Problem 14 - 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. $$\frac{2}{5}$$

Step-by-Step Solution

Answer

The number \(\frac{2}{5}\) belongs to the set of rational numbers \(\mathbf{Q}\).

Step 1: Identify if it's a Natural Number

Natural numbers \(\mathbf{N}\) are the set \(\{1, 2, 3, ...\}\). Since \frac{2}{5}\ is not a positive whole number in this set, it is not a natural number.

Step 2: Check if it's a Whole Number

Whole numbers \(\mathbf{W}\) include all natural numbers and 0, i.e., \(\{0, 1, 2, 3, ...\}\). Since \frac{2}{5}\ is not a whole number or zero, it does not belong to whole numbers either.

Step 3: Determine if it's an Integer

Integers \(\mathbf{Z}\) include all whole numbers and their negative counterparts, i.e., \(\{..., -3, -2, -1, 0, 1, 2, 3, ...\}\). Since \frac{2}{5}\ is not a whole number or its negative, it is not an integer.

Step 4: Evaluate if it's a Rational Number

Rational numbers \(\mathbf{Q}\) are numbers that can be expressed as a fraction of two integers, with a non-zero denominator. Since \frac{2}{5}\ is already in fraction form with integers 2 and 5, it is a rational number.

Step 5: Confirm if it's an Irrational Number

Irrational numbers \(\mathbf{I}\) are those numbers that cannot be expressed as a simple fraction of two integers, such as \(\pi\) and \(\sqrt{2}\). Since \frac{2}{5}\ can be expressed as a fraction of integers, it is not an irrational number.

Key Concepts

Natural Numbers

Natural numbers are the foundational building blocks in mathematics. They are the simplest set of numbers used for counting and ordering.
Natural numbers start from 1 and go upwards like 1, 2, 3, and so on. These numbers do not include zero, negative numbers, or fractions.
Here are some key points about natural numbers:

• They are positive numbers.
• Natural numbers are used when counting discrete items like apples or students.
• The set of natural numbers is denoted by \( \mathbf{N} \).

Given these points, fractions like \( \frac{2}{5} \) do not qualify as natural numbers because they are not whole, positive numbers.

Whole Numbers

Whole numbers expand on natural numbers by adding a single element: zero. This makes it easy to perform subtraction operations within the set without leaving the set.
Whole numbers are comprised of the numbers 0, 1, 2, 3, and so forth.
Let's summarize the key aspects of whole numbers:

• The set begins with zero and continues with all natural numbers.
• Whole numbers are non-negative and do not include fractions, decimals, or negative numbers.
• This set is symbolically represented as \( \mathbf{W} \).

Since fractions like \( \frac{2}{5} \) are neither whole nor zero, they cannot be considered whole numbers.

Integers

Integers are a broader category, including both positive and negative numbers along with zero. They allow for all kinds of counting, including debts and credits.
Integers cover numbers like -3, -2, -1, 0, 1, 2, 3, and continue indefinitely in both directions.
Key features of integers include:

• The integer set, \( \mathbf{Z} \), comprises positive numbers, their negatives, and zero.
• They do not consist of fractions or decimals.
• Integers can be used in a variety of arithmetic operations.

Given that \( \frac{2}{5} \) is a fraction, it does not fit into the set of integers.

Rational Numbers

Rational numbers are any numbers that can be expressed as the quotient of two integers, where the denominator is not zero.
This includes simple numbers like 1 or 0 (which can be written as \( \frac{1}{1} \) or \( \frac{0}{1} \)), as well as fractions like \( \frac{2}{5} \).
Some important points about rational numbers:

• The set \( \mathbf{Q} \) includes both integers and fractions.
• Decimal numbers that only repeat after a while or stop are also rational because they can be turned into fractions.
• Rational numbers cover a wide range of values, from negatives to positives.

Thus, a fraction such as \( \frac{2}{5} \) fits perfectly into the category of rational numbers, making this the appropriate set for \( \frac{2}{5} \).