Name all subsets of the real number system that each number belongs to. (Use r=real, I=irrational, q=rational, z=integer, w=whole, n=natural)

The real number system can be divided into several subsets, each with its own defining characteristics. The subsets include natural numbers (\( n \)), whole numbers (\( w \)), integers (\( z \)), rational numbers (\( q \)), and irrational numbers (\( I \)). Every real number (\( r \)) belongs to at least one of these subsets. Below are the subsets of the real number system along with their relationships:

1. **Natural Numbers (\( n \))**:
   - This subset includes all positive counting numbers starting from 1, i.e., \( \{1, 2, 3, \dots\} \).

2. **Whole Numbers (\( w \))**:
   - This subset includes all natural numbers and zero, i.e., \( \{0, 1, 2, 3, \dots\} \).

3. **Integers (\( z \))**:
   - This subset includes all whole numbers and their negatives, i.e., \( \{\dots, -3, -2, -1, 0, 1, 2, 3, \dots\} \).

4. **Rational Numbers (\( q \))**:
   - This subset includes all numbers that can be expressed as a fraction \( \frac{a}{b} \) where \( a \) and \( b \) are integers and \( b \neq 0 \). It includes integers and fractions, as well as terminating and repeating decimals.

5. **Irrational Numbers (\( I \))**:
   - This subset includes all real numbers that cannot be expressed as a fraction of two integers. Their decimal expansions are non-terminating and non-repeating, such as \( \sqrt{2} \), \( \pi \), and \( e \).

6. **Real Numbers (\( r \))**:
   - This is the set of all possible numbers on the number line, including both rational and irrational numbers.

### Relationship Between the Subsets

- \( n \subset w \subset z \subset q \subset r \)
- \( I \subset r \)

### Example Number and Their Subset Memberships:

1. **\( 3 \)**:
   - Subsets: \( n, w, z, q, r \)

2. **\( 0 \)**:
   - Subsets: \( w, z, q, r \)

3. **\( -5 \)**:
   - Subsets: \( z, q, r \)

4. **\( \frac{1}{2} \)**:
   - Subsets: \( q, r \)

5. **\( \sqrt{2} \)**:
   - Subsets: \( I, r \)

6. **\( \pi \)**:
   - Subsets: \( I, r \)

### Summary
- **Natural Numbers (\( n \))**: \( \{1, 2, 3, \dots\} \)
- **Whole Numbers (\( w \))**: \( \{0, 1, 2, 3, \dots\} \)
- **Integers (\( z \))**: \( \{\dots, -3, -2, -1, 0, 1, 2, 3, \dots\} \)
- **Rational Numbers (\( q \))**: Numbers expressible as \( \frac{a}{b} \) where \( b \neq 0 \)
- **Irrational Numbers (\( I \))**: Numbers that cannot be expressed as fractions
- **Real Numbers (\( r \))**: All numbers on the number line

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Do you have any questions or need further explanation on any of these subsets?

Here are five related questions:
1. What is the difference between rational and irrational numbers?
2. How can you determine whether a number is natural, whole, or an integer?
3. What are some examples of irrational numbers other than \( \pi \) and \( \sqrt{2} \)?
4. Can a number be both rational and irrational? Why or why not?
5. How do rational numbers relate to repeating and terminating decimals?

**Tip:** To identify the subset to which a number belongs, always start with the smallest subset and check upward (e.g., check if it's natural first, then whole, etc.).

Explore the subsets of the real number system including natural numbers, whole numbers, integers, rational numbers, and irrational numbers. Understand how each real number fits into these subsets with examples and detailed explanations. Suitable for students in Grades 7-9 studying number theory and real number properties.

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