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Introduction
In chemistry and other scientific disciplines, accurate measurements are crucial for obtaining reliable results. One of the fundamental concepts in measurement is the significance of digits, which refers to the number of digits that are known to be reliable in a measurement. When measurements are multiplied or divided, the number of significant digits in the result depends on the number of significant digits in the original measurements. In this article, we will discuss how to count significant digits when measurements are multiplied or divided.
Significant Digits in Measurement Multiplication
When two or more measurements are multiplied together, the number of significant digits in the result is determined by the measurement with the fewest significant digits. This is because the result of a multiplication is only as reliable as the least reliable measurement.
Example 1: Multiplying Measurements with the Same Number of Significant Digits
Suppose we have two measurements: 4.69 g and 3.73 g. Both measurements have 3 significant digits. When we multiply these measurements together, the result will also have 3 significant digits.
4.69 g × 3.73 g = 17.45 g²
In this example, the result has 3 significant digits, which is the same as the number of significant digits in the original measurements.
Example 2: Multiplying Measurements with Different Numbers of Significant Digits
Suppose we have two measurements: 4.69 g (3 significant digits) and 0.373 g (3 significant digits). When we multiply these measurements together, the result will have 3 significant digits, which is determined by the measurement with the fewest significant digits.
4.69 g × 0.373 g = 1.75 g²
In this example, the result has 3 significant digits, which is the same as the number of significant digits in the measurement with the fewest significant digits.
Significant Digits in Measurement Division
When two measurements are divided, the number of significant digits in the result is determined by the measurement with the fewest significant digits. This is because the result of a division is only as reliable as the least reliable measurement.
Example 1: Dividing Measurements with the Same Number of Significant Digits
Suppose we have two measurements: 4.69 g and 1.73 g. Both measurements have 3 significant digits. When we divide these measurements together, the result will also have 3 significant digits.
4.69 g ÷ 1.73 g = 2.70
In this example, the result has 3 significant digits, which is the same as the number of significant digits in the original measurements.
Example 2: Dividing Measurements with Different Numbers of Significant Digits
Suppose we have two measurements: 4.69 g (3 significant digits) and 0.173 g (3 significant digits). When we divide these measurements together, the result will have 3 significant digits, which is determined by the measurement with the fewest significant digits.
4.69 g ÷ 0.173 g = 27.0
In this example, the result has 3 significant digits, which is the same as the number of significant digits in the measurement with the fewest significant digits.
Conclusion
In conclusion, when measurements are multiplied or divided, the number of significant digits in the result depends on the number of significant digits in the original measurements. The measurement with the fewest significant digits determines the number of significant digits in the result. By following these rules, scientists and researchers can ensure that their measurements are accurate and reliable.
Practice Problems
- Multiply the following measurements: 2.45 g and 3.69 g. Ensure each answer contains the correct number of significant digits.
- Divide the following measurements: 4.69 g and 1.73 g. Ensure each answer contains the correct number of significant digits.
- Multiply the following measurements: 4.69 g and 0.373 g. Ensure each answer contains the correct number of significant digits.
- Divide the following measurements: 2.45 g and 0.173 g. Ensure each answer contains the correct number of significant digits.
Answers
- 2.45 g × 3.69 g = 9.07 g²
- 4.69 g ÷ 1.73 g = 2.70
- 4.69 g × 0.373 g = 1.75 g²
- 2.45 g ÷ 0.173 g = 14.2
Frequently Asked Questions (FAQs) on Significant Digits in Measurement Multiplication and Division =============================================================================================
Q: What is the rule for counting significant digits when measurements are multiplied together?
A: When measurements are multiplied together, the number of significant digits in the result is determined by the measurement with the fewest significant digits.
Q: Can you provide an example of multiplying measurements with the same number of significant digits?
A: Suppose we have two measurements: 4.69 g and 3.73 g. Both measurements have 3 significant digits. When we multiply these measurements together, the result will also have 3 significant digits.
4.69 g × 3.73 g = 17.45 g²
Q: What happens when we multiply measurements with different numbers of significant digits?
A: When we multiply measurements with different numbers of significant digits, the result will have the same number of significant digits as the measurement with the fewest significant digits.
Q: Can you provide an example of multiplying measurements with different numbers of significant digits?
A: Suppose we have two measurements: 4.69 g (3 significant digits) and 0.373 g (3 significant digits). When we multiply these measurements together, the result will have 3 significant digits.
4.69 g × 0.373 g = 1.75 g²
Q: What is the rule for counting significant digits when measurements are divided together?
A: When measurements are divided together, the number of significant digits in the result is determined by the measurement with the fewest significant digits.
Q: Can you provide an example of dividing measurements with the same number of significant digits?
A: Suppose we have two measurements: 4.69 g and 1.73 g. Both measurements have 3 significant digits. When we divide these measurements together, the result will also have 3 significant digits.
4.69 g ÷ 1.73 g = 2.70
Q: What happens when we divide measurements with different numbers of significant digits?
A: When we divide measurements with different numbers of significant digits, the result will have the same number of significant digits as the measurement with the fewest significant digits.
Q: Can you provide an example of dividing measurements with different numbers of significant digits?
A: Suppose we have two measurements: 4.69 g (3 significant digits) and 0.173 g (3 significant digits). When we divide these measurements together, the result will have 3 significant digits.
4.69 g ÷ 0.173 g = 27.0
Q: How do I determine the number of significant digits in a measurement?
A: To determine the number of significant digits in a measurement, look for the following:
- Non-zero digits: These are the digits that are known to be reliable.
- Zeros between non-zero digits: These zeros are also considered significant.
- Leading zeros: These zeros are not considered significant.
- Trailing zeros: These zeros are only considered significant if the measurement has a decimal point.
Q: Can you provide an example of determining the number of significant digits in a measurement?
A: Suppose we have a measurement: 0.00469 g. This measurement has 3 significant digits because the non-zero digits (4, 6, and 9) are known to be reliable.
Q: What is the importance of significant digits in measurement?
A: Significant digits are important in measurement because they determine the accuracy and reliability of the result. By following the rules for counting significant digits, scientists and researchers can ensure that their measurements are accurate and reliable.
Q: Can you provide a summary of the rules for counting significant digits in measurement multiplication and division?
A: Here is a summary of the rules:
- When measurements are multiplied together, the number of significant digits in the result is determined by the measurement with the fewest significant digits.
- When measurements are divided together, the number of significant digits in the result is determined by the measurement with the fewest significant digits.
- The measurement with the fewest significant digits determines the number of significant digits in the result.
- Leading zeros are not considered significant.
- Trailing zeros are only considered significant if the measurement has a decimal point.