Problem 22 - Prealgebra

Question

Name all of the sets of numbers to which each real number belongs. Let \(\mathbf{N}=\) natural numbers, \(\mathbf{W}=\) whole numbers, \(\mathbf{Z}=\) integers, \(\mathbf{Q}=\) rational numbers, and I = irrational numbers. $$0 . \overline{2}$$

Step-by-Step Solution

Answer

The number \(0.\overline{2}\) belongs to the set of rational numbers \(\mathbf{Q}\).

Step 1: Identify Decimal Representation

The number given is \(0.\overline{2}\), which represents a repeating decimal. The overline indicates that the digit '2' repeats indefinitely, so the number is equivalent to \(0.2222\ldots\).

Step 2: Classify as Rational or Irrational

A number is classified as rational if it can be expressed as the quotient of two integers with a non-zero denominator. Since a repeating decimal can be represented as a fraction, \(0.\overline{2}\) is a rational number.

Step 3: Check for Natural Numbers \(\mathbf{N}\)

The natural numbers \(\mathbf{N}\) include all positive integers starting from 1 (e.g., 1, 2, 3, ...). Since \(0.\overline{2}\) is a decimal and not a positive integer, it is not a natural number.

Step 4: Check for Whole Numbers \(\mathbf{W}\)

Whole numbers \(\mathbf{W}\) include all non-negative integers (e.g., 0, 1, 2, ...). Since \(0.\overline{2}\) is a decimal between 0 and 1, it is not a whole number.

Step 5: Check for Integers \(\mathbf{Z}\)

Integers \(\mathbf{Z}\) include all whole numbers and their negatives (e.g., -2, -1, 0, 1, 2, ...). \(0.\overline{2}\) is not an integer because it is not a complete unit; it is between 0 and 1.

Step 6: Finalize Number Sets Classification

Given the classifications above, \(0.\overline{2}\) fits only in the set of rational numbers \(\mathbf{Q}\), as it cannot be placed in \(\mathbf{N}\), \(\mathbf{W}\), or \(\mathbf{Z}\), and it is not irrational.

Key Concepts

Sets of Numbers

Numbers can be classified into several different sets, each with its own unique properties. Understanding these sets is key to determining the nature of any given number. Here are the most commonly discussed sets of numbers:

• Natural Numbers (]
• Whole Numbers (]
• Integers (]
• Rational Numbers (]
• Irrational Numbers (]

Knowing these sets helps in the identification and classification of numbers based on their characteristics.

Classification of Numbers

Classifying numbers involves determining which set or sets a number belongs to based on its properties. For the number \(0.\overline{2}\), we can follow a simple process:
First, identify the number's form: \(0.\overline{2}\) is a repeating decimal, indicating it is a rational number. Rational numbers are everything that can be written as fractions, like \(\frac{2}{9}\), with non-zero denominators.
Once we affirm it's rational, we proceed to check other classifications:

• It is not a natural number because it's not a positive integer starting from 1.
• It is not a whole number as it is not a non-negative integer.
• It is not an integer because integers don't have decimal or fractional parts.
• Since it is exactly representable by a fraction \(\frac{2}{9}\), it isn't irrational.

Understanding classification helps recognize if a number belongs exclusively to one set or multiple sets of numbers.

Real Numbers

The set of real numbers is an extensive classification encompassing both rational and irrational numbers. Real numbers include any value that can be represented on the number line, filling every possible gap without exception. This broad classification means they aren't limited to whole numbers or fractions but cover decimals extensively.
Real numbers include:

• Rational Numbers: As mentioned, these are numbers capable of being expressed as fractions \(\left(\frac{a}{b}\right)\), where both \(a\) and \(b\) are integers.
• Irrational Numbers: Which have non-repeating, non-terminating decimals, such as \(\pi\) and \(\sqrt{2}\).

A number like \(0.\overline{2}\), although not frequently framed as real by nature, still belongs to this classification due to its rationality. Any real number can inhabit multiple subcategories (like being both an integer and a rational number) or stay within one, depending on its properties. Thus, real numbers offer an all-encompassing way to file away every conceivable numerical figure.