Number system classification worksheet for identifying number types.

To solve the problem, we need to classify each number into the appropriate sets: Natural Numbers, Whole Numbers, Integers, Rational Numbers, Irrational Numbers, and Real Numbers. Let's go through each number step by step.

Definitions:

1. Natural Numbers (ℕ): Positive integers starting from 1 (e.g., 1, 2, 3, ...).
2. Whole Numbers: Non-negative integers starting from 0 (e.g., 0, 1, 2, 3, ...).
3. Integers (ℤ): All whole numbers and their negatives (e.g., ..., -3, -2, -1, 0, 1, 2, 3, ...).
4. Rational Numbers (ℚ): Numbers that can be expressed as a ratio of two integers \( \frac{p}{q} \) where \( q \neq 0 \). This includes all integers, fractions, and terminating or repeating decimals.
5. Irrational Numbers: Numbers that cannot be expressed as a ratio of two integers. They are non-terminating, non-repeating decimals (e.g., \( \sqrt{2}, \pi \)).
6. Real Numbers (ℝ): All rational and irrational numbers combined.

Classification of Each Number:

#### 1. \( -\sqrt{17} \)
- Natural Numbers: No (negative and irrational).
- Whole Numbers: No (negative and irrational).
- Integers: No (irrational).
- Rational Numbers: No (irrational).
- Irrational Numbers: Yes (non-terminating, non-repeating decimal).
- Real Numbers: Yes (all irrational numbers are real).

#### 2. \( -2 \)
- Natural Numbers: No (negative).
- Whole Numbers: No (negative).
- Integers: Yes.
- Rational Numbers: Yes (can be written as \( \frac{-2}{1} \)).
- Irrational Numbers: No (rational).
- Real Numbers: Yes.

#### 3. \( -\frac{9}{37} \)
- Natural Numbers: No (negative and fraction).
- Whole Numbers: No (fraction).
- Integers: No (fraction).
- Rational Numbers: Yes (ratio of two integers).
- Irrational Numbers: No (rational).
- Real Numbers: Yes.

#### 4. \( 0 \)
- Natural Numbers: No (0 is not a natural number).
- Whole Numbers: Yes.
- Integers: Yes.
- Rational Numbers: Yes (can be written as \( \frac{0}{1} \)).
- Irrational Numbers: No (rational).
- Real Numbers: Yes.

#### 5. \( -6.06 \)
- Natural Numbers: No (negative and decimal).
- Whole Numbers: No (decimal).
- Integers: No (decimal).
- Rational Numbers: Yes (terminating decimal).
- Irrational Numbers: No (rational).
- Real Numbers: Yes.

#### 6. \( 4.\overline{56} \)
- Natural Numbers: No (decimal).
- Whole Numbers: No (decimal).
- Integers: No (decimal).
- Rational Numbers: Yes (repeating decimal).
- Irrational Numbers: No (rational).
- Real Numbers: Yes.

#### 7. \( 3.050050005... \)
- Natural Numbers: No (decimal).
- Whole Numbers: No (decimal).
- Integers: No (decimal).
- Rational Numbers: No (non-repeating, non-terminating decimal).
- Irrational Numbers: Yes.
- Real Numbers: Yes.

#### 8. \( 18 \)
- Natural Numbers: Yes.
- Whole Numbers: Yes.
- Integers: Yes.
- Rational Numbers: Yes (can be written as \( \frac{18}{1} \)).
- Irrational Numbers: No (rational).
- Real Numbers: Yes.

#### 9. \( \frac{-43}{0} \)
- Natural Numbers: No (undefined).
- Whole Numbers: No (undefined).
- Integers: No (undefined).
- Rational Numbers: No (undefined).
- Irrational Numbers: No (undefined).
- Real Numbers: No (undefined).

#### 10. \( \pi \)
- Natural Numbers: No (irrational).
- Whole Numbers: No (irrational).
- Integers: No (irrational).
- Rational Numbers: No (irrational).
- Irrational Numbers: Yes.
- Real Numbers: Yes.

#### 11. \( .\overline{634} \)
- Natural Numbers: No (decimal).
- Whole Numbers: No (decimal).
- Integers: No (decimal).
- Rational Numbers: Yes (repeating decimal).
- Irrational Numbers: No (rational).
- Real Numbers: Yes.

#### 12. \( \sqrt{225} \)
- Natural Numbers: Yes (since \( \sqrt{225} = 15 \)).
- Whole Numbers: Yes.
- Integers: Yes.
- Rational Numbers: Yes (can be written as \( \frac{15}{1} \)).
- Irrational Numbers: No (rational).
- Real Numbers: Yes.

#### 13. \( .634 \)
- Natural Numbers: No (decimal).
- Whole Numbers: No (decimal).
- Integers: No (decimal).
- Rational Numbers: Yes (terminating decimal).
- Irrational Numbers: No (rational).
- Real Numbers: Yes.

#### 14. \( \sqrt{\frac{4}{49}} \)
- Natural Numbers: No (fractional result).
- Whole Numbers: No (fractional result).
- Integers: No (fractional result).
- Rational Numbers: Yes (simplifies to \( \frac{2}{7} \)).
- Irrational Numbers: No (rational).
- Real Numbers: Yes.

#### 15. \( -\sqrt{64} \)
- Natural Numbers: No (negative).
- Whole Numbers: No (negative).
- Integers: Yes (since \( -\sqrt{64} = -8 \)).
- Rational Numbers: Yes (can be written as \( \frac{-8}{1} \)).
- Irrational Numbers: No (rational).
- Real Numbers: Yes.

Final Answer:

\[
\boxed{
\begin{array}{|c|c|c|c|c|c|c|c|}
\hline
& \text{Number} & \text{Natural Numbers} & \text{Whole Numbers} & \text{Integers} & \text{Rational Numbers} & \text{Irrational Numbers} & \text{Real Numbers} \\
\hline
1. & -\sqrt{17} & & & & & X & X \\
2. & -2 & & & X & X & & X \\
3. & -\frac{9}{37} & & & & X & & X \\
4. & 0 & & X & X & X & & X \\
5. & -6.06 & & & & X & & X \\
6. & 4.\overline{56} & & & & X & & X \\
7. & 3.050050005... & & & & & X & X \\
8. & 18 & X & X & X & X & & X \\
9. & \frac{-43}{0} & & & & & & \\
10. & \pi & & & & & X & X \\
11. & .\overline{634} & & & & X & & X \\
12. & \sqrt{225} & X & X & X & X & & X \\
13. & .634 & & & & X & & X \\
14. & \sqrt{\frac{4}{49}} & & & & X & & X \\
15. & -\sqrt{64} & & & X & X & & X \\
\hline
\end{array}
}
\]