Problem 23 - Intermediate Algebra

Question

Describe the set of rational numbers using set-builder notation.

Step-by-Step Solution

Answer

The set of rational numbers is \( \{ \frac{a}{b} \mid a, b \in \mathbb{Z}, b \neq 0 \} \).

Step 1: Understanding Rational Numbers

Rational numbers are numbers that can be expressed as the quotient or fraction \( \frac{a}{b} \) of two integers, where the numerator \( a \) is an integer and the denominator \( b \) is a non-zero integer.

Step 2: Introducing Set-Builder Notation

Set-builder notation uses a common structure like \( \{ x \mid \text{property of } x \} \) to describe a set in terms of the properties its members must satisfy.

Step 3: Defining Conditions for Rational Numbers

For rational numbers, we need to express the condition that each number \( x \) belongs to the set if and only if \( x = \frac{a}{b} \) where \( a \) and \( b \) are integers, and \( b eq 0 \).

Step 4: Writing Set-Builder Notation

We translate the conditions into set-builder notation: \( \mathbb{Q} = \left\{ \frac{a}{b} \mid a, b \in \mathbb{Z}, b eq 0 \right\} \). Here, \( \mathbb{Z} \) denotes the set of all integers.

Key Concepts

Rational Numbers

Rational numbers are a special group of numbers that can be expressed as a fraction of two integers. This means that any rational number can be represented in the form \( \frac{a}{b} \), where \( a \) and \( b \) are both integers, and importantly, \( b \) cannot be zero.
The reason \( b \) must be non-zero is that division by zero is undefined in mathematics, leading to nonsensical results.
Here's a quick breakdown of rational numbers:

• **Integers** like -2, 0, and 5 are also rational numbers, as they can be expressed as \( \frac{-2}{1}, \frac{0}{1}, \frac{5}{1} \).
• **Fractions** like \( \frac{3}{4} \) or \( \frac{-7}{2} \) are rational because they meet the requirement of being a fraction of two integers.
• **Decimals** that terminate or repeat are also considered rational, such as 0.75 (which is \( \frac{3}{4} \)) and 0.666... (which is \( \frac{2}{3} \)).

Thus, whenever you are dealing with rational numbers, remember that they are versatile and appear in various forms throughout mathematics.

Integers

Integers are a set of numbers composed of whole numbers and their negatives, including zero. They are represented by the set \( \mathbb{Z} \). This set is fundamental when working with rational numbers, as both the numerator (\( a \)) and denominator (\( b \)) in rational numbers come from integers. Let's take a closer look:

• **Whole Numbers**: These are numbers like 0, 1, 2, 3, and so on, without any fractional or decimal parts.
• **Negative Numbers**: Integers also include the negatives of whole numbers, like -1, -2, -3, etc.
• **Zero**: Zero is included as it's neither positive nor negative, making it unique among integers.

Knowing that integers form the basis for rational numbers, they are integral to many areas of math. They are especially pivotal when expressing numerical relationships, setting up equations, and performing arithmetic operations.

Properties of Numbers

Understanding the properties of numbers is crucial for working with set-builder notation and identifying rational numbers. These properties include commutativity, associativity, distributivity, identity, and inverse, which are often applied to both integers and rational numbers.- **Commutativity**: This property states that numbers can be added or multiplied in any order without changing the result, for example, \( a + b = b + a \).- **Associativity**: According to this property, the grouping of numbers does not affect their sum or product, like \( (a + b) + c = a + (b + c) \).- **Distributivity**: This property links addition and multiplication, such as \( a(b + c) = ab + ac \).- **Identity**: The identity property shows that with addition, any number plus zero remains the same (\( a + 0 = a \)), and with multiplication, any number times one remains unchanged (\( a\times 1 = a \)).- **Inverse**: This property indicates that every number has an opposite that results in an identity element, like \( a + (-a) = 0 \) for addition.These properties help in understanding and manipulating numbers within mathematical expressions. For rational numbers, these properties guide how they interact in equations and logical setups.