Math Project for Class 8: Understanding the Hierarchy of Rational Numbers.

Actually, there is no math problem or task written in the image you uploaded — it’s a diagram showing the hierarchy of number systems, likely for educational purposes (Class 8 Math Project, Activity 6).

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📌 What the Image Shows:

It visually represents how different sets of numbers are related:

1. Natural Numbers:
→ `1, 2, 3, 4, 5, 6, ...`
These are the counting numbers starting from 1.

2. Whole Numbers:
→ `0, 1, 2, 3, 4, 5, 6, ...`
Whole numbers = Natural numbers + zero.

3. Integers:
→ `..., -4, -3, -2, -1, 0, 1, 2, 3, 4, ...`
Integers = Whole numbers + negative whole numbers.

4. Rational Numbers:
→ All numbers that can be written as a fraction `p/q`, where `p` and `q` are integers and `q ≠ 0`.
This includes all integers, fractions, terminating decimals, and repeating decimals.

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✔ So, if this is “Activity 6” for Class 8 Math, here’s what you’re likely being asked to do:

> “Explain the relationship between Natural Numbers, Whole Numbers, Integers, and Rational Numbers using the diagram.”

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## 📘 Solution & Explanation:

1. Natural Numbers (N):

- Start from 1: `1, 2, 3, 4, ...`
- Used for counting.
- Subset of Whole Numbers.

2. Whole Numbers (W):

- Include 0 and all natural numbers: `0, 1, 2, 3, ...`
- 0 is the only whole number not in natural numbers.
- Subset of Integers.

3. Integers (Z):

- Include positive numbers, negative numbers, and zero: `..., -3, -2, -1, 0, 1, 2, 3, ...`
- Subset of Rational Numbers.

4. Rational Numbers (Q):

- Any number that can be expressed as `a/b`, where `a` and `b` are integers, and `b ≠ 0`.
- Examples: `1/2`, `-3/4`, `5` (since 5 = 5/1), `0.75` (since 0.75 = 3/4), `-2.5` (since -2.5 = -5/2).
- All integers are rational because any integer `n` can be written as `n/1`.

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## 🔁 Hierarchy (from smallest to largest set):

> Natural Numbers ⊂ Whole Numbers ⊂ Integers ⊂ Rational Numbers

This means:
- Every natural number is a whole number.
- Every whole number is an integer.
- Every integer is a rational number.
- But NOT vice versa!

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## 🎯 Example to Clarify:

| Number | Natural? | Whole? | Integer? | Rational? |
|--------|----------|--------|----------|-----------|
| 5 | ✔ | ✔ | ✔ | ✔ |
| 0 | ✘ | ✔ | ✔ | ✔ |
| -3 | ✘ | ✘ | ✔ | ✔ |
| 1/2 | ✘ | ✘ | ✘ | ✔ |

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## 💡 Conclusion:

The diagram helps visualize how each number system expands to include more types of numbers. Understanding these relationships is foundational for higher math concepts like real numbers, irrational numbers, and algebra.

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✔ Final Answer for Activity 6:

> The diagram shows the inclusion relationship among number systems:
> Natural Numbers are contained within Whole Numbers, which are contained within Integers, which are contained within Rational Numbers.
> Each larger set includes all the numbers of the smaller sets plus additional numbers (like 0, negatives, and fractions).
> This hierarchy helps us understand how numbers are classified and related in mathematics.

Let me know if you need to turn this into a poster, worksheet, or presentation! 🧮📚