Problem 70 - Chemistry The Central Science

Question

Is the use of significant figures in each of the following statements appropriate? (a) The 2005 circulation of National Geo graphic 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. (\boldsymbol{d} ) ~ Y o u ~ c a l c u l a t e ~ y o u r ~ g r a d e ~ p o i n t ~ a v e r a g e ~ t o ~ b e ~ 3.87562

Step-by-Step Solution

Answer

The use of significant figures in statements (a) and (b) is not appropriate, as they provide an unnecessarily precise count of circulation and population. It would be more suitable to round these numbers (e.g., 7.81 million and 5.30 million, respectively). For statement (c), the given percentage (0.621%) is appropriate as it may be relevant to have more precise data when dealing with percentages. However, in statement (d), the calculated grade point average should be rounded to two decimal points (3.88) based on common grading system practices.

Step 1: Statement (a): National Geographic circulation in 2005

The given circulation number is 7,812,564. This number implies that the count is precise down to the last reader. In this situation, however, there is no need for reporting such a high degree of precision. Circulations are usually given in whole numbers with a few leading digits being more significant. Appropriate significant figures: 7.81 million or 7,810,000.

Step 2: Statement (b): Population of Cook County, Illinois, on July 1, 2005

The given population number is 5,303,683. Similar to statement (a), the population count does not need to be this precise. Population numbers are often approximate and generally rounded up to thousands or tens of thousands. Appropriate significant figures: 5.30 million or 5,304,000.

Step 3: Statement (c): Percentage of the U.S. population with the surname Brown

The given percentage is 0.621%. While it may seem unnecessary to have three significant figures in this case, percentages are often reported with a decimal point. Given that the percentage deals with the count of people, being precise might be relevant. Appropriate significant figures: 0.621% (the original number is appropriate).

Step 4: Statement (d): Grade point average calculation

The given grade point average is calculated to be 3.87562. A grade point average must not have that many significant figures since most grading systems themselves do not possess that level of precision. The grade point averages are typically reported up to two decimal points. Appropriate significant figures: 3.88.

Key Concepts

Precision in Measurements

Precision in measurements is crucial when reporting data and results because it provides an understanding of how reliable and accurate the figures are. Significant figures play a vital role in representing the precision of a number. They indicate the certainty of measured quantities and help in maintaining consistency and accuracy in data representation.

In our daily life and scientific studies, different measurements require varying levels of precision. For example, population counts or social statistics, like those mentioned in statements (a) and (b) regarding the circulation and population, typically do not need excessive precision, as these numbers are subject to fluctuation.

On the other hand:

• For critical calculations in sciences or engineering, more significant digits may be essential to maintain precision.
• In different contexts, precision serves different purposes—reflecting accuracy in scientific processes or providing a ballpark figure in everyday applications.

Recognizing when and how much precision is needed is important since overly detailed figures can imply a false sense of accuracy.

Scientific Notation

Scientific notation is a practical system for expressing very large or very small numbers in a compact form. This notation is particularly useful in scientific, engineering, and mathematical fields where extreme values are common. It takes the form of:

\( a \times 10^n \)

Where \(a\) is a number from 1 to 10, and \(n\) is an integer. This form simplifies complex calculations and data representation by focusing on the significant digits.

Consider the significant numbers in examples from the exercise, such as populations and percentages. These quantities can be unwieldy to work with unless expressed using scientific notation. For instance, National Geographic's circulation could be presented as \( 7.81 \times 10^6 \), emphasizing significant figures and omitting unnecessary detail.

The primary benefits of scientific notation include:

• Simplified Arithmetic: Facilitates easier multiplication and division.
• Enhanced Accuracy: Preserves significant figures without extraneous zeros.

Using scientific notation allows you to handle numbers systematically and maintain precision where it's most needed.

Rounding Numbers

Rounding numbers is a mathematical practice used to make numbers simpler and more manageable. It reduces a number to fewer digits while retaining a value that is close to the original, aiding in comparison or mental arithmetic. This technique is vital in maintaining proper significant figures.

For instance:

• In statement (d) from the Original Solution, a GPA of 3.87562 should be rounded to 3.88. This reflects the standard level of precision used in educational settings.
• In everyday reporting, whole numbers are often sufficient for figures involving large populations or long-term results, as seen in statements (a) and (b).

To round a number, follow these steps:

• Identify which decimal place is to be rounded to (e.g., tenths, hundredths).
• Look at the number immediately after. If it's 5 or higher, round up. Otherwise, round down or leave it unchanged.

Rounding helps focus on meaningful data points and discards insignificant "noise." Knowing how to round correctly ensures that the information remains relevant and precise to the context it is used in.