Problem 5 - 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. \(19,900,037,000,000\) (The federal debt in dollars at the end of January 2017 )

Step-by-Step Solution

Answer

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

Step 1: Identify if the number is a Natural Number

A natural number is a positive whole number. Since \(19,900,037,000,000\) is a positive whole number, it can be classified as a natural number.

Step 2: Identify if the number is an Integer

An integer includes all whole numbers, both positive and negative, as well as zero. Since \(19,900,037,000,000\) is a positive whole number, it qualifies as an integer.

Step 3: Identify if the number is a Rational Number

A rational number is any number that can be expressed as a fraction \(\frac{a}{b}\), where \(a\) and \(b\) are integers and \(b eq 0\). The number \(19,900,037,000,000\) can be expressed as \(\frac{19,900,037,000,000}{1}\), which makes it a rational number.

Step 4: Identify if the number is a Real Number

Real numbers include all numbers on the number line, encompassing natural numbers, integers, rational numbers, and irrational numbers. Since \(19,900,037,000,000\) is a rational number, it is also a real number.

Key Concepts

Natural Numbers

Natural numbers are the simplest kind of numbers we use in day-to-day life. They are positive whole numbers that start from 1 and increase one by one without end. Think of these like the numbers you use to count objects, such as 1, 2, 3, and so on.
Here are some important characteristics of natural numbers:

• Natural numbers are always positive.
• They do not include 0.
• No fractions or decimals are involved, just whole numbers.

Since the number we are considering, \(19,900,037,000,000\), is a positive whole number, it is classified as a natural number. Remember, whenever you see whole numbers that are greater than zero, you'll want to think about natural numbers.

Integers

Integers expand upon natural numbers by including zero and negative numbers as well. They form a larger set that includes:

• Positive whole numbers (just like natural numbers)
• Zero
• Negative whole numbers

Integers are helpful in many situations, like when dealing with temperatures or heights above and below sea level.
For example, -5, 0, and 21 are all integers. Because \(19,900,037,000,000\) is a positive whole number, it naturally fits into the category of integers as well.
When thinking of integers, keep in mind they have no fractional or decimal parts and can extend infinitely in both positive and negative directions.

Rational Numbers

Rational numbers are a broad category, which include integers, and refer to any number that can be represented as a fraction \(\frac{a}{b}\) where \(a\) and \(b\) are integers and \(b eq 0\). This means:

• They can be whole numbers, like 5 or -3 (since they can be expressed as \(\frac{5}{1}\) or \(\frac{-3}{1}\)).
• Fractions like \(\frac{1}{2}\) or \(\frac{-7}{4}\) are rational because they can still be represented by integers.
• Decimals that end or repeat, like 0.75 or 6.333..., also qualify as rational numbers.

Since \(19,900,037,000,000\) can be written as a fraction \(\frac{19,900,037,000,000}{1}\), it's confirmed to be a rational number. Rational numbers give us the flexibility to describe values more precisely.

Real Numbers

Real numbers are a huge category that includes all possible numbers you could find on a number line. They encompass:

• Natural numbers
• Integers
• Rational numbers
• Irrational numbers (like \(\pi\) and \(\sqrt{2}\), which cannot be written as exact fractions)

Essentially, any number you've dealt with in everyday life is a real number. They form the complete spectrum of numbers used in arithmetic and calculus.
Since \(19,900,037,000,000\) is a rational number (which means it has a place on the number line), it is also a real number. When thinking about real numbers, imagine a continuous line where every possible number has a point.