Problem 114 - Chemistry Matter and Change

Question

Infer Which of these measurements was made with the most precise measuring device: \(8.1956 \mathrm{m}, 8.20 \mathrm{m},\) or 8.196 \(\mathrm{m} ?\) Explain your answer.

Step-by-Step Solution

Answer

The measurement 8.1956 m is the most precise, as it has the most decimal places.

Step 1: Understand Precision

Precision in measurement refers to the number of digits in a measurement, indicating the smallest unit by which the device can measure accurately. More digits imply more precision.

Step 2: Analyze the Measurements

Let's look at each measurement: 1. **8.1956 m** - This measurement is given to four decimal places. 2. **8.20 m** - This measurement is given to two decimal places. 3. **8.196 m** - This measurement is given to three decimal places.

Step 3: Compare the Precision

By comparing the decimal places: - **8.1956 m** is the most precise as it has the most decimal places (four). - **8.196 m** comes next with three decimal places. - **8.20 m** is the least precise with only two decimal places.

Step 4: Conclusion

The measurement **8.1956 m** is made with the most precise measuring device, as it provides the greatest number of significant digits, indicating smaller and more accurate measurements.

Key Concepts

Understanding Significant Digits

Significant digits are the digits in a measurement that carry meaningful information about its precision. For example, in the number 8.1956, all five digits are significant. This means each digit contributes to how detailed and precise the measurement is. On the other hand, in the number 8.20, there are only three significant digits, but they are crucial as well in indicating the level of precision. To determine how many digits are significant, consider:

• Non-zero digits are always significant.
• Any zeros between significant digits are also significant.
• Trailing zeros to the right of the decimal point are significant because they show precision.

Understanding significant digits is essential because they show the exactness of a measurement. A larger number of significant digits usually implies a more precise measurement.

Exploring Decimal Places

Decimal places in a measurement refer to the number of digits showing to the right of the decimal point. The more decimal places a measurement has, the finer the detail it provides. For instance:

• The measurement 8.1956 m has four decimal places. This suggests a very fine precision level.
• The measurement 8.20 m shows two decimal places, indicating a lesser degree of precision.

Decimal places are crucial because they help us understand the level of precision an instrument can provide. When two measurements are close, the one with more decimal places is generally the more precise because it uses smaller divisions in measurements.

Defining Accuracy in Measurement

Accuracy in measurement refers to how close a measured value is to the true or accepted value. Precision and accuracy are related but distinct concepts. A measurement can be precise, having many decimal places and significant digits, but not accurate if it is not close to the actual value. Consider the analogy of throwing darts:

• If all darts land close to each other but far from the bullseye, the throws are precise but not accurate.
• If darts land near the bullseye but spread apart, they are accurate but not precise.

In measurement, achieving both precision and accuracy is ideal. Precision helps ensure the same result is achieved every time, while accuracy ensures that the result is close to the true value. Together, they contribute to reliable and trustworthy measurements.