If a and b are two negative integers, then state the sign of the given expressions in each of the following, if knows definitely.
i. a+b `" ii. "a-b" iii. "a times b" iv. "a div b`

Answer

To solve the problem, we need to determine the signs of the expressions involving two negative integers \( a \) and \( b \). Let's analyze each expression step by step.

### Step-by-Step Solution:

1. **Expression: \( a + b \)**
- Since both \( a \) and \( b \) are negative integers, we can represent them as \( a = -m \) and \( b = -n \) where \( m \) and \( n \) are positive integers.
- Therefore, \( a + b = -m + (-n) = -(m + n) \).
- The result is negative because the sum of two positive integers \( m + n \) is positive, and the negative sign in front makes it negative.
- **Sign:** Negative

2. **Expression: \( a - b \)**
- Again, substituting \( a = -m \) and \( b = -n \), we have \( a - b = -m - (-n) = -m + n = n - m \).
- The sign of \( n - m \) depends on the values of \( m \) and \( n \):
- If \( n > m \), then \( n - m \) is positive.
- If \( n < m \), then \( n - m \) is negative.
- If \( n = m \), then \( n - m = 0 \).
- **Sign:** Either positive, negative, or zero (depends on the values of \( a \) and \( b \)).

3. **Expression: \( a \times b \)**
- Here, we multiply \( a \) and \( b \): \( a \times b = (-m) \times (-n) = mn \).
- The product of two negative integers is always positive.
- **Sign:** Positive

4. **Expression: \( a \div b \)**
- For division, we have \( a \div b = (-m) \div (-n) = \frac{m}{n} \).
- Dividing two negative integers also results in a positive value.
- **Sign:** Positive

### Summary of Signs:
- **i. \( a + b \)**: Negative
- **ii. \( a - b \)**: Either positive, negative, or zero (depends on values)
- **iii. \( a \times b \)**: Positive
- **iv. \( a \div b \)**: Positive