Problem 7 - A Graphical Approach to Precalculus with Limits

Question

Classify each number as one or more of the following: natural number, integer, rational number, or real number. \(-24\) (The change in the number of football fields of area of land loss in Louisiana each day)

Step-by-Step Solution

Answer

\(-24\) is an integer, a rational number, and a real number.

Step 1: Identify the number

The number in question is \(-24\). This is a negative integer.

Step 2: Determine natural number status

Natural numbers are all positive counting numbers starting from 1, such as 1, 2, 3, etc. Since \(-24\) is negative, it is not a natural number.

Step 3: Determine integer status

Integers are all whole numbers, positive and negative, including zero. Since \(-24\) is a whole number and negative, it is indeed an integer.

Step 4: Determine rational number status

A rational number is any number that can be expressed as a fraction \(\frac{p}{q}\) where \(p\) and \(q\) are integers, and \(qeq0\). The number \(-24\) can be written as \(-\frac{24}{1}\), which satisfies the conditions for being a rational number.

Step 5: Determine real number status

Real numbers include all numbers on the number line, encompassing both rational and irrational numbers. Since \(-24\) is a rational number, it is also a real number.

Key Concepts

Natural Numbers

Natural numbers are the set of positive numbers that we typically use for counting. You can think of them as the numbers you would count on your fingers, starting from 1. They include numbers like 1, 2, 3, and continue infinitely upwards.

Natural numbers do not include negative numbers, zero, or any fractions. They are strictly the building blocks of the counting numbers that we use in everyday life. Therefore, any number less than 1, including zero and negative numbers like \(-24\), cannot be categorized as natural numbers.

Integers

Integers are a broader category than natural numbers. They include all whole numbers, both positive and negative, as well as zero.

Here’s a quick overview of integers:

• Positive integers: 1, 2, 3, ...
• Negative integers: -1, -2, -3, ...
• Zero: 0

Because integers encompass negative whole numbers, \(-24\) is an integer due to its completeness and lack of fractional component. Any whole negative or positive number, along with zero, comfortably fits into the integer category.

Rational Numbers

Rational numbers are numbers that can be expressed as the quotient or fraction of two integers, where the denominator is not zero. \( rac{p}{q} \) is the general form representing rational numbers, where both \( p \) and \( q \) are integers.

For example, the number \(-24\) can be written as \(-\frac{24}{1}\), which fits the definition, confirming that \(-24\) is indeed a rational number. Since rational numbers include integers, fractions like \( rac{1}{2}, -rac{3}{4}, \) and even repeating decimals, they represent a vast set of numbers.

Real Numbers

Real numbers encompass all possible numbers that you can find on the number line, including both rational and irrational numbers. This vast umbrella covers almost any number you will encounter in basic math.

Real numbers include:

• Real rational numbers such as integers and fractions.
• Real irrational numbers like \( \pi \) and \( \sqrt{2} \), which cannot be expressed as precise fractions.

Since \(-24\) is a rational number, it automatically makes it a part of the real number system. In essence, if a number can be plotted on the number line, it is a real number.