Problem 13 - College Algebra with Modeling and Visualization

Question

For the measured quantity, state the set of numbers that most appropriately describes it. Choose from the natural numbers, integers, and rational numbers. Explain your answer Gallons of gasoline

Step-by-Step Solution

Answer

Rational numbers are most appropriate because they include both whole and fractional amounts.

Step 1: Understanding the Measurement

The measurement in question is 'gallons of gasoline'. This refers to the amount of gasoline in terms of whole gallons, which are discrete quantities rather than continuous.

Step 2: Consideration of Natural Numbers

Natural numbers are the set of positive whole numbers starting from 1, 2, 3, etc. Since gasoline could be measured as whole gallons (1 gallon, 2 gallons), natural numbers could be a possible fit.

Step 3: Consideration of Integers

Integers include positive and negative whole numbers, as well as zero. However, negative quantities do not make sense in the context of measuring the amount of gasoline, so integers are not appropriate.

Step 4: Consideration of Rational Numbers

Rational numbers are numbers that can be expressed as the quotient of two integers, which includes fractions. Gallons of gasoline can be measured in whole and fractional amounts (e.g., 1.5 gallons), making rational numbers a suitable set for this measurement.

Step 5: Selecting the Most Appropriate Set

Since gallons of gasoline can involve fractional amounts, the most suitable set of numbers to describe the measurement is the rational numbers. This set accommodates both whole numbers and fractions.

Key Concepts

Natural Numbers

Natural numbers are the foundation of mathematics and consist of all positive whole numbers starting from 1. They are often used to count distinct and complete items like apples, pens, or books. Natural numbers are labeled as \(\mathbb{N}\) and the series is 1, 2, 3, and so on.

• They start from 1 and go to infinity.
• They do not include zero.
• They are not suitable for dealing with fractions or negative numbers.

Natural numbers can be used in contexts where only whole units are counted or required. For example, if gasoline was only ever sold in complete gallons, natural numbers would be the suitable set to use. But when you are considering partial gallons, such as 1.5 gallons, then natural numbers fall short as they do not accommodate fractions.

Integers

Integers broaden the landscape beyond natural numbers by including zero and negative numbers. They are represented as \(\mathbb{Z}\) and cover a wide range of values such as -3, -2, -1, 0, 1, 2, and 3.

• The set includes all whole numbers, both positive and negative.
• Zero is also an integer.
• They cannot represent fractional values or decimals.

Integers are useful in scenarios where you may need to express a concept of loss, debt, or deficiency, which is why negative numbers are part of the set. In the context of measuring gasoline, negative numbers and zero have no meaningful application since you cannot have a negative or zero quantity of gasoline to measure. This makes integers not suitable for the precise measurement of gas when fractional values are involved.

Fractional Measurements

Fractional measurements unlock the possibility of expressing parts of a whole, expanding the applicability of our calculations. Rational numbers, represented as \(\mathbb{Q}\), cover all numbers that can be expressed as a ratio or quotient of two integers, where the denominator is not zero.

• Includes whole numbers (which are fractions with denominators of one).
• Allows for fractions and decimal representations.
• Can precisely represent quantities like 1.5 or 3.75 gallons.

Fractional measurements are particularly important in contexts such as measuring liquids, where amounts are rarely whole. In the case of gasoline, a car's tank could need a precise amount like 5.75 gallons, or you might purchase 3.5 gallons. Rational numbers are perfect for these scenarios as they provide the flexibility necessary for representing both whole numbers and any fractional amount.