Question

Describe all the sets and subsets the number 2/3 belong

Answer 1

The number \( \frac{2}{3} \) can belong to different sets and subsets based on various mathematical contexts. Here’s a breakdown:

1. **Natural Numbers (N)**: The set of natural numbers includes positive integers starting from 1 (i.e., {1, 2, 3, ...}). Since \( \frac{2}{3} \) is not a natural number, it does not belong to this set.

2. **Whole Numbers (W)**: Whole numbers include all natural numbers and zero (i.e., {0, 1, 2, 3, ...}). \( \frac{2}{3} \) is not a whole number, so it does not belong to this set either.

3. **Integers (Z)**: The set of integers includes positive and negative whole numbers, as well as zero (i.e., {..., -2, -1, 0, 1, 2, ...}). Since \( \frac{2}{3} \) is not an integer, it does not belong to this set.

4. **Rational Numbers (Q)**: The set of rational numbers consists of all numbers that can be expressed as the quotient of two integers, where the denominator is not zero. Since \( \frac{2}{3} \) can be expressed in this form, it belongs to the set of rational numbers.

5. **Real Numbers (R)**: The set of real numbers includes all rational and irrational numbers. Since \( \frac{2}{3} \) is a rational number, it also belongs to the set of real numbers.

6. **Complex Numbers (C)**: The set of complex numbers includes all numbers of the form \( a + bi \), where \( a \) and \( b \) are real numbers, and \( i \) is the imaginary unit. Since \( \frac{2}{3} \) can be expressed as \( \frac{2}{3} + 0i \), it belongs to the set of complex numbers.

### Summary of Sets Containing \( \frac{2}{3} \):
- **Rational Numbers (Q)**
- **Real Numbers (R)**
- **Complex Numbers (C)**

### Subsets:
1. **Subsets of Rational Numbers**:
- Decimals (e.g., \( \frac{2}{3} = 0.666...\))
- Proper Fractions (as \( \frac{2}{3} \) is less than 1)

2. **Subsets of Real Numbers**:
- Irrational Numbers (that would not include \( \frac{2}{3} \))
- Decimal Numbers (including both rational and irrational decimals)
- Positive Real Numbers (since \( \frac{2}{3} > 0 \))

In summary, \( \frac{2}{3} \) belongs to the sets: rational numbers, real numbers, and complex numbers, with several relevant subsets in each category.