Problem 16 - 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. Prices paid (in dollars and cents) for gasoline tank fill-ups

Step-by-Step Solution

Answer

Prices are best described by rational numbers.

Step 1: Define the measurement

The prices paid for gasoline tank fill-ups are typically given in dollars and cents. This means they can be fractions (for example, $15.75) where the cents represent parts of a whole dollar.

Step 2: Determine characteristics of the numbers

These prices need to allow for both whole dollars and fractions of a dollar, as prices can be in cents as well.

Step 3: Choose the most appropriate set of numbers

Since prices can include cents (fractions of a dollar), they aren't represented well by just natural numbers or integers. The appropriate set of numbers to represent values that include fractions is the set of rational numbers.

Key Concepts

Natural Numbers

Natural numbers are one of the first sets of numbers introduced to students when learning mathematics. They are the simple counting numbers like 1, 2, 3, and so on. Natural numbers are used for counting objects or entities that do not include any negative numbers or fractions.
One way to think about natural numbers is to consider them as the numbers you would use if you were counting objects that you cannot divide, such as apples or books. For example, you might say, "I have 3 apples," which is an exact whole number without any need for parts of a whole.

Natural numbers are represented by the symbol \( \mathbb{N} \). It's important to note that some sources include zero in the set of natural numbers, but traditionally, they begin with 1. When using natural numbers, always remember that they can't represent fractions of a whole. Hence, when dealing with monetary values that include cents, such as prices for gasoline, natural numbers aren't a suitable choice, since they cannot capture fractional parts like cents.

Integers

Integers extend the concept of natural numbers by including their negative counterparts and zero. Essentially, integers consist of positive whole numbers, their opposites, and zero. An example of integers would be -3, 0, and 4. They are represented by the symbol \( \mathbb{Z} \).
Integers are useful in scenarios where it is necessary to account for values below zero, such as in financial situations where one might encounter debts or overdrafts. However, integers are still incapable of representing fractions or decimal values.

When it comes to expressing quantities like prices, which might include parts of a whole, integers fall short. You cannot accurately describe \(3.50 or \)15.75 using integers alone because there is no way to include the cents or fractional parts without moving to a more inclusive number set. In the context of gasoline prices, which include decimal components, integers are not suitable because they strip away that crucial fractional value.

Measurement of Quantities

Measuring quantities often requires selecting the most appropriate set of numbers to describe the value accurately. In the case of prices, like those for gasoline fill-ups, precision is essential.
Generally, these prices are detailed to the nearest cent, such as $15.75, where the '.75' represents cents, a fraction of the whole dollar value. To accurately measure and communicate such quantities, rational numbers are the most appropriate set to use.

Rational numbers can express both whole and fractional parts because they are numbers that can be expressed as fractions \( \frac{a}{b} \), where \( a \) and \( b \) are integers and \( b eq 0 \). This flexibility makes rational numbers ideal for dealing with exact values in transactions and measurements like pricing.

• They include natural numbers and integers, thus covering whole numbers and their opposites.
• They also account for fractions and decimals, which makes them perfect for precise measurements.

So, when dealing with any measurement requiring granularity or detail beyond whole numbers, such as monetary values with cents, rational numbers provide the necessary level of detail and accuracy.