Diagram of the Real Number System with classification of numbers and examples.

Problem Overview:

The task involves classifying numbers into the appropriate subsets of the Real Number System. The Real Number System is organized as follows:

1. Real Numbers (ℝ)
- Rational Numbers (ℚ): Numbers that can be expressed as a ratio of two integers, \( \frac{p}{q} \), where \( q \neq 0 \).
- Integers (ℤ): Whole numbers and their negatives.
- Whole Numbers (ℕ₀): Non-negative integers (0, 1, 2, ...).
- Natural Numbers (ℕ): Positive integers (1, 2, 3, ...).
- Irrational Numbers: Numbers that cannot be expressed as a ratio of two integers and have non-repeating, non-terminating decimal expansions.

The goal is to classify each given number into its correct subset(s) of the Real Number System.

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Given Numbers:

1. \( 30 \)
2. \( -11 \)
3. \( \frac{5}{7} \)
4. \( \sqrt{21} \)
5. \( 3.0 \)
6. \( -\pi \)
7. \( \frac{7}{3} \)
8. \( \pi \)
9. \( 3.\overline{3} \)

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Solution:

#### Step 1: Classify Each Number
We will analyze each number and determine which subset(s) it belongs to.

1. \( 30 \):
- Classification: Natural Number, Whole Number, Integer, Rational Number, Real Number.
- Reason: \( 30 \) is a positive integer, so it fits all these categories.

2. \( -11 \):
- Classification: Integer, Rational Number, Real Number.
- Reason: \( -11 \) is a negative integer, so it is not a natural or whole number but is rational and real.

3. \( \frac{5}{7} \):
- Classification: Rational Number, Real Number.
- Reason: \( \frac{5}{7} \) is a fraction, so it is rational and real but not an integer, whole number, or natural number.

4. \( \sqrt{21} \):
- Classification: Irrational Number, Real Number.
- Reason: \( \sqrt{21} \) is not a perfect square, so it cannot be expressed as a fraction and has a non-repeating, non-terminating decimal expansion. Thus, it is irrational and real.

5. \( 3.0 \):
- Classification: Natural Number, Whole Number, Integer, Rational Number, Real Number.
- Reason: \( 3.0 \) is equivalent to \( 3 \), which is a positive integer, so it fits all these categories.

6. \( -\pi \):
- Classification: Irrational Number, Real Number.
- Reason: \( \pi \) is an irrational number, and multiplying it by \(-1\) does not change its irrationality. Thus, \( -\pi \) is irrational and real.

7. \( \frac{7}{3} \):
- Classification: Rational Number, Real Number.
- Reason: \( \frac{7}{3} \) is a fraction, so it is rational and real but not an integer, whole number, or natural number.

8. \( \pi \):
- Classification: Irrational Number, Real Number.
- Reason: \( \pi \) is a well-known irrational number with a non-repeating, non-terminating decimal expansion. Thus, it is irrational and real.

9. \( 3.\overline{3} \):
- Classification: Rational Number, Real Number.
- Reason: \( 3.\overline{3} \) represents \( 3.333\ldots \), which is equivalent to \( \frac{10}{3} \). Since it can be expressed as a fraction, it is rational and real.

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#### Step 2: Summarize the Results
Here is the classification for each number:

1. \( 30 \): Natural Number, Whole Number, Integer, Rational Number, Real Number.
2. \( -11 \): Integer, Rational Number, Real Number.
3. \( \frac{5}{7} \): Rational Number, Real Number.
4. \( \sqrt{21} \): Irrational Number, Real Number.
5. \( 3.0 \): Natural Number, Whole Number, Integer, Rational Number, Real Number.
6. \( -\pi \): Irrational Number, Real Number.
7. \( \frac{7}{3} \): Rational Number, Real Number.
8. \( \pi \): Irrational Number, Real Number.
9. \( 3.\overline{3} \): Rational Number, Real Number.

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Final Answer:

\[
\boxed{
\begin{array}{ll}
1. & \text{Natural Number, Whole Number, Integer, Rational Number, Real Number} \\
2. & \text{Integer, Rational Number, Real Number} \\
3. & \text{Rational Number, Real Number} \\
4. & \text{Irrational Number, Real Number} \\
5. & \text{Natural Number, Whole Number, Integer, Rational Number, Real Number} \\
6. & \text{Irrational Number, Real Number} \\
7. & \text{Rational Number, Real Number} \\
8. & \text{Irrational Number, Real Number} \\
9. & \text{Rational Number, Real Number} \\
\end{array}
}
\]