In each of the following find `r+s, r- s, rs, (r )/(s)` if r denotes the first complex number and s denotes the second complex number
`-7, -1-3i`

Answer

To solve the problem, we need to find \( r+s \), \( r-s \), \( rs \), and \( \frac{r}{s} \) for the complex numbers \( r = -7 \) and \( s = -1 - 3i \).

### Step 1: Find \( r + s \)

Given:
- \( r = -7 \)
- \( s = -1 - 3i \)

We can calculate \( r + s \):
\[ r + s = -7 + (-1 - 3i) = -7 - 1 - 3i = -8 - 3i \]

### Step 2: Find \( r - s \)

Now, we calculate \( r - s \):
\[ r - s = -7 - (-1 - 3i) = -7 + 1 + 3i = -6 + 3i \]

### Step 3: Find \( rs \)

Next, we calculate \( rs \):
\[ rs = (-7)(-1 - 3i) = 7(1 + 3i) = 7 + 21i \]

### Step 4: Find \( \frac{r}{s} \)

Now we calculate \( \frac{r}{s} \):
\[ \frac{r}{s} = \frac{-7}{-1 - 3i} \]
To simplify this, we multiply the numerator and denominator by the conjugate of the denominator:
\[ \frac{-7}{-1 - 3i} \cdot \frac{-1 + 3i}{-1 + 3i} = \frac{7(-1 + 3i)}{(-1)^2 - (3i)^2} = \frac{7(-1 + 3i)}{1 - (-9)} = \frac{7(-1 + 3i)}{10} \]
This simplifies to:
\[ \frac{-7 + 21i}{10} = -\frac{7}{10} + \frac{21}{10}i \]

### Summary of Results

- \( r + s = -8 - 3i \)
- \( r - s = -6 + 3i \)
- \( rs = 7 + 21i \)
- \( \frac{r}{s} = -\frac{7}{10} + \frac{21}{10}i \)