Subsets of Real Numbers | Overview & Practice Problems | Study.com - Free Printable

Actually, you did not upload an image — the text you provided is a *description* or *text representation* of a diagram titled “Real Number System,” showing nested sets: Natural → Whole → Integer → Rational → Real, with Irrational numbers alongside Rational to make up the Real numbers.

Since no actual image was uploaded, I’ll solve the implied task: Explain the structure and relationships shown in this diagram of the Real Number System.

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## ✔ SOLUTION & EXPLANATION

The diagram illustrates the hierarchical classification of real numbers, showing how different subsets are contained within larger sets.

📌 Key Sets Shown:

1. Natural Numbers (ℕ)
- Examples: 1, 2, 3, 4, 5, ...
- These are the counting numbers starting from 1.
- *Note: Sometimes 0 is included, but here it’s not — 0 is in Whole numbers.*

2. Whole Numbers (𝕎)
- Examples: 0, 1, 2, 3, 4, ...
- Includes all natural numbers + zero.
- So: Natural ⊂ Whole

3. Integers (ℤ)
- Examples: ..., -3, -2, -1, 0, 1, 2, 3, ...
- Includes all whole numbers and their negatives.
- So: Whole ⊂ Integer

4. Rational Numbers (ℚ)
- Any number that can be written as a fraction a/b, where a and b are integers and b ≠ 0.
- Includes:
- Integers (e.g., -17 = -17/1)
- Fractions (e.g., -5/6, 1/2)
- Terminating decimals (e.g., -3.25 = -13/4)
- Repeating decimals (e.g., 76.333... = 76 + 1/3 = 229/3)
- So: Integer ⊂ Rational

5. Irrational Numbers (𝕀)
- Numbers that cannot be expressed as a simple fraction.
- Non-repeating, non-terminating decimals.
- Examples: π, e, √7, -√3, 16.2467453315... (non-repeating)
- Not part of Rational numbers.

6. Real Numbers (ℝ)
- The union of Rational and Irrational numbers.
- Every point on the number line is a real number.
- So: ℝ = ℚ ∪ 𝕀, and ℚ ∩ 𝕀 = ∅ (they don’t overlap)

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## 🔍 Visual Hierarchy (Nested Boxes):

The diagram uses nested boxes to show inclusion:

```
Real Numbers
├── Rational Numbers
│ └── Integer
│ └── Whole
│ └── Natural
└── Irrational Numbers (separate box, same level as Rational)
```

This means:
- All Natural numbers are Whole.
- All Whole numbers are Integers.
- All Integers are Rational.
- All Rational and Irrational numbers together form Real numbers.

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## 🧠 Important Notes:

- 0 is a Whole number, but not a Natural number (in this diagram).
- -1.678 is Rational because it’s a terminating decimal → can be written as -1678/1000.
- π and √7 are Irrational — they cannot be written as fractions and their decimal expansions never repeat or terminate.
- 16.2467453315... is listed under Irrational — we assume the ellipsis (...) means it continues without repeating.

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## ✔ Final Answer Summary:

> The diagram correctly shows the hierarchy of real numbers:
> Natural ⊂ Whole ⊂ Integer ⊂ Rational ⊂ Real,
> and Irrational numbers are disjoint from Rational, but together they make up All Real Numbers.
> This classification helps us understand properties like whether a number can be written as a fraction, or if it has a repeating/terminating decimal expansion.

Let me know if you’d like to classify specific numbers using this system!