Problem 14 - A Graphical Approach to College Algebra

Question

For each measured quantity, state the set of numbers that is most appropriate to describe it. Choose from the natural numbers, integers, and rational numbers. Distances to nearby cities on road signs

Step-by-Step Solution

Answer

Natural numbers are most appropriate for distances on road signs.

Step 1: Understand the Question

The task is asking us which set of numbers is the most appropriate for describing the distances to nearby cities as shown on road signs.

Step 2: Consider Natural Numbers

Natural numbers are the set of positive integers starting from 1, 2, 3, etc. They are used for counting objects. Consider if distances can be counted as whole numbers without fractions or negative numbers.

Step 3: Consider Integers

Integers include natural numbers, zero, and negative numbers (-3, -2, -1, 0, 1, 2, 3, etc.). Reflect on whether distances to cities could potentially be negative or require zero.

Step 4: Consider Rational Numbers

Rational numbers include all numbers that can be expressed as a fraction or ratio of two integers, including integers and non-integer decimals or fractions (like 1.5, 2.75, etc.). Think about whether distances are often expressed in exact decimals.

Step 5: Determine the Appropriate Set

Distances on road signs are often rounded to whole numbers, thus natural numbers tend to be preferred. Both integers and rational numbers may not be necessary because distances aren't negative or typically in fractional form.

Key Concepts

Natural Numbers

Natural numbers are one of the most fundamental sets in mathematics. They consist of positive whole numbers starting from 1, extending infinitely upward: 1, 2, 3, and so on. Natural numbers do not include zero or any negative numbers. These numbers are primarily used for counting and ordering objects.

When it comes to measuring distances, especially such as those to nearby cities shown on road signs, natural numbers are often employed. This is because distances in this context are typically approximated as whole numbers, making them easy to read and understand. You wouldn't find half a city or -3 kilometers on such signs. Thus, when using natural numbers, you ensure clarity and simplicity in conveying distances.

Integers

Integers extend the scope of natural numbers by including zero and the negative equivalents of positive numbers, forming a continuous set:

• Negative integers: -3, -2, -1, and so on.
• Zero.
• Positive integers (natural numbers): 1, 2, 3, and so forth.

The set of integers is crucial for mathematical operations and solving equations, where both adding and subtracting numbers in full increments is common.

While integers include both positive and negative numbers, they are less practical for expressing distances on road signs. This is because distance is inherently a non-negative measure. You would not expect to see negative distances in this context, as it would be misleading and confusing. Therefore, while integers are significant in various mathematical scenarios, they are not as applicable for displaying city distances.

Rational Numbers

Rational numbers are a more comprehensive set that includes all numbers that can be articulated as a fraction, a quotient of two integers \(\frac{a}{b}\), where \(b eq 0\). For instance, \(\frac{1}{2}, 2.75, -3.4\) are rational numbers. They have the capability to represent fractions, decimals, and both non-integer and integer numbers, making them versatile in mathematics.

In the context of distances, rational numbers would allow for precise measurements, accommodating fractions and decimals. However, road signs don't generally display distances as fractions or decimals for simplicity and ease of understanding. Typically, they are rounded to the nearest whole number, aligning with the ease of use found in natural numbers rather than the preciseness of rational numbers. While rational numbers extend the utility for detailed measurements, they are less utilized in contexts like road signs where immediate clarity and straightforwardness are prioritized.