Problem 15 - 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. $$4$$

Step-by-Step Solution

Answer

4 belongs to \\(\\mathbf{N}, \\mathbf{W}, \\mathbf{Z}, \\mathbf{Q}\\).

Step 1: Identify if 4 is a Natural Number

Natural numbers \(\mathbf{N}\) are the positive integers starting from 1. Since 4 is a positive integer greater than 0, it belongs to the set of natural numbers.

Step 2: Identify if 4 is a Whole Number

Whole numbers \(\mathbf{W}\) include all natural numbers along with 0. Since 4 is a natural number, it also belongs to the set of whole numbers.

Step 3: Identify if 4 is an Integer

Integers \(\mathbf{Z}\) include all whole numbers and their negative counterparts. Since 4 is a whole number, it is also an integer.

Step 4: Identify if 4 is a Rational Number

Rational numbers \(\mathbf{Q}\) can be expressed as a fraction \(\frac{a}{b}\) with integers \(a\) and \(beq 0\). The number 4 can be written as \(\frac{4}{1}\), so it is a rational number.

Step 5: Determine if 4 is an Irrational Number

Irrational numbers (I) cannot be expressed as a fraction. Since 4 can be expressed as the fraction \(\frac{4}{1}\), it is not an irrational number.

Key Concepts

Natural Numbers

Natural numbers are the most basic set of numbers in mathematics. They are the counting numbers you use every day. Imagine the numbers on your fingers: 1, 2, 3, 4, and so on. These are natural numbers. They start from 1 and go on to infinity.
Natural numbers don't include 0 or negative numbers. They do not include fractions or decimals either. So, if someone asks for a natural number, you're looking at the numbers you'd usually count physical objects with.

• Examples of natural numbers: 1, 5, 10, 100
• Not natural numbers: 0, -5, 3.5, \( \frac{1}{2} \)

Whole Numbers

Whole numbers are just like natural numbers but with an extra friend: the number zero. This set still includes all the natural numbers—think 0, 1, 2, 3, and so on to infinity.
Whole numbers extend natural numbers by simply adding this zero, which makes them more complete for certain mathematical operations. You can think of them as a step up from natural numbers.

• Examples of whole numbers: 0, 3, 15, 1000
• Not whole numbers: -3, 2.5, \( \frac{2}{3} \)

Integers

Integers broaden the concept of numbers a lot further. They include all whole numbers as well as their negative counterparts. If you can think of a number line, integers cover every point on that line which isn’t a fraction or decimal.
Imagine the number line with zeros in the middle, stretching infinitely in both the positive and negative directions.

• Examples of integers: -3, 0, 4, 100
• Not integers: 2.7, \( \frac{3}{4} \), -0.5

Rational Numbers

Rational numbers are a bit more flexible. If a number can be expressed as a fraction of two integers, it's a rational number. This set not only includes integers and whole numbers, but also numbers like 1.5 or 0.75 because they can be rewritten as fractions.
If you can take any integer as a numerator and a non-zero integer as a denominator, you'll have a rational number.

• Examples of rational numbers: \( \frac{1}{2} \), 5, -3.4
• Not rational numbers: \( \pi \), \( \sqrt{2} \)