Question

For complex numbers \( z \), if \( |z| = 1 \), then the value of \( \frac{z-1}{z+1} \) is

• A. 0
• B. real
• C. purely imaginary
• D. complex numbers

Hint:

When \( |z| = 1 \), the ratio \( \frac{z-1}{z+1} \) will always be purely imaginary, as the real part cancels out.

Answer 1

The Correct Option is C

Step 1: Understand the given condition.
We are given that \( |z| = 1 \), which means that \( z \) lies on the unit circle in the complex plane. This implies that \( z \) can be written in the form \( z = e^{i\theta} \), where \( \theta \) is a real number.

Step 2: Analyze the expression.
The expression \( \frac{z-1}{z+1} \) represents a complex number. By substituting \( z = e^{i\theta} \), we can simplify this expression to show that it results in a purely imaginary number. The real part cancels out due to the properties of complex numbers on the unit circle.

Step 3: Conclusion.
Thus, the value of \( \frac{z-1}{z+1} \) is purely imaginary, which corresponds to option (C).