Problem 109 - Chemistry Matter and Change

Question

When multiplying 602.4 \(\mathrm{m}\) by 3.72 \(\mathrm{m},\) which value determines the number of significant figures in the answer? Explain.

Step-by-Step Solution

Answer

3.72 determines the number of significant figures in the answer, which is three.

Step 1: Identify Significant Figures

First, determine the number of significant figures in each of the given numbers. For 602.4, there are four significant figures, and for 3.72, there are three significant figures.

Step 2: Select the Limiting Factor

The number of significant figures in the final result of a multiplication problem is determined by the number with the least significant figures among the numbers being multiplied. In this case, 3.72 has the least significant figures, which is three.

Step 3: Apply the Rule

When multiplying numbers, the answer should be rounded to have the same number of significant figures as the number in the operation with the fewest significant figures. Therefore, the final answer should have three significant figures.

Key Concepts

Understanding Multiplication Rules in Significant Figures

In mathematics and science, understanding how to handle significant figures during multiplication is crucial for producing accurate results. When multiplying two numbers, the result must be rounded to match the number with the least significant figures. This process ensures that the precision of the calculated answer reflects the least precise measurement involved in the multiplication.

• For example, when multiplying 602.4 (which has four significant figures) by 3.72 (which has three significant figures), the precision of the final result is based on 3.72, because it has fewer significant figures.
• This rule prevents overstating the precision of the answer. By recognizing the number with the fewest significant figures, we can accurately reflect the limitations of the measurement tools used in gathering the data.

If we multiply the two numbers, let's breakdown:
602.4 \(\times\) 3.72 equals 2,240.928. According to multiplication rules, the result should have three significant figures, ensuring the answer aligns with the precision limitations of the input values.

Grasping Measurement Precision

Measurement precision refers to the degree of exactness of a measurement. It quantifies how repeatable a measurement is if taken multiple times under identical conditions.
In analyzing measurements, it's crucial to recognize the inherent limitations in precision, which are expressed through significant figures.

• For instance, 602.4 has a higher level of precision compared to 3.72 due to having one more significant figure. In real-world applications, precision is often limited by the measuring instruments, which is why this concept is vital in scientific calculations.
• Using significant figures helps ensure that the results of such calculations are realistically attainable, rather than offering a false representation of accuracy beyond the measurement's capability.

This context highlights why identifying the number with fewer significant figures is essential for accurately portraying precision in any computation.

Exploring Scientific Notation and Its Application

Scientific notation is a method used to express very large or very small numbers efficiently. It involves representing numbers as a product of a decimal and a power of ten. This format not only simplifies calculations but also emphasizes significant figures.
When applying scientific notation in multiplication, the focus remains on maintaining precision according to the number of significant figures.

• In practice, converting numbers like 602.4 to scientific notation would look like: \(6.024 \times 10^2\), retaining its four significant figures.
• Similarly, 3.72 becomes \(3.72 \times 10^0\), maintaining three significant figures.

Multiplying these in scientific notation ensures that the product respects the same rule of rounding to the least number of significant figures, resulting in a product that's concise and precise. This technique is invaluable in scientific fields where base units and powers of ten are frequently used.