Problem 169 - Introductory Chemistry Atoms First

Question

True or false? If any statement is false, rewrite it to make it true. (a) When multiplying or dividing a series of measured values, the number of significant figures in the answer is determined by the measured value having the fewest significant figures. (b) When adding or subtracting a series of measured values, the number of significant figures in the answer is determined by the measured value having the fewest significant figures.

Step-by-Step Solution

Answer

(a) True. (b) False. Corrected statement: "When adding or subtracting a series of measured values, the number of decimal places in the answer is determined by the measured value having the fewest decimal places."

Step 1: (a) Analyzing Statement a

Let's first analyze the statement regarding multiplying or dividing a series of measured values: "When multiplying or dividing a series of measured values, the number of significant figures in the answer is determined by the measured value having the fewest significant figures." This statement is true. When we multiply or divide measurements, the final answer should have the same number of significant figures as the measurement with the smallest number of significant figures. Therefore, this statement is correct.

Step 2: (b) Analyzing Statement b

Now we will analyze the statement regarding adding or subtracting a series of measured values: "When adding or subtracting a series of measured values, the number of significant figures in the answer is determined by the measured value having the fewest significant figures." This statement is false. When we add or subtract measurements, the final answer should have the same number of decimal places as the measurement with the smallest number of decimal places. The rule stated in the exercise is for multiplication and division, not for addition and subtraction.

Step 3: (b) Rewriting Statement b

Since Statement b is false, we will rewrite it correctly: "When adding or subtracting a series of measured values, the number of decimal places in the answer is determined by the measured value having the fewest decimal places." Now, both statements are correct: (a) When multiplying or dividing a series of measured values, the number of significant figures in the answer is determined by the measured value having the fewest significant figures. (b) When adding or subtracting a series of measured values, the number of decimal places in the answer is determined by the measured value having the fewest decimal places.

Key Concepts

Multiplication and Division Rules

When performing multiplication or division with numbers that have different significant figures, it is crucial to preserve precision correctly. Let's break it down to make it more digestible. The rule here is to use the quantity with the fewest significant figures to determine the number of significant figures in the final answer.
For example, if you are multiplying 5.12 (which has three significant figures) by 2.1 (which has two significant figures), your answer should be reported with two significant figures. Even though the mathematical result might appear as 10.752, it should be written as 11 to reflect the correct significant figures limitation.

• In multiplication, multiply as usual and then apply the significant figure rule.
• In division, divide as normal and ensure the answer has correct significant figures based on the same rule.

By following this principle, you are ensuring your result reflects the precision of your least precise measurement, maintaining scientific honesty and consistency in reporting.

Addition and Subtraction Rules

The process for adding or subtracting measurements involves a different rule for significant figures. Here, the number of decimal places in your result is determined by the measurement with the fewest decimal places.
This means you completely ignore the number of significant figures in each number when adding or subtracting. What matters is the decimal precision. For example, when adding 12.67 and 3.8, despite the differing decimal places, your result should be expressed as 16.5, aligning with one decimal place, because 3.8 has the fewest decimal places.

• Align decimal points at the start for easy comparison.
• The result cannot be more precise than the least precise measurement.

This rule ensures that your answers are not misleading by suggesting a precision that your data does not support, allowing for more reliable data interpretation.

Measurement Precision

Understanding measurement precision is key in scientific calculations. Precision refers to how exact a measurement is, which is often reflected in significant figures or decimal places.
The choice of significant figures or decimal places helps scientists and engineers communicate the reliability of their measurements and the degree of certainty in their calculations.

• Significant figures are used to convey the precision of a measured or calculated quantity.
• Decimal places indicate the exactness in cases of additions and subtractions.
• Precision levels differ according to measurement tools and methods employed.

Maintaining measurement precision during calculations is vital. It prevents exaggeration of the reliability of measured data, ensures consistency, and preserves the integrity of scientific results. Through this understanding, we respect the limitations and capabilities of our measurements, ultimately strengthening our findings.