Which of the following are rational numbers?
(i) `frac(5)(-8)`
(ii) `frac(-6)(11)`
(iii) `frac(7)(15)`
(iv) `frac(-8)(-12)`
(v) 6
(vi) -3
(vii) 0
(viii) `frac(0)(1)`
(ix) `frac(1)(0)`
(x) `frac(0)(0)`

Answer

To determine which of the given numbers are rational numbers, we need to remember that a rational number is defined as any number that can be expressed in the form \( \frac{p}{q} \), where \( p \) and \( q \) are integers and \( q \neq 0 \).

Let's analyze each option step by step:

1. **Option (i): \( \frac{5}{-8} \)**
- This is in the form \( \frac{p}{q} \) where \( p = 5 \) and \( q = -8 \).
- Since \( q \neq 0 \), it is a rational number.

2. **Option (ii): \( \frac{-6}{11} \)**
- This is in the form \( \frac{p}{q} \) where \( p = -6 \) and \( q = 11 \).
- Since \( q \neq 0 \), it is a rational number.

3. **Option (iii): \( \frac{7}{15} \)**
- This is in the form \( \frac{p}{q} \) where \( p = 7 \) and \( q = 15 \).
- Since \( q \neq 0 \), it is a rational number.

4. **Option (iv): \( \frac{-8}{-12} \)**
- This is in the form \( \frac{p}{q} \) where \( p = -8 \) and \( q = -12 \).
- Since \( q \neq 0 \), it is a rational number.

5. **Option (v): 6**
- We can express 6 as \( \frac{6}{1} \).
- Since this is in the form \( \frac{p}{q} \) where \( p = 6 \) and \( q = 1 \), it is a rational number.

6. **Option (vi): -3**
- We can express -3 as \( \frac{-3}{1} \).
- Since this is in the form \( \frac{p}{q} \) where \( p = -3 \) and \( q = 1 \), it is a rational number.

7. **Option (vii): 0**
- We can express 0 as \( \frac{0}{1} \).
- Since this is in the form \( \frac{p}{q} \) where \( p = 0 \) and \( q = 1 \), it is a rational number.

8. **Option (viii): \( \frac{0}{1} \)**
- This is in the form \( \frac{p}{q} \) where \( p = 0 \) and \( q = 1 \).
- Since \( q \neq 0 \), it is a rational number.

9. **Option (ix): \( \frac{1}{0} \)**
- This is in the form \( \frac{p}{q} \) where \( p = 1 \) and \( q = 0 \).
- Since \( q = 0 \), it is not a rational number (division by zero is undefined).

10. **Option (x): \( \frac{0}{0} \)**
- This is in the form \( \frac{p}{q} \) where \( p = 0 \) and \( q = 0 \).
- Since \( q = 0 \), it is not a rational number (division by zero is undefined).

### Summary of Rational Numbers:
- Rational numbers from the options are: (i), (ii), (iii), (iv), (v), (vi), (vii), and (viii).
- Not rational numbers: (ix) and (x).