Classification of rational numbers and their subsets.

It looks like you've uploaded an image containing educational content about rational numbers, but there is no specific problem or question included in the image. The image provides a definition and explanation of rational numbers, including:

- Definition: A number that can be written as $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$.
- Examples of rational numbers: $\frac{3}{7}, -\frac{17}{19}, -3, 0, 4.33, 7.123123123...$
- Classification:
- Positive rational numbers (both numerator and denominator have same sign).
- Negative rational numbers (opposite signs).
- Equivalent rational numbers: $\frac{a}{b} = \frac{a \times m}{b \times m}$ for non-zero integer $m$.
- A hierarchy diagram showing how numbers are classified:
- Numbers → Rational & Irrational
- Rational → Fractions, Integers
- Integers → Whole Numbers → Natural Numbers
- Also includes Negatives and Zero

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✔ But — what's missing?

You mentioned "solve the problem", but there is no explicit question or task in the image provided.

To help you properly, I need to know:

👉 What is the specific problem you'd like solved? For example:
- Simplify a rational number?
- Find equivalent fractions?
- Determine if a number is rational?
- Classify a given number?
- Solve a word problem involving rational numbers?

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📌 Example Problem (for illustration):

Let’s say the hidden task was:

> "Is $0.123123123...$ a rational number? Justify."

#### ✔ Solution:
Yes, $0.123123123...$ is a rational number because it is a repeating decimal.

Any repeating decimal can be expressed as a fraction of two integers.

Let $x = 0.\overline{123}$

Then:
$$
1000x = 123.\overline{123}
$$
$$
x = 0.\overline{123}
$$
Subtract:
$$
1000x - x = 123.\overline{123} - 0.\overline{123} \Rightarrow 999x = 123
\Rightarrow x = \frac{123}{999}
$$

Simplify:
$$
\frac{123}{999} = \frac{41}{333}
$$

So, $0.\overline{123} = \frac{41}{333}$ → Rational number.

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🔍 Conclusion:

Please clarify what problem or question you want me to solve based on this image. Once you provide it, I’ll give you a detailed solution with explanation.

For now, here’s a summary of what the image teaches:

| Concept | Description |
|--------|-------------|
| Rational Number | Can be written as $\frac{p}{q}$, $q \neq 0$, $p,q \in \mathbb{Z}$ |
| Examples | $\frac{3}{7}, -3, 0, 4.33, 7.123123...$ |
| Positive Rational | Both numerator and denominator same sign |
| Negative Rational | Opposite signs |
| Equivalent Rational | Multiply numerator and denominator by same non-zero integer |
| Number Hierarchy | Rational ⊆ Real; Includes fractions, integers, whole numbers, natural numbers |

Let me know the actual problem!