Problem 6 - 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. \(700,000,000,000\) (The federal 2008 bailout fund in dollars)

Step-by-Step Solution

Answer

The number 700,000,000,000 is a natural number, integer, rational number, and real number.

Step 1: Identifying Natural Numbers

Natural numbers are the set of positive integers starting from 1, such as 1, 2, 3, and so on. The number \(700,000,000,000\) is a positive whole number and is part of the set of natural numbers. Therefore, \(700,000,000,000\) is a natural number.

Step 2: Identifying Integers

Integers include all whole numbers, both positive and negative, as well as zero. Since \(700,000,000,000\) is a positive whole number, it clearly falls within the category of integers.

Step 3: Identifying Rational Numbers

Rational numbers are numbers that can be expressed as the quotient of two integers, where the denominator is not zero. The number \(700,000,000,000\) can be represented as \(\frac{700,000,000,000}{1}\), thereby categorizing it as a rational number.

Step 4: Identifying Real Numbers

Real numbers include all the numbers along the continuous number line, encompassing rational numbers, integers, and irrational numbers. Since \(700,000,000,000\) is both a rational and an integer, it is automatically a real number.

Key Concepts

Natural Numbers

When we think about counting the things around us, we use natural numbers. These are the numbers we learn as children, starting from 1 and going upwards: 1, 2, 3, and so forth.
Natural numbers are fundamentally the set of positive whole numbers we use in everyday life for counting and ordering objects.
They do not include zero, negatives, or fractions. Here are some essential characteristics of natural numbers:

• Always positive
• Do not include fractions or decimals
• Start from 1 and move up (1, 2, 3...)

A number like 700,000,000,000 easily qualifies as a natural number because it's a large positive number.
It fits right into this group, making it a perfect example of natural numbers.

Integers

Integers expand the idea of natural numbers by adding zero and negative whole numbers. It’s like taking the number line backwards and forwards without jumping off the track to decimals or fractions.
This means integers include:

• Positive numbers like 1, 2, 3...
• The number zero (0)
• Negative numbers like -1, -2, -3...

The integer set creates an unbroken line of whole numbers. Since 700,000,000,000 is a positive whole number, it comfortably fits in the category of integers.
It shares space with all the numbers you can count on your fingers, as well as every minus version of those.

Rational Numbers

Rational numbers take a broader view and include numbers that can be expressed as a ratio of two integers.
The key feature here is the fraction form, represented as \( \frac{a}{b} \), where both \(a\) and \(b\) are integers, and \(b eq 0\).

• They can be positive or negative.
• Include integers (e.g., \( \frac{5}{1} = 5 \))
• Include true fractions (e.g., \( \frac{3}{4} \))

The number 700,000,000,000 can easily be written as \( \frac{700,000,000,000}{1} \), making it a solid example of a rational number because it meets the fraction condition perfectly.
Thus, every integer and every natural number is, by default, a rational number because it can be expressed in this way.

Real Numbers

Real numbers are like clouds in the sky that cover all that is beneath. They include every conceivable number type we encounter in math.
These numbers create a full, continuous line on the number line, covering:

• Rational numbers (like integers and fractions)
• Irrational numbers (like \( \sqrt{2} \), \( \pi \))

Real numbers do not include complex numbers, but they encompass a wide variety of numerical forms comprising both rational and irrational numbers.
Since 700,000,000,000 falls under both rational numbers and integers, it is indeed a real number. Real numbers exemplify the rich variety of values covering all conceivable points along the numerical continuum.