Write the following as complex numbers:
(i) `1+sqrt(-1)`
(ii) `4-sqrt(-5)`
(iii) `(sqrt(3))/(2)-(sqrt(-2))/(7)`

Answer

To express the given expressions as complex numbers, we will use the fact that the square root of a negative number can be represented using the imaginary unit \( i \), where \( i = \sqrt{-1} \).

Let's solve each part step by step:

### (i) \( 1 + \sqrt{-1} \)

1. **Identify the square root of -1**:
\[ \sqrt{-1} = i \]

2. **Substitute into the expression**:
\[ 1 + \sqrt{-1} = 1 + i \]

Thus, the complex number is:
\[ 1 + i \]

### (ii) \( 4 - \sqrt{-5} \)

1. **Identify the square root of -5**:
\[ \sqrt{-5} = i\sqrt{5} \]

2. **Substitute into the expression**:
\[ 4 - \sqrt{-5} = 4 - i\sqrt{5} \]

Thus, the complex number is:
\[ 4 - i\sqrt{5} \]

### (iii) \( \frac{\sqrt{3}}{2} - \frac{\sqrt{-2}}{7} \)

1. **Identify the square root of -2**:
\[ \sqrt{-2} = i\sqrt{2} \]

2. **Substitute into the expression**:
\[ \frac{\sqrt{3}}{2} - \frac{\sqrt{-2}}{7} = \frac{\sqrt{3}}{2} - \frac{i\sqrt{2}}{7} \]

Thus, the complex number is:
\[ \frac{\sqrt{3}}{2} - \frac{i\sqrt{2}}{7} \]

### Final Answers:
1. \( 1 + i \)
2. \( 4 - i\sqrt{5} \)
3. \( \frac{\sqrt{3}}{2} - \frac{i\sqrt{2}}{7} \)