The next digit (3) < 5, so the second digit remains 2.

Solution: 12371 rounded to 2 significant figures is 12000.

Q3. Rounding to 3 Significant Figures

a) 9281

Expressed as 9.281 × 10³, focus on the first three digits.

The next digit (1) < 5, so the third digit remains 8.

Solution: 9281 rounded to 3 significant figures is 9280.

b) 3736

Expressed as 3.736 × 10³, focus on the first three digits.

The next digit (6) ≥ 5, so the third digit is rounded up from 3 to 4.

Solution: 3736 rounded to 3 significant figures is 3740.

c) 123.59

Focus on the first three significant digits: 1, 2, and 3.

The next digit (5) ≥ 5, so the third digit is rounded up from 3 to 4.

Solution: 123.59 rounded to 3 significant figures is 124.

d) 13.892

Focus on the first three significant digits: 1, 3, and 8.

The next digit (9) ≥ 5, so the third digit is rounded up from 8 to 9.

Solution: 13.892 rounded to 3 significant figures is 13.9.

Understanding the Rounding Process

Rounding is the process of reducing the number of digits in a number while maintaining its value as close as possible to the original number. The key to effective rounding lies in understanding the position of each digit and the rules that determine how each digit should be handled based on its value and position.

Rounding Rules Explained

1. Identify the Digit to Round To

Determine the position of the digit that will remain after rounding. All digits to the right of this digit are subject to rounding.

2. Examine the Next Digit

Look at the digit immediately to the right of the digit you're rounding to:

If this digit is 5 or greater, increase the rounding digit by 1.
If this digit is less than 5, leave the rounding digit unchanged.

3. Adjust the Number Accordingly

After determining whether to round up or maintain the digit, adjust the number by replacing all subsequent digits with zeros (for whole numbers) or removing them (for decimal numbers).

Significant Figures in Scientific Contexts

In scientific measurements, significant figures are crucial for conveying the precision of measurements and ensuring that calculations reflect the uncertainty inherent in the measurements.

Why Significant Figures Matter

Significant figures provide a way to express the precision of measurements. They are essential in ensuring that numerical results are not overstated in terms of precision.

Application in Calculations

When performing calculations involving multiplication, division, addition, or subtraction, the number of significant figures in the result should reflect the precision of the least precise measurement.

Multiplication and Division

The result should have the same number of significant figures as the measurement with the fewest significant figures.

Addition and Subtraction

The result should be rounded to the same decimal place as the measurement with the fewest decimal places.

Comprehensive Summary of Answers

Question	Answer
Q1. First significant figure in 81049	8
Q2. Second significant figure in 73.29	3
Q3. Second significant figure in 1.7328	7
Q4. Third significant figure in 2.60153	0
Q5a. Significant figures in 34902	5
Q5b. Significant figures in 0.014	2
Q6a. 1628 to 3 sig figs	1630
Q6b. 1628 to 2 sig figs	1600
Q7a. 523 to 1 sig fig	500
Q7b. 27 to 1 sig fig	30
Q7c. 345 to 1 sig fig	300
Q7d. 0.684 to 1 sig fig	0.7
Q8a. 568 to 2 sig figs	570
Q8b. 932 to 2 sig figs	930
Q8c. 0.629 to 2 sig figs	0.63
Q8d. 6566 to 2 sig figs	6600
Part 2 Q1a. 32 to 1 sig fig	30
Part 2 Q1b. 1024 to 1 sig fig	1000
Part 2 Q1c. 0.00257 to 1 sig fig	0.003
Part 2 Q1d. 9399 to 1 sig fig	9000
Part 2 Q2a. 3478 to 2 sig figs	3500
Part 2 Q2b. 0.2149 to 2 sig figs	0.21
Part 2 Q2c. 25693 to 2 sig figs	26000
Part 2 Q2d. 12371 to 2 sig figs	12000
Part 2 Q3a. 9281 to 3 sig figs	9280
Part 2 Q3b. 3736 to 3 sig figs	3740
Part 2 Q3c. 123.59 to 3 sig figs	124
Part 2 Q3d. 13.892 to 3 sig figs	13.9

Conclusion

Mastering significant figures and the associated rounding rules is essential for ensuring precision and accuracy in numerical calculations. Whether handling simple numbers or complex measurements, adhering to these principles guarantees that results are both reliable and meaningful. By understanding and applying significant figure rules effectively, one can communicate numerical data with clarity and confidence.