Question

If \( m, n \) are respectively the least positive and greatest negative integer values of such that \( (\frac{1-i}{1+i})^k = -i\), then \( m - n = \):

• A. 4
• B. 0
• C. 6
• D. 2

Hint:

To find the least positive and greatest negative integer values of \( k \), consider the boundaries of the range defined by \( m \leq k \leq n \). The least positive integer is the smallest possible positive value, and the greatest negative integer is the largest negative value within the range.

Answer 1

The Correct Option is A

We are given the equation: \[ \left( \frac{1 - i}{1 + i} \right)^k = -i \] Step 1: Express the complex fraction in polar form
We know that: \[ \frac{1 - i}{1 + i} \] Dividing numerator and denominator by the modulus of \(1 + i\) (which is \( \sqrt{2} \)), \[ \frac{1 - i}{1 + i} = \frac{(1 - i)(1 - i)}{(1 + i)(1 - i)} = \frac{(1 - i)^2}{2} \] Now, \[ (1 - i)^2 = 1 - 2i + i^2 = 1 - 2i - 1 = -2i \] Thus, \[ \frac{1 - i}{1 + i} = \frac{-2i}{2} = -i \] Step 2: Equating the powers
From the original equation, \[ (-i)^k = -i \] Using the polar form of \( -i = e^{-i \pi/2} \), we have: \[ (-i)^k = e^{-i \frac{\pi}{2} k} \] Equating the arguments, \[ -\frac{\pi}{2} k = -\frac{\pi}{2} + 2n\pi \] Dividing both sides by \( -\frac{\pi}{2} \), \[ k = 1 + 4n \] Step 3: Finding the values of \( k \)
For the least positive integer value, setting \(n = 0\), \[ k = 1 \] For the greatest negative integer value, set \(n = -1\), \[ k = 1 + 4(-1) = -3 \] Step 4: Compute \( m - n \)
\[ m = 1, \quad n = -3 \] \[ m - n = 1 - (-3) = 4 \]