To determine the best answer for the question, we need to analyze the definition provided.
The numbers in question are of the form [tex]\(\left\{\left.\frac{a}{b} \right\rvert\, a, b \in \mathbb{Z} , b \neq 0\right\}\)[/tex]. Let's break down what this means:
1. [tex]\(\frac{a}{b}\)[/tex]: This notation refers to a fraction, where [tex]\(a\)[/tex] is the numerator and [tex]\(b\)[/tex] is the denominator.
2. [tex]\(a, b \in \mathbb{Z}\)[/tex]: This means that both the numerator [tex]\(a\)[/tex] and the denominator [tex]\(b\)[/tex] are integers.
3. [tex]\(b \neq 0\)[/tex]: This stipulates that the denominator [tex]\(b\)[/tex] cannot be zero, as division by zero is undefined.
Given these three points, we can identify the type of numbers being described. They are fractions where both the numerator and the denominator are integers (with the denominator non-zero). These are known as rational numbers.
Rational numbers are commonly designated with the letter [tex]\(Q\)[/tex].
Let's evaluate the other options briefly to ensure they're not a fit:
- Natural numbers are non-negative integers, typically starting from 1 (or sometimes 0). They do not include fractions.
- Irrational numbers are numbers that cannot be expressed as a fraction of two integers. Their decimal expansions are non-repeating and non-terminating.
- Real numbers include both rational and irrational numbers, so it's a broader category than just rational numbers.
Therefore, the best answer is:
Rational numbers.
