Problem 13 - 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. $$\frac{1}{2}$$

Step-by-Step Solution

Answer

\( \frac{1}{2} \) belongs to the set of rational numbers \( \mathbf{Q} \).

Step 1: Identifying Type of Number

The number \( \frac{1}{2} \) is a fraction, which is a ratio of two integers (1 divided by 2).

Step 2: Determine Rationality

Since \( \frac{1}{2} \) can be expressed as a fraction of two integers, it is classified as a rational number, denoted \( \mathbf{Q} \).

Step 3: Check for Other Number Sets

Since \( \frac{1}{2} \) is not a whole number, natural number, or integer, it does not belong to the sets \( \mathbf{N} \), \( \mathbf{W} \), or \( \mathbf{Z} \). Additionally, it is not an irrational number, so it doesn't belong to set \( I \).

Key Concepts

Real Numbers

The set of real numbers is perhaps the most inclusive number set used in mathematics. Real numbers encompass all possible numbers you could plot on a number line. This includes both rational and irrational numbers.
Real numbers are denoted by the symbol \( \mathbb{R} \). They include:

• Rational Numbers: which are numbers that can be expressed as the ratio of two integers.
• Irrational Numbers: which cannot be expressed as a simple fraction - their decimal form goes on forever without repeating.

Whether a number is a whole number, a fraction, or a repeating or non-repeating decimal, if it can be placed on a number line, it’s a real number. It's important to understand that while this set is quite extensive, it does not include complex numbers, which incorporate imaginary numbers.

Rational Numbers

Rational numbers are a subset of real numbers and encompass any number that can be expressed as a fraction \( \frac{a}{b} \) where \( a \) and \( b \) are integers, and \( b eq 0 \). This set is denoted \( \mathbb{Q} \). What makes a number rational is its ability to be written in this fractional form.
There are a few characteristics that define rational numbers:

• They can be positive or negative.
• They can be expressed in decimal form as either terminating or repeating.
• Examples include \( \frac{1}{2} \), \(-5\), and \(0.75\).

For instance, the number \( \frac{1}{2} \) mentioned in the exercise is rational because it is the ratio of two integers (1 and 2). Rational numbers make up a significant portion of the number line, filling in the spaces between whole and natural numbers.

Integers

Integers are the number set denoted by \( \mathbb{Z} \) and include all whole numbers, both positive and negative, as well as zero. This set does not include fractions or decimals that do not result in whole numbers.
Key attributes of integers include:

• They extend infinitely in both directions on the number line.
• They include ...,-3, -2, -1, 0, 1, 2, 3,... and so on.
• While integers include all natural numbers (\( \mathbb{N} \)), not all rational numbers are integers unless they can be written as fractions where the denominator is 1.

So, while \( \frac{1}{2} \) is a rational number, it is not an integer, because you cannot express it as a whole number. Understanding integers helps form the foundation of more complex number sets, and they are essential in basic arithmetic and algebra.