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).

Answer 2

Instead of substituting \( z = e^{i\theta} \), let's use the conjugate/algebraic property that for \( |z|=1 \), \( \bar z = 1/z \), and test whether \( w = \dfrac{z-1}{z+1} \) satisfies \( \bar w = -w \) (the defining property of a purely imaginary number).

Compute \( \bar w = \dfrac{\bar z - 1}{\bar z + 1} = \dfrac{1/z - 1}{1/z+1} = \dfrac{1-z}{1+z} = -\dfrac{z-1}{z+1} = -w \). Since \( \bar w = -w \), the number \( w \) is purely imaginary (this is exactly the algebraic condition that characterizes numbers with zero real part).

1. 0: \( w=0 \) would require \( z=1 \), a single specific point on the unit circle, not the value for a general \( z \) with \( |z|=1 \); this is too restrictive to be the general answer.
2. real: A real number satisfies \( \bar w = w \), but we've shown \( \bar w=-w \), which is the opposite condition (unless \( w=0 \)); so "real" does not describe \( w \) in general.
3. purely imaginary: This matches the condition \( \bar w=-w \) derived directly from \( |z|=1 \), confirming this option.
4. complex numbers: While technically every number (including purely imaginary ones) is complex, this option is too vague and doesn't capture the specific structural property (\( \bar w=-w \)) that \( w \) actually satisfies; "purely imaginary" is the precise classification.

The conjugate relation confirms \( w=\dfrac{z-1}{z+1} \) is purely imaginary whenever \( |z|=1 \) (and \( z\ne-1 \)).

Therefore, the correct answer is purely imaginary.