Question

If \( z \) is a complex number of unit modulus, then \[ \left| \frac{1+z}{1+ \overline{z}} \right| \] equals:

• A. 2
• B. 1
• C. \( \frac{1}{2} \)
• D. 4
• E. 6

Hint:

When simplifying expressions involving complex numbers, remember that the modulus of a complex number is its distance from the origin. If a complex number has unit modulus, its conjugate will also have unit modulus, which can simplify calculations.

Answer 1

The Correct Option is B

Let \( z \) be a complex number of unit modulus, which means that \[ |z| = 1. \] We need to evaluate: \[ \left| \frac{1 + z}{1 + \overline{z}} \right|. \] Step 1: Express \( \overline{z} \) in terms of \( z \) The complex conjugate of \( z \) is denoted by \( \overline{z} \), and for any complex number \( z = x + iy \), its conjugate is given by \( \overline{z} = x - iy \). Since \( z \) has unit modulus, we know that:
\[ |z|^2 = 1 \Rightarrow z \overline{z} = 1. \] Step 2: Simplify the expression
We can simplify the expression as follows: \[ \left| \frac{1+z}{1+\overline{z}} \right| = \frac{|1+z|}{|1+\overline{z}|}. \] Step 3: Evaluate \( |1+z| \) and \( |1+\overline{z}| \) Using the fact that \( |z| = 1 \), we compute both the modulus of \( 1+z \) and \( 1+\overline{z} \). 
Since the modulus of a complex number is the distance from the origin, and \( |z| = 1 \), the expressions for \( |1+z| \) and \( |1+\overline{z}| \) are equal.
Thus: \[ \left| \frac{1+z}{1+\overline{z}} \right| = 1. \] 
Therefore, the correct answer is option (B), which is 1.