Problem 52 - Prealgebra

Question

Name all of the sets of numbers to which each real number belongs. Let \(N=\) natural numbers, \(\mathbf{W}=\) whole numbers, \(\mathbf{Z}=\) integers, \(\mathbf{Q}=\) rational numbers, and $\mathbf{I}=\text { irrational numbers. $$0 . \overline{4}$$

Step-by-Step Solution

Answer

The number \(0.\overline{4}\) belongs to the set of rational numbers \(\mathbf{Q}\).

Step 1: Understanding the Number

Recognize that \(0.\overline{4}\) refers to a repeating decimal, which means \(0.4444\ldots\) with the digit 4 repeating infinitely.

Step 2: Determine if Natural Number

Natural numbers \(N\) are positive integers starting from 1. Since \(0.\overline{4}\) is not a positive whole number, it does not belong to \(N\).

Step 3: Determine if Whole Number

Whole numbers \(W\) include all of the natural numbers plus zero. Since \(0.\overline{4}\) is not a whole number (it's not an integer), it does not belong to \(W\).

Step 4: Determine if Integer

Integers \(Z\) include all whole numbers and their negatives. Since \(0.\overline{4}\) is not an integer, it does not belong to \(Z\).

Step 5: Determine if Rational Number

Rational numbers \(Q\) can be expressed as a fraction \( \frac{a}{b} \) where \(a\) and \(b\) are integers and \(b eq 0\). Since \(0.\overline{4}\) is a repeating decimal, it can be expressed as \( \frac{4}{9} \). Thus, it belongs to \(Q\).

Step 6: Determine if Irrational Number

Irrational numbers \(I\) cannot be expressed as a fraction of two integers and have non-terminating, non-repeating decimals. Since \(0.\overline{4}\) is repeating, it does not belong to \(I\).

Key Concepts

Understanding Rational Numbers

Rational numbers are an essential part of our number system. They are numbers that can be expressed as a fraction \( \frac{a}{b} \), where \(a\) and \(b\) are integers and \(b eq 0\). Rational numbers include integers, fractions, and terminating or repeating decimals.

• An integer like 5 can be considered as \( \frac{5}{1} \), making it a rational number.
• A simple fraction such as \( \frac{3}{4} \) is clearly rational.
• A decimal like 0.75, which terminates, also qualifies as a rational number since it equals \( \frac{3}{4} \).
• Repeating decimals, such as 0.333... (which equals \( \frac{1}{3} \)), are rational numbers since they can be converted into fractions.

Understanding that both terminating and repeating decimals fall into this category is crucial. This means that if you ever encounter a number that can be expressed as a fraction, it is guaranteed to be rational.

Exploring Number Sets

In mathematics, number sets categorize different types of numbers with shared properties. These help us understand and work with numbers more effectively.

• **Natural Numbers (\(N\))**: These are the simplest set, starting from 1 and increasing incrementally by 1 (i.e., 1, 2, 3, ...).
• **Whole Numbers (\(W\))**: This set includes all natural numbers and also the number 0 (i.e., 0, 1, 2, 3, ...).
• **Integers (\(Z\))**: Integers include all whole numbers and their negative counterparts (i.e., ..., -3, -2, -1, 0, 1, 2, 3, ...).
• **Rational Numbers (\(Q\))**: As previously explained, these numbers can be represented as a fraction.
• **Irrational Numbers (\(I\))**: Numbers that cannot be expressed as a simple fraction fall into this set. They have non-repeating, non-terminating decimals, like \(\pi\) and \(\sqrt{2}\).

Each set builds upon the one before it, creating a hierarchy of number types used in mathematics.

Decoding Repeating Decimals

Repeating decimals are fascinating as they show another aspect of rational numbers. These decimals never end but follow a repeating pattern. For example, \(0.\overline{4}\), or 0.4444..., has the digit 4 repeating endlessly.
To express a repeating decimal as a fraction, we can use a simple algebraic trick:1. Let \(x = 0.\overline{4}\).
2. Multiply both sides of the equation by 10 to shift the decimal point. So, \(10x = 4.\overline{4}\).
3. Subtract the original equation from this new equation: \(10x - x = 4.\overline{4} - 0.\overline{4}\).
4. This simplifies to \(9x = 4\), leading to \(x = \frac{4}{9}\).
Repeating decimals like \(0.\overline{4}\) are rational because they can be expressed as fractions. Understanding this conversion helps demystify the nature of repeating decimals and their role in the wider set of rational numbers.