Problem 16 - A Graphical Approach to Precalculus with Limits

Question

For each measured quantity, state the set of numbers that is most appropriate 10 describe it. Choose from the natural numbers, integers, and rational numbers. Prices paid (in dollars and cents) for gasoline tank fill-ups

Step-by-Step Solution

Answer

Rational numbers are most appropriate for describing prices paid for gasoline.

Step 1: Understanding the Context

When paying for gasoline, the amount you pay is often in dollars and cents. Understanding this context, we need to identify the nature of this quantity in terms of number sets.

Step 2: Identify the Measurement

Prices paid for gasoline can include cents, meaning these prices are typically not whole numbers. They often include fractional values that are expressed in decimal form.

Step 3: Define Rational Numbers

Rational numbers are numbers that can be expressed as a fraction or a quotient of two integers. Decimal numbers that terminate or repeat can also be considered rational numbers.

Step 4: Select the Most Appropriate Number Set

Since gasoline prices can be fractional (in dollars and cents), it is most appropriate to describe them using rational numbers, as they capture both the integer and decimal components.

Key Concepts

Natural Numbers

Natural numbers are the numbers we use for counting objects. They start from 1 and go on infinitely like this: 1, 2, 3, 4, and so on. Natural numbers can be used to answer questions like "How many apples are in the basket?" because they are always whole and positive. They do not include zero or any fractions or decimals.

• Natural numbers are a subset of the integers.
• They do not have any negative numbers.
• They are perfect for counting items.

Understanding natural numbers is essential because they form the foundation of more advanced mathematical concepts. However, since gasoline prices often include cents, natural numbers are not the right fit for expressing these amounts as they cannot represent fractions or decimals.

Integers

Integers extend beyond natural numbers to include zero and negative numbers. Examples of integers are -3, 0, 1, and 4. They form a number line where you can move in both directions from zero: to the right for positive numbers and to the left for negative numbers. This makes integers very helpful for representing gains and losses or increases and decreases.

Integers include:

• Positive whole numbers like 1, 2, 3.
• Negative whole numbers like -1, -2, -3.
• Zero.

While integers are great for expressing values in many situations, like temperatures above and below zero, they still can't capture the detail of decimals or fractions. Thus, they aren't the right choice for representing gasoline prices paid in dollars and cents, which often include a fraction of a dollar.

Rational Numbers

Rational numbers are the most inclusive among the three number sets when it comes to representing real-world quantities like money. A rational number can be expressed as the quotient of two integers, where the denominator is not zero. Any number that can be written as a fraction is a rational number. This means all integers and natural numbers are also rational numbers.

What's cool about rational numbers is:

• They include all fractions, such as 1/2, 3/4, or 7/8.
• They include all decimal numbers that either terminate (like 0.75) or repeat (like 0.333...).
• They are very useful for recording precise values, like prices.

In the context of paying for gasoline, rational numbers are perfect. They allow us to describe prices that are not whole numbers, accommodating both dollars and the fractional cents that often appear on receipts.