Significant Figures Worksheet 1 Answer Key

Significant Figures Worksheet 1: Answer Key and Comprehensive Guide

Welcome to a deep dive into the world of significant figures! This comprehensive guide will not only provide you with the answers to a sample significant figures worksheet but also equip you with a thorough understanding of the rules and applications of significant figures in scientific calculations and data representation. Understanding significant figures is crucial for accurately representing data and performing calculations with appropriate precision in science, engineering, and other quantitative fields.

Understanding Significant Figures

Before we jump into the answer key, let's solidify our understanding of significant figures. Significant figures (sig figs) are the digits in a number that carry meaning contributing to its measurement resolution. They represent the precision of a measurement. Knowing how to identify and use significant figures is fundamental to accurate scientific reporting and calculations.

Rules for Determining Significant Figures

Here's a recap of the essential rules for determining the number of significant figures in a given number:

• Non-zero digits are always significant. For example, in the number 123, there are three significant figures.
• Zeros between non-zero digits are always significant. For example, in the number 1002, there are four significant figures.
• Leading zeros (zeros to the left of the first non-zero digit) are never significant. They only serve to place the decimal point. For example, in the number 0.0012, there are only two significant figures (1 and 2).
• Trailing zeros (zeros to the right of the last non-zero digit) are significant only if the number contains a decimal point. For example:
  • 120 has two significant figures.
  • 120. has three significant figures.
  • 120.0 has four significant figures.
• In scientific notation (a x 10^b), all digits in 'a' are significant. For example, 1.23 x 10⁴ has three significant figures.

Sample Significant Figures Worksheet and Answer Key

Let's consider a sample worksheet. Remember, the context of the problem is crucial; you should always consider the precision of your measuring instruments.

Worksheet:

Instructions: Determine the number of significant figures in each of the following numbers:

1. 250
2. 0.005
3. 10.02
5. 3.00 x 10^-2
6. 70000
7. 0.04050
8. 1200
9. 25.050
10. 0.000008

Calculations and the Application of Rules:

Let's break down how to determine the number of significant figures for each number in the worksheet.

1. 250: Two significant figures. The trailing zero is not significant because there's no decimal point.
2. 0.005: One significant figure. The leading zeros are not significant.
3. 10.02: Four significant figures. Zeros between non-zero digits are significant.
4. 2000.: Four significant figures. The trailing zeros are significant because of the decimal point.
5. 3.00 x 10^-2: Three significant figures. All digits in the coefficient are significant.
6. 70000: One significant figure. Trailing zeros without a decimal are not significant.
7. 0.04050: Four significant figures. Leading zeros are not significant, but the trailing zero after the decimal is significant.
8. 1200: Two significant figures. Trailing zeros without a decimal point are not significant.
9. 25.050: Five significant figures. All zeros after the decimal point are significant.
10. 0.000008: One significant figure. All leading zeros before the 8 are insignificant.

Answer Key:

1. 2
2. 1
3. 4
4. 4
5. 3
6. 1
7. 4
8. 2
9. 5
10. 1

Rounding and Significant Figures

Once you've determined the number of significant figures in your measurements, you will often need to round your results during calculations to reflect the appropriate level of precision. Here are the rules for rounding:

• If the digit to be dropped is less than 5, round down. For example, rounding 3.141 to three significant figures results in 3.14.
• If the digit to be dropped is greater than or equal to 5, round up. For example, rounding 3.146 to three significant figures results in 3.15.

Significant Figures in Calculations

The rules for significant figures extend to mathematical operations. The precision of the result of a calculation is limited by the least precise measurement involved.

Addition and Subtraction

When adding or subtracting numbers, the result should have the same number of decimal places as the number with the fewest decimal places.

Example: 12.34 + 5.6 + 1.234 = 19.174 rounds to 19.2 (one decimal place).

Multiplication and Division

When multiplying or dividing numbers, the result should have the same number of significant figures as the number with the fewest significant figures.

Example: 12.34 x 5.6 = 69.104 rounds to 69 (two significant figures).

Advanced Applications and Considerations

Let's explore some more complex scenarios and considerations regarding significant figures.

Scientific Notation and Significant Figures

Using scientific notation greatly simplifies working with very large or very small numbers, and it clearly displays the significant figures. Always express your final answer using scientific notation when dealing with very large or very small numbers to maintain clarity and precision.

Error Analysis and Propagation of Uncertainty

Significant figures are intrinsically linked to uncertainty in measurements. Every measurement has an associated uncertainty, often expressed as a plus or minus value. Understanding this uncertainty is crucial for accurate interpretation of experimental data. When performing calculations, you need to consider how the uncertainty in your measurements propagates through the calculations to affect the final result. This is known as error propagation.

Exact Numbers and Significant Figures

Exact numbers, such as counting numbers (e.g., 3 apples) or defined constants (e.g., there are exactly 100 centimeters in a meter), are considered to have an infinite number of significant figures and do not limit the number of significant figures in a calculation.

Conclusion

Mastering significant figures is a cornerstone of accurate scientific work. By understanding the rules for determining significant figures, performing calculations, and applying rounding techniques correctly, you will significantly improve the precision and reliability of your scientific reports and analyses. This detailed guide, including the example worksheet and answer key, serves as a comprehensive resource to aid you in your journey to mastering this essential skill. Remember to always focus on the context of your work and the precision inherent in your measurements for the most accurate results. Continue practicing with different numbers and calculations to solidify your understanding and build confidence in your abilities. Consistent practice is key to developing proficiency in working with significant figures.