Real numbers classification diagram with answer key.

It looks like you've uploaded an image of a completed worksheet titled "REAL NUMBERS" with a Venn diagram and an answer key. The task appears to be classifying numbers as rational, irrational, integers, whole numbers, or natural/counting numbers based on their properties.

Let me walk through the solution and explanation step by step, assuming this is what you're asking for:

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🔷 Understanding the Number System Hierarchy (Venn Diagram)

The diagram shows nested sets:
- Natural/Counting Numbers: {1, 2, 3, ...}
- Whole Numbers: {0, 1, 2, 3, ...} → includes natural numbers + zero
- Integers: {..., -3, -2, -1, 0, 1, 2, 3, ...} → whole numbers + negatives
- Rational Numbers: Any number that can be written as a fraction \( \frac{a}{b} \) where \( a \) and \( b \) are integers and \( b \neq 0 \). Includes terminating decimals, repeating decimals, and fractions.
- Irrational Numbers: Cannot be written as a fraction. Non-repeating, non-terminating decimals (e.g., \( \pi \), \( \sqrt{2} \)).

> Note: Rational and Irrational numbers together make up the Real Numbers.

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✔ Answer Key Explanation

Here’s how each number from the table should be classified:

| Number | Classification |
|--------|----------------|
| 1. 45 | Natural, Whole, Integer, Rational |
| 2. \( \sqrt{2} \) | Irrational (non-repeating, non-terminating) |
| 3. \( \frac{3}{4} \) | Rational (fraction) |
| 4. 0 | Whole, Integer, Rational (not natural) |
| 5. \( \pi \) | Irrational (well-known constant) |
| 6. \( \sqrt{9} = 3 \) | Natural, Whole, Integer, Rational |
| 7. \( -\frac{1}{2} \) | Rational (negative fraction) |
| 8. \( \sqrt{10} \) | Irrational (not a perfect square) |
| 9. 1.25 | Rational (terminating decimal = \( \frac{5}{4} \)) |
| 10. \( \frac{22}{7} \) | Rational (approximation of π, but still rational) |
| 11. \( \sqrt{16} = 4 \) | Natural, Whole, Integer, Rational |
| 12. \( -3 \) | Integer, Rational (not whole or natural) |
| 13. \( \sqrt{5} \) | Irrational |
| 14. \( 0.\overline{3} \) | Rational (repeating decimal = \( \frac{1}{3} \)) |
| 15. \( 0.75 \) | Rational (terminating decimal = \( \frac{3}{4} \)) |

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🧠 How to Classify Each Number

#### ✔️ Rational Numbers
- Can be expressed as a fraction.
- Include:
- Integers: e.g., 45, -3
- Fractions: e.g., \( \frac{3}{4} \)
- Terminating decimals: e.g., 1.25
- Repeating decimals: e.g., \( 0.\overline{3} \)

#### ✘ Irrational Numbers
- Cannot be written as a fraction.
- Examples:
- \( \sqrt{2}, \sqrt{10}, \sqrt{5} \) → not perfect squares
- \( \pi \) → known irrational
- Non-repeating, non-terminating decimals

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✔ Final Answer Placement in the Diagram

Based on the classifications above, here's how they fit into the Venn diagram:

- Natural Numbers: 45, 3, 4, 4 (from √9, √16)
- Whole Numbers: 45, 0, 3, 4
- Integers: 45, 0, 3, 4, -3
- Rational Numbers: All except irrational ones
- Irrational Numbers: \( \sqrt{2}, \pi, \sqrt{10}, \sqrt{5} \)

> So, only irrational numbers go outside the Rational circle.

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✔ Summary of Key Points

- Rational vs. Irrational: The main split.
- All integers are rational (they can be written as \( \frac{n}{1} \)).
- Not all rationals are integers (e.g., \( \frac{3}{4} \)).
- 0 is whole and integer, but not natural (some definitions include it, but traditionally natural numbers start at 1).
- Negative numbers are integers and rational, but not whole or natural.

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📝 Conclusion

The answer key correctly places:
- Rational numbers inside the Rational oval (including integers, whole, natural).
- Irrational numbers in the Irrational section.
- Natural numbers in the innermost circle.
- Whole numbers in the next circle (includes 0).
- Integers in the larger circle (includes negatives).

This classification aligns perfectly with standard mathematical definitions.

If you'd like, I can generate a printable version or help create a similar worksheet!