Consider The Following Equation:$\[ 0.0136 \, G + 2.70 \times 10^{-4} \, G - 4.21 \times 10^{-3} \, G = ? \\]How Many Digits To The Right Of The Decimal Point Should Be Used To Report The Result? \[$\square\$\]

by ADMIN 211 views

Introduction

Significant figures are a crucial concept in mathematics, particularly in scientific calculations. They help determine the accuracy and precision of a result. In this article, we will explore the concept of significant figures and how to apply them to mathematical equations.

What are Significant Figures?

Significant figures are the digits in a number that are known to be reliable and certain. They are used to express the precision of a measurement or calculation. The number of significant figures in a result depends on the number of significant figures in the input values.

Rules for Significant Figures

There are several rules to follow when working with significant figures:

  • Non-zero digits are always significant: Any non-zero digit in a number is considered significant.
  • Zeros between non-zero digits are significant: Zeros that appear between non-zero digits are also significant.
  • Leading zeros are not significant: Zeros that appear at the beginning of a number are not significant.
  • Trailing zeros are significant only if the number contains a decimal point: If a number contains a decimal point, trailing zeros are significant. Otherwise, they are not significant.
  • Rounding numbers affects significant figures: When rounding a number, the number of significant figures in the result is determined by the number of significant figures in the input value.

Significant Figures in 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 1

Consider the equation: 0.0136 g+2.70×10−4 g−4.21×10−3 g=?0.0136 \, g + 2.70 \times 10^{-4} \, g - 4.21 \times 10^{-3} \, g = ?

To determine the number of significant figures in the result, we need to examine the input values.

  • 0.0136 g0.0136 \, g has 3 significant figures.
  • 2.70×10−4 g2.70 \times 10^{-4} \, g has 3 significant figures.
  • 4.21×10−3 g4.21 \times 10^{-3} \, g has 3 significant figures.

Since all input values have the same number of significant figures, the result should also have 3 significant figures.

Significant Figures in 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 2

Consider the equation: 0.0136 g×2.70×10−4 g=?0.0136 \, g \times 2.70 \times 10^{-4} \, g = ?

To determine the number of significant figures in the result, we need to examine the input values.

  • 0.0136 g0.0136 \, g has 3 significant figures.
  • 2.70×10−4 g2.70 \times 10^{-4} \, g has 3 significant figures.

Since both input values have the same number of significant figures, the result should also have 3 significant figures.

Conclusion

Significant figures are an essential concept in mathematics, particularly in scientific calculations. By understanding the rules for significant figures, we can determine the accuracy and precision of a result. In this article, we explored the concept of significant figures and how to apply them to mathematical equations.

Final Answer

Based on the discussion above, the correct answer is:

  • The result should have 3 significant figures.

Additional Resources

For more information on significant figures, please refer to the following resources:

Practice Problems

Try the following practice problems to test your understanding of significant figures:

  • What is the number of significant figures in the result of the equation: 0.0136 g+2.70×10−4 g−4.21×10−3 g=?0.0136 \, g + 2.70 \times 10^{-4} \, g - 4.21 \times 10^{-3} \, g = ?
  • What is the number of significant figures in the result of the equation: 0.0136 g×2.70×10−4 g=?0.0136 \, g \times 2.70 \times 10^{-4} \, g = ?
    Significant Figures Q&A =========================

Q: What are significant figures?

A: Significant figures are the digits in a number that are known to be reliable and certain. They are used to express the precision of a measurement or calculation.

Q: How do I determine the number of significant figures in a number?

A: To determine the number of significant figures in a number, follow these steps:

  • Non-zero digits are always significant: Any non-zero digit in a number is considered significant.
  • Zeros between non-zero digits are significant: Zeros that appear between non-zero digits are also significant.
  • Leading zeros are not significant: Zeros that appear at the beginning of a number are not significant.
  • Trailing zeros are significant only if the number contains a decimal point: If a number contains a decimal point, trailing zeros are significant. Otherwise, they are not significant.

Q: What are the rules for significant figures in addition and subtraction?

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

Q: What are the rules for significant figures in multiplication and division?

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

Q: How do I round numbers to the correct number of significant figures?

A: To round numbers to the correct number of significant figures, follow these steps:

  • Determine the number of significant figures in the input value: Identify the number of significant figures in the input value.
  • Round the result to the correct number of significant figures: Round the result to the correct number of significant figures based on the number of significant figures in the input value.

Q: What is the difference between significant figures and decimal places?

A: Significant figures and decimal places are related but distinct concepts.

  • Significant figures: Significant figures are the digits in a number that are known to be reliable and certain.
  • Decimal places: Decimal places are the positions of the decimal point in a number.

Q: Why are significant figures important in scientific calculations?

A: Significant figures are important in scientific calculations because they help determine the accuracy and precision of a result.

Q: How do I apply significant figures to real-world problems?

A: To apply significant figures to real-world problems, follow these steps:

  • Identify the input values: Identify the input values and their corresponding significant figures.
  • Determine the number of significant figures in the result: Determine the number of significant figures in the result based on the input values.
  • Round the result to the correct number of significant figures: Round the result to the correct number of significant figures based on the number of significant figures in the input values.

Q: What are some common mistakes to avoid when working with significant figures?

A: Some common mistakes to avoid when working with significant figures include:

  • Not considering the number of significant figures in the input values: Failing to consider the number of significant figures in the input values can lead to inaccurate results.
  • Not rounding the result to the correct number of significant figures: Failing to round the result to the correct number of significant figures can lead to inaccurate results.
  • Not following the rules for significant figures in addition and subtraction: Failing to follow the rules for significant figures in addition and subtraction can lead to inaccurate results.

Conclusion

Significant figures are an essential concept in mathematics, particularly in scientific calculations. By understanding the rules for significant figures and applying them to real-world problems, you can ensure accurate and precise results.