Problem 5 - A Graphical Approach to College Algebra

Question

Classify each number as one or more of the following: natural number, integer, rational number, or real number. \(16,351,000,000,000\) (The federal debt in dollars in January 2013 )

Step-by-Step Solution

Answer

\(16,351,000,000,000\) is a natural number, integer, rational number, and real number.

Step 1: Identify the Number Type

The given number is \(16,351,000,000,000\). It is a whole number and can be written without a fraction or decimal, so it fits the category of an integer.

Step 2: Check for Natural Number Status

Natural numbers are positive integers that begin from 1, 2, 3, and so on. Since \(16,351,000,000,000\) is a positive integer, it is also a natural number.

Step 3: Determine if the Number is Rational

A rational number is any number that can be expressed as a fraction \(\frac{a}{b}\), where \(a\) and \(b\) are integers and \(b eq 0\). Since \(16,351,000,000,000\) can be expressed as \(\frac{16,351,000,000,000}{1}\), it is a rational number.

Step 4: Confirm it is a Real Number

Real numbers include all the rational numbers, including integers and natural numbers, as well as irrational numbers. Since \(16,351,000,000,000\) is a rational number, it is also a real number.

Key Concepts

Natural Number

Natural numbers are the basic counting numbers that everyone starts learning early on. They are like the first friends you meet when you start counting things around you. These numbers are positive and don't include any zero or negative numbers. An easy way to remember natural numbers is to think of them as numbers you would use to count apples, books, or toys—starting from 1 and going upward.

• Start from 1 and go up: 1, 2, 3, 4,...
• No zero or negative numbers involved
• Also known as counting numbers

In the context of the original exercise, the number 16,351,000,000,000 is considered a natural number since it is a positive whole number that you can easily fit into your counting. This makes it a part of the natural numbers set.

Integer

Integers extend the concept of natural numbers by including zero and negative numbers. Think of integers as a broader family that includes all your natural number friends and their opposites.

• Includes positive numbers (e.g., 1, 2, 3,...)
• Includes zero (0)
• Includes negative numbers (e.g., -1, -2, -3,...)

So, the set of integers looks like: ..., -3, -2, -1, 0, 1, 2, 3, .... This means that 16,351,000,000,000 is an integer because it is a whole number without fractions or decimals involved, and it fits comfortably into the line of integers, right beside other positive numbers.

Rational Number

Rational numbers are numbers that can be written as a fraction. This means you can express them in the form \( \frac{a}{b} \), where both \(a\) and \(b\) are integers, and \(b\) is not zero.

• Can be positive or negative
• Can be a simple fraction, a whole number, or even a terminating or repeating decimal
• Examples include: \( \frac{1}{2} \), 0.75, 7, -3

Even large numbers like 16,351,000,000,000 qualify as rational because you can write them as \( \frac{16,351,000,000,000}{1} \). So as long as you can express it in fractional form without \(b\) being zero, you've got yourself a rational number. It's like having the reassurance that every whole number can slip into the rational set.

Real Number

Real numbers is the biggest umbrella category that encompasses all the numbers we deal with in everyday life. This category includes both rational numbers, like integers and fractions, and irrational numbers, which cannot be neatly expressed as fractions.

• Includes both rational numbers (1, 2.5, \(\frac{3}{4}\))
• Also includes irrational numbers (\(\pi\), \(\sqrt{2}\))
• They can be positive, negative, or zero

Essentially, any number you can find on the number line is a real number. Since 16,351,000,000,000 is a rational number, it automatically becomes a part of the real number family. Whether you're dealing with very big numbers or numbers with endless decimal expansions, they all find a home here among real numbers.