ercise 1 llustrate the real number system using a tree diagram.

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Certainly! A tree diagram is a great way to visualize the hierarchical structure of the real number system. Below is an illustrative tree diagram of the real numbers, broken down into their various subsets:

```
Real Numbers (ℝ)
│
├── Rational Numbers (ℚ)
│   │
│   ├── Integers (ℤ)
│   │   │
│   │   ├── Positive Integers (Natural Numbers, ℕ): 1, 2, 3, …
│   │   ├── Negative Integers: -1, -2, -3, …
│   │   └── Zero: 0
│   │
│   └── Non-Integer Rationals
│       │
│       ├── Proper Fractions: Numerator < Denominator (e.g., 1/2, 3/4)
│       └── Improper Fractions: Numerator ≥ Denominator (e.g., 5/3, 7/4)
│
└── Irrational Numbers
    │
    ├── Algebraic Irrationals
    │   │
    │   ├── Square Roots of Non-Perfect Squares (e.g., √2, √3)
    │   └── Other Roots and Solutions to Polynomial Equations (e.g., ∛5)
    │
    ├── Transcendental Numbers
    │   │
    │   ├── π (Pi)
    │   └── e (Euler’s Number)
    │
    └── Other Irrationals
        └── Numbers like φ (Golden Ratio), etc.
```

### **Explanation of the Tree Diagram:**

1. **Real Numbers (ℝ):**
   - The broadest category containing all the numbers that can be represented on an infinite number line, including both rational and irrational numbers.

2. **Rational Numbers (ℚ):**
   - **Definition:** Numbers that can be expressed as the quotient of two integers (a fraction) where the denominator is not zero.
   - **Subcategories:**
     - **Integers (ℤ):** Whole numbers and their negatives, including zero.
       - **Positive Integers (Natural Numbers, ℕ):** 1, 2, 3, …
       - **Negative Integers:** -1, -2, -3, …
       - **Zero:** 0
     - **Non-Integer Rationals:** Fractions that are not whole numbers.
       - **Proper Fractions:** The numerator is less than the denominator (e.g., 1/2).
       - **Improper Fractions:** The numerator is equal to or greater than the denominator (e.g., 5/3).

3. **Irrational Numbers:**
   - **Definition:** Numbers that cannot be expressed as a simple fraction; their decimal expansions are non-repeating and non-terminating.
   - **Subcategories:**
     - **Algebraic Irrationals:** Solutions to polynomial equations that cannot be simplified to rational numbers.
       - **Square Roots of Non-Perfect Squares:** √2, √3, etc.
       - **Other Roots and Polynomial Solutions:** Cube roots like ∛5.
     - **Transcendental Numbers:** A subset of irrationals that are not roots of any non-zero polynomial equation with rational coefficients.
       - **π (Pi):** The ratio of the circumference of a circle to its diameter.
       - **e (Euler’s Number):** The base of natural logarithms.
     - **Other Irrationals:** Numbers like the Golden Ratio (φ), which do not fit neatly into the above categories but are still irrational.

### **Key Points to Remember:**

- **Real Numbers** encompass both **Rational** and **Irrational** numbers.
- **Rational Numbers** include **Integers** and **Non-Integer Rationals** (proper and improper fractions).
- **Irrational Numbers** are divided into **Algebraic** and **Transcendental** categories, among others.
- This hierarchical structure helps in understanding how different types of numbers relate to each other within the real number system.

### **Visual Representation:**

If you prefer a more visual representation, imagine starting with a single box labeled "Real Numbers" at the top. This box branches into two main boxes: "Rational Numbers" and "Irrational Numbers." Each of these boxes further branches into their respective subcategories as detailed above.

This tree diagram framework helps illustrate the composition and classification within the real number system, making it easier to understand how various numbers are categorized based on their properties.
Here's a simplified tree diagram of the real number system: ``` Real Numbers (ℝ) ├── Rational Numbers (ℚ) │ ├── Integers (ℤ) │ │ ├── Natural Numbers (ℕ): 1, 2, 3, … │ │ └── Negative Integers: -1, -2, -3, … │ └── Non-Integer Rationals │ ├── Proper Fractions: 1/2, 3/4, etc. │ └── Improper Fractions: 5/3, 7/4, etc. └── Irrational Numbers ├── Algebraic Irrationals │ ├── √2, √3, etc. │ └── ∛5, etc. └── Transcendental Numbers ├── π └── e ``` This diagram shows the main categories and their subcategories within the real numbers.

ercise 1 llustrate the real number system using a tree diagram.

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