Problem 70 - Chemistry: The Central Science in SI Units, Global Edition

Question

Is the use of significant figures in each of the following statements appropriate? (a) The 2005 circulation of National Geographic was \(7,812,564 .\) (b) On July 1, 2005, the population of Cook County, Illinois, was \(5,303,683 .(\mathbf{c})\) In the United States, \(0.621 \%\) of the population has the surname Brown. (d) You calculate your grade point average to be \(3.87562 .\)

Step-by-Step Solution

Answer

For each statement: (a) The use of significant figures in the National Geographic circulation statement is appropriate since the circulation is a countable quantity. (b) The use of significant figures in the population of Cook County statement is appropriate as population should be reported with all significant figures for accuracy. (c) The use of significant figures in the percentage of Brown surnames statement is not appropriate; it would be better to round the percentage to one or two decimal places, e.g., \(0.62\%\) or \(0.621\%\). (d) The use of significant figures in the GPA calculation statement is not appropriate; it is recommended to round the value to one or two decimal places, e.g., \(3.88\) or \(3.9\).

Step 1: (a) National Geographic Circulation

In this statement, the given circulation number is \(7,812,564\). All non-zero digits (7, 8, 1, 2, 5, and 4) are considered significant figures. Since the circulation is a countable quantity, it is appropriate to use all the significant figures.

Step 2: (b) Population of Cook County, Illinois

The population of Cook County, Illinois, is given as \(5,303,683\). Similarly as in the first case, all non-zero digits (5, 3, 6, and 8) are considered significant figures. When dealing with population, we need to maintain accuracy, and it is typical to report populations with all their significant figures.

Step 3: (c) Percentage of Brown surnames in the United States

In this statement, \(0.621\%\) of the population has the surname Brown. Percentages are not countable quantities, and using the three significant figures as provided may not be appropriate. When dealing with percentages, it is usually better to round the number to one or two decimal places for easier comprehension and interpretation. So, it would be more appropriate to report the percentage as \(0.62\%\) or \(0.621\%\), depending on the desired precision.

Step 4: (d) Grade Point Average Calculation

The stated grade point average (GPA) is \(3.87562\). GPAs are usually calculated based on a scale, such as from 0 to 4, and this number has many decimal places, which suggests a very high level of precision. In general, GPAs should be reported with one or two decimal places only (e.g., \(3.88\) or \(3.9\)) for better interpretation and communication. Using the given value with five decimal places is not appropriate, and rounding to a fewer decimal places is recommended.

Key Concepts

Scientific Notation

When handling very large or very small numbers in mathematics or science, scientific notation becomes a powerful tool. It helps simplify numbers by expressing them with powers of ten. For instance, a large number like 7,812,564 can be written as \(7.812564 \times 10^6\) in scientific notation.
This notation makes it easier to read, comprehend, and communicate numerical values, especially where there are many zeros. It reduces the length of numbers and often emphasizes the significant figures, which are central in maintaining the precision of measurements.
Scientific notation is crucial for:

• Understanding and comparing very large or small numbers
  
• Maintaining consistency in how data is reported
  
• Simplifying calculations by easily adding, subtracting, multiplying, or dividing numbers expressed in powers of ten

Accuracy and Precision

Accuracy and precision are often mentioned together, but they refer to different qualities of measurement. **Accuracy** is about how close a measurement is to the true value, while **precision** refers to the consistency of repeated measurements.
In the context of the given exercise, using all significant figures when expressing population numbers reflects a high level of precision. This is important because:

• Precision allows for clearer and more consistent data reporting, which is vital in demography and statistical analysis
  
• Accuracy ensures that measurements or data reflect the true value or condition at that time, adding reliability
  

It is important to balance both precision and accuracy depending on the context you are working in. Overly precise numbers may not be necessary when approximate values are sufficient for a given purpose, such as when conveying a percentage of a common surname.

Decimal Places

The number of decimal places in a number reflects its level of precision in terms of its measurement or calculation. Decimal places are particularly crucial in contexts where minor differences can significantly alter the outcome or interpretation of data.
For example, in the calculation of GPA, using too many decimal places, like in the example of 3.87562, suggests an unnecessary level of precision. Generally, a GPA should be reported with fewer decimal places, like 3.88.
Here are some things to consider when deciding how many decimal places to use:

• Determining the level of precision needed based on the context and importance of the data
  
• Understanding that too many decimal places can make data more cumbersome to interpret
  
• Supporting easier communication and comprehension of results by rounding to an appropriate number of decimal places

By sticking to a reasonable number of decimal places, data becomes clearer and more user-friendly without sacrificing necessary accuracy.