To solve the problem, we need to classify the given real numbers into four categories: rational numbers, irrational numbers, positive integers, and negative integers. Here’s how we can do that step by step:
### Step 1: Identify Rational Numbers
A rational number is any number that can be expressed in the form \( \frac{p}{q} \), where \( p \) and \( q \) are integers and \( q \neq 0 \).
1. **-8**: Can be expressed as \( \frac{-8}{1} \) (rational)
2. **0**: Can be expressed as \( \frac{0}{1} \) (rational)
3. **\( \sqrt{5} \)**: Not a rational number (irrational)
4. **\( \frac{5}{7} \)**: Already in the form \( \frac{p}{q} \) (rational)
5. **-\( \sqrt{18} \)**: Not a rational number (irrational)
6. **\( \sqrt{32} \)**: Not a rational number (irrational)
7. **4.28**: Can be expressed as \( \frac{428}{100} \) (rational)
8. **\( \pi \)**: Not a rational number (irrational)
9. **3**: Can be expressed as \( \frac{3}{1} \) (rational)
10. **-\( \frac{8}{15} \)**: Already in the form \( \frac{p}{q} \) (rational)
11. **0.07**: Can be expressed as \( \frac{7}{100} \) (rational)
**Rational Numbers**: -8, 0, \( \frac{5}{7} \), 4.28, 3, -\( \frac{8}{15} \), 0.07
### Step 2: Identify Irrational Numbers
An irrational number cannot be expressed as a fraction of two integers.
- **\( \sqrt{5} \)**: (irrational)
- **-\( \sqrt{18} \)**: (irrational)
- **\( \sqrt{32} \)**: (irrational)
- **\( \pi \)**: (irrational)
**Irrational Numbers**: \( \sqrt{5} \), -\( \sqrt{18} \), \( \sqrt{32} \), \( \pi \)
### Step 3: Identify Positive Integers
Positive integers are whole numbers greater than zero.
- **-8**: Not a positive integer
- **0**: Not a positive integer
- **\( \sqrt{5} \)**: Not an integer
- **\( \frac{5}{7} \)**: Not an integer
- **-\( \sqrt{18} \)**: Not an integer
- **\( \sqrt{32} \)**: Not an integer
- **4.28**: Not an integer
- **\( \pi \)**: Not an integer
- **3**: Positive integer
- **-\( \frac{8}{15} \)**: Not an integer
- **0.07**: Not an integer
**Positive Integers**: 3
### Step 4: Identify Negative Integers
Negative integers are whole numbers less than zero.
- **-8**: Negative integer
- **0**: Not a negative integer
- **\( \sqrt{5} \)**: Not an integer
- **\( \frac{5}{7} \)**: Not an integer
- **-\( \sqrt{18} \)**: Not an integer
- **\( \sqrt{32} \)**: Not an integer
- **4.28**: Not an integer
- **\( \pi \)**: Not an integer
- **3**: Not a negative integer
- **-\( \frac{8}{15} \)**: Not an integer
- **0.07**: Not an integer
**Negative Integers**: -8
### Final Classification
- **Rational Numbers**: -8, 0, \( \frac{5}{7} \), 4.28, 3, -\( \frac{8}{15} \), 0.07
- **Irrational Numbers**: \( \sqrt{5} \), -\( \sqrt{18} \), \( \sqrt{32} \), \( \pi \)
- **Positive Integers**: 3
- **Negative Integers**: -8
