Problem 1 - 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. $$7$$

Step-by-Step Solution

Answer

The number 7 belongs to the sets: \(\mathbf{N}, \mathbf{W}, \mathbf{Z}, \mathbf{Q}\).

Step 1: Identify as Natural Number

The number \(7\) is a natural number because natural numbers are positive integers starting from 1. Thus, \(7 \in \mathbf{N}\).

Step 2: Identify as Whole Number

Whole numbers include all natural numbers and zero. Since \(7\) is a positive integer, \(7 \in \mathbf{W}\).

Step 3: Identify as Integer

Integers include all whole numbers and their negative counterparts. Since \(7\) is a positive whole number, \(7 \in \mathbf{Z}\).

Step 4: Identify as Rational Number

Rational numbers are numbers that can be expressed as a fraction, where the numerator and denominator are integers and the denominator is not zero. The number \(7\) can be expressed as \(\frac{7}{1}\), so \(7 \in \mathbf{Q}\).

Step 5: Identify as a Non-Irrational Number

Irrational numbers cannot be expressed as fractions of integers. Since \(7\) is a rational number, it cannot belong to the set of irrational numbers.

Key Concepts

Natural Numbers

Natural numbers are the numbers that we all start counting with when we first learn about numbers. They are the simplest kind of numbers and are typically used for counting objects. The main characteristics of natural numbers are:

• They start from 1 and go upward, like 1, 2, 3, 4, and so forth.
• They do not include zero, any fractions, or any negative numbers.

This means that if you have a number of objects that you can count, natural numbers are used. For instance, if you have 7 apples, you would use the natural number 7 to count them. Thus, 7 is indeed a natural number.

Whole Numbers

Whole numbers are an extension of natural numbers. They include everything that natural numbers do, plus the number zero. This set of numbers adds another layer of understanding to our concept of counting. Key features of whole numbers include:

• All natural numbers.
• The number zero.
• No fractions or negatives.

Basically, whole numbers are simply the natural numbers you know, with an extra zero. So 0, 1, 2, 3, 4, and so on are whole numbers. Again, since 7 is part of this counting process and not negative, it fits right into whole numbers.

Integers

Integers expand the scope of whole numbers. They are a broader category that includes not only whole numbers but also their negative counterparts. In essence, integers are:

• All whole numbers.
• All negative whole numbers (like -1, -2, -3, etc.).
• Zero is also an integer.

Integers do not include fractions or decimals. Instead, they offer a way to describe both the positive and negative directions on the number line. Since 7 is a whole number, it is also an integer—just without a negative sign. Integers are quite useful for mathematical operations that involve negative numbers.

Rational Numbers

Rational numbers get a little more complex. They include almost all numbers we use regularly—including fractions. A key definition of rational numbers is that they can be expressed as a fraction, where both the numerator and the denominator are integers. Crucial points about rational numbers include:

• Perfect examples include numbers like \frac{1}{2}, 3, or \frac{-7}{4}.
• Every integer is a rational number itself, since it can be written with a denominator of 1.
• The denominator must never be zero, as that makes the fraction undefined.

So for the number 7, you can express it as \( \frac{7}{1} \), which clearly shows it can be a rational number. This flexibility allows rational numbers to cover a lot of ground, including many common number scenarios.