Identify the sets to which each number belongs by marking an "X" in the appropriate boxes.

Let's solve this step by step. We need to classify each number into the appropriate sets:

Definitions:

- Natural Numbers (ℕ): Positive integers starting from 1: {1, 2, 3, ...}
- Whole Numbers (W): Natural numbers including 0: {0, 1, 2, 3, ...}
- Integers (ℤ): All whole numbers and their negatives: {..., -3, -2, -1, 0, 1, 2, 3, ...}
- Rational Numbers (ℚ): Numbers that can be expressed as a fraction $\frac{p}{q}$ where $p$ and $q$ are integers and $q \ne 0$. Includes terminating and repeating decimals.
- Irrational Numbers: Numbers that cannot be written as a fraction. Non-repeating, non-terminating decimals (e.g., $\pi$, $\sqrt{2}$).
- Real Numbers (ℝ): All rational and irrational numbers.

Now let’s analyze each number:

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1. $-\sqrt{17}$
- $\sqrt{17}$ is irrational (not a perfect square), so $-\sqrt{17}$ is also irrational.
- Not natural, not whole, not integer, not rational.
- But it is real.
- ✔ Irrational, ✔ Real

2. $-2$
- Negative integer.
- ✘ Natural (negative), ✘ Whole (negative), ✔ Integer, ✔ Rational (can be written as $-\frac{2}{1}$), ✘ Irrational, ✔ Real
- ✔ Integer, ✔ Rational, ✔ Real

3. $-\frac{9}{37}$
- Fraction of two integers → rational.
- Negative → not natural, not whole.
- Not integer (not a whole number).
- ✔ Rational, ✔ Real

4. $0$
- Zero is a whole number and an integer.
- Not natural (natural starts at 1).
- ✔ Whole, ✔ Integer, ✔ Rational ($\frac{0}{1}$), ✔ Real

5. $-6.06$
- Terminating decimal → can be written as $-\frac{606}{100} = -\frac{303}{50}$ → rational.
- Not natural, not whole, not integer.
- ✔ Rational, ✔ Real

6. $4.\overline{56}$
- Repeating decimal → rational.
- Can be written as a fraction.
- Not natural? Wait: 4.56... > 4, but not integer.
- So: ✘ Natural, ✘ Whole, ✘ Integer, ✔ Rational, ✔ Real

7. $3.050050005...$
- Pattern: 3.050050005... — the number of zeros increases between 5s.
- This is non-repeating, non-terminating → irrational.
- Not rational, not integer, etc.
- ✔ Irrational, ✔ Real

8. $18$
- Positive integer.
- ✔ Natural, ✔ Whole, ✔ Integer, ✔ Rational, ✔ Real

9. $\frac{-43}{0}$
- Division by zero → undefined.
- Not a number at all → doesn't belong to any set.
- ✘ All boxes (but note: sometimes this is considered undefined, so no X anywhere)

10. $\pi$
- Famous irrational number.
- ✔ Irrational, ✔ Real

11. $.6\overline{34}$
- Repeating decimal → rational.
- ✔ Rational, ✔ Real

12. $\sqrt{225}$
- $\sqrt{225} = 15$ → positive integer.
- ✔ Natural, ✔ Whole, ✔ Integer, ✔ Rational, ✔ Real

13. $.634$
- Terminating decimal → rational.
- ✔ Rational, ✔ Real

14. $\sqrt{\frac{4}{49}}$
- $\sqrt{\frac{4}{49}} = \frac{\sqrt{4}}{\sqrt{49}} = \frac{2}{7}$ → rational.
- ✔ Rational, ✔ Real

15. $-\sqrt{64}$
- $\sqrt{64} = 8$, so $-\sqrt{64} = -8$
- Integer, negative → not natural, not whole.
- ✔ Integer, ✔ Rational, ✔ Real

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Final Answer Table:

| Number | Natural Numbers | Whole Numbers | Integers | Rational Numbers | Irrational Numbers | Real Numbers |
|--------|------------------|----------------|-----------|--------------------|----------------------|----------------|
| 1. $-\sqrt{17}$ | | | | | ✔ | ✔ |
| 2. $-2$ | | | ✔ | ✔ | | ✔ |
| 3. $-\frac{9}{37}$ | | | | ✔ | | ✔ |
| 4. $0$ | | ✔ | ✔ | ✔ | | ✔ |
| 5. $-6.06$ | | | | ✔ | | ✔ |
| 6. $4.\overline{56}$ | | | | ✔ | | ✔ |
| 7. $3.050050005...$ | | | | | ✔ | ✔ |
| 8. $18$ | ✔ | ✔ | ✔ | ✔ | | ✔ |
| 9. $\frac{-43}{0}$ | | | | | | | *(undefined)* |
| 10. $\pi$ | | | | | ✔ | ✔ |
| 11. $.6\overline{34}$ | | | | ✔ | | ✔ |
| 12. $\sqrt{225}$ | ✔ | ✔ | ✔ | ✔ | | ✔ |
| 13. $.634$ | | | | ✔ | | ✔ |
| 14. $\sqrt{\frac{4}{49}}$ | | | | ✔ | | ✔ |
| 15. $-\sqrt{64}$ | | | ✔ | ✔ | | ✔ |

> Note: For #9, since division by zero is undefined, it does not belong to any number set, so no "X" is placed.

✔ All numbers are real, except possibly #9 (which isn't a number at all). So every number listed (except #9) is a real number.

Let me know if you'd like this in a printable format or need explanations for specific ones!