If `|z| = 1`, then `((1 + z)/(1 + z))` equals :

Options

A. `bar(z)`
B. z
C. `z^(-1)`
D. none of these

Answer

To solve the problem, we need to analyze the expression given that \( |z| = 1 \).

### Step-by-Step Solution:

1. **Understanding the Given Condition**:
We are given that \( |z| = 1 \). This means that \( z \) lies on the unit circle in the complex plane.

**Hint**: Remember that the modulus of a complex number \( z = x + yi \) is given by \( |z| = \sqrt{x^2 + y^2} \).

2. **Analyzing the Expression**:
The expression we need to evaluate is \( \frac{1 + z}{1 + z} \).

**Hint**: Simplifying expressions can often lead to a clearer understanding of their value.

3. **Simplifying the Expression**:
Since the numerator and the denominator are identical, we can simplify this expression:
\[ \frac{1 + z}{1 + z} = 1 \quad \text{(as long as \( 1 + z \neq 0 \))} \]

**Hint**: Check if the denominator can be zero. If \( 1 + z = 0 \), then \( z = -1 \), which is a special case.

4. **Considering the Special Case**:
If \( z = -1 \), then \( |z| = 1 \) still holds true. In this case:
\[ \frac{1 + (-1)}{1 + (-1)} = \frac{0}{0} \]
This is undefined. However, for all other values of \( z \) on the unit circle, the expression simplifies to 1.

**Hint**: Always consider edge cases when dealing with algebraic expressions.

5. **Conclusion**:
Since the expression \( \frac{1 + z}{1 + z} \) equals 1 for all \( z \) on the unit circle except when \( z = -1 \), we conclude that the value of the expression is generally 1, but it is undefined when \( z = -1 \).

Therefore, the answer to the question is:
\[ \text{The expression equals } 1 \text{ (undefined for } z = -1\text{).} \]

### Final Answer:
The expression \( \frac{1 + z}{1 + z} \) equals 1 (except when \( z = -1 \), where it is undefined).