Problem 3 - College Algebra with Modeling and Visualization

Question

Classify the number as one or more of the following: natural number, integer, rational number, or real number. 7.5 (Average number of gallons of water used each minute while taking a shower)

Step-by-Step Solution

Answer

7.5 is a rational and real number.

Step 1: Understand the categories

We need to classify the number 7.5 into one or more of these categories: natural numbers, integers, rational numbers, and real numbers. Let's understand what each category means:1. **Natural Numbers**: Natural numbers are positive integers starting from 1, 2, 3, etc.2. **Integers**: Integers are whole numbers and include negative numbers, zero, and positive numbers, such as ..., -3, -2, -1, 0, 1, 2, 3, etc.3. **Rational Numbers**: A rational number can be expressed as a ratio of two integers (i.e., fractions), such as \(\frac{a}{b}\) where \(b eq 0\).4. **Real Numbers**: Real numbers include all the rational and irrational numbers. They can be identified on the number line.

Step 2: Check each category

Now analyze 7.5 against each of these categories:1. **Natural Number**: 7.5 is not a natural number because natural numbers are whole numbers without fractions or decimals.2. **Integer**: 7.5 is not an integer because it is not a whole number; it has a decimal component.3. **Rational Number**: 7.5 is a rational number because it can be expressed as the fraction \(\frac{75}{10}\), which simplifies to \(\frac{15}{2}\).4. **Real Number**: 7.5 is a real number as it exists on the number line.

Key Concepts

Natural Numbers

Natural numbers are the simplest form of numbers that we first learn when we start counting. These numbers include all the positive whole numbers starting from 1 onwards. If you ever count apples, bicycles, or people, you're using natural numbers! They do not include zero, fractions, or decimals.

To summarize, natural numbers include:

• Positive integers
• Whole numbers starting from 1

It's clear from this definition that any number with a decimal or fractional component, like 7.5, does not fit in the category of natural numbers.

Integers

Integers expand the world of natural numbers by including zero and negative numbers as well. This means that integers consist of all whole numbers and do not include any decimal or fractional parts.

Here is what integers include:

• Positive whole numbers (1, 2, 3, ...)
• Zero (0)
• Negative whole numbers (-1, -2, -3, ...)

Once again, since 7.5 is not a whole number due to its decimal, it cannot be classified as an integer either.

Rational Numbers

Rational numbers open up the realm of possibilities beyond integers by including fractions and decimals that can be expressed as a fraction. A rational number is essentially any number that can be written as the ratio of two integers. For example, 7.5 can be expressed as \[ rac{75}{10} = rac{15}{2} \]which is a ratio of two integers.

This means that 7.5 is a rational number because:

• It has a fractional representation.
• It can be identified as \[ rac{a}{b} \]
• Where both the numerator and denominator are integers and the denominator is not zero.

Real Numbers

Real numbers are possibly the most inclusive set of numbers, encompassing everything from rational numbers to the mysterious realm of irrational numbers (which cannot be expressed as fractions). Real numbers can be represented on a number line, going from negative infinity to positive infinity.

Real numbers include:

• All integers
• All rational numbers (like 7.5)
• All irrational numbers (like \( \pi \) and \( \sqrt{2} \))

Since 7.5 is a number you can find on the number line as well as being rational, it easily fits into the classification of real numbers, making it both rational and real at the same time.