If ` z ne 0,z' ne 0 ` be any two complex numbers such that `| z| = | z'| = 1 ` and arg z = - arg z' then zz' =

Options

A. 1
B. `-1`
C. 2
D. none

Answer

To solve the problem, we start by using the properties of complex numbers given in the question.

### Step-by-Step Solution:

1. **Understanding the Given Conditions**:
- We have two complex numbers \( z \) and \( z' \) such that \( |z| = |z'| = 1 \).
- The argument of \( z \) is the negative of the argument of \( z' \), i.e., \( \arg(z) = -\arg(z') \).

2. **Expressing the Complex Numbers**:
- Since \( |z| = 1 \), we can express \( z \) in exponential form:
\[ z = e^{i\theta} \]
where \( \theta = \arg(z) \).
- Given \( \arg(z') = -\arg(z) \), we can express \( z' \) as:
\[ z' = e^{-i\theta} \]

3. **Multiplying the Complex Numbers**:
- Now, we need to find the product \( zz' \):
\[ zz' = e^{i\theta} \cdot e^{-i\theta} \]

4. **Using the Property of Exponents**:
- By the property of exponents, we can combine the terms:
\[ zz' = e^{i\theta - i\theta} = e^{0} = 1 \]

5. **Conclusion**:
- Therefore, we conclude that:
\[ zz' = 1 \]

### Final Answer:
\[ zz' = 1 \]

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