Question

If \(\left(\frac{1-i}{1+i}\right)^{n/2} = -1\), where \(i = \sqrt{-1}\), one possible value of n is

• A. 2
• B. 4
• C. 6
• D. 8

Hint:

The complex number \(\frac{1-i}{1+i}\) simplifies to \(-i\), and its reciprocal \(\frac{1+i}{1-i}\) simplifies to \(i\). Memorizing these common simplifications can save time on exams.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
This problem involves operations with complex numbers, specifically simplifying a ratio of complex numbers and then solving an equation involving powers of a complex number. The key is to first simplify the base of the power and then use the properties of powers of \(i\).
Step 2: Key Formula or Approach:
1. To simplify a fraction of complex numbers like \(\frac{z_1}{z_2}\), multiply the numerator and denominator by the complex conjugate of the denominator, \(\overline{z_2}\). 2. Use Euler's formula \(e^{i\theta} = \cos\theta + i\sin\theta\) or standard powers of \(i\) to solve the final equation. We know that \(-1 = e^{i\pi}\). Also, \(-i = e^{-i\pi/2}\).
Step 3: Detailed Explanation:
1. Simplify the base of the power: Let's simplify the complex number \(z = \frac{1-i}{1+i}\). Multiply the numerator and denominator by the conjugate of the denominator, which is \(1-i\): \[ z = \frac{1-i}{1+i} \times \frac{1-i}{1-i} = \frac{(1-i)^2}{(1)^2 - (i)^2} \] The numerator is \((1-i)^2 = 1^2 - 2(1)(i) + i^2 = 1 - 2i - 1 = -2i\). The denominator is \(1 - (-1) = 2\). \[ z = \frac{-2i}{2} = -i \] 2. Solve the equation: Substitute the simplified base back into the original equation: \[ (-i)^{n/2} = -1 \] 3. Find the required power: We need to find an exponent \(k = n/2\) such that \((-i)^k = -1\). Let's check the powers of \(-i\):

\((-i)^1 = -i\)
\((-i)^2 = (-1)^2 (i)^2 = 1 \cdot (-1) = -1\)
\((-i)^3 = (-1)^3 (i)^3 = -1 \cdot (-i) = i\)
\((-i)^4 = (-1)^4 (i)^4 = 1 \cdot (1) = 1\)
We see that the smallest positive integer power that gives \(-1\) is 2. So, we must have the exponent \(k = n/2\) equal to 2 (or \(2+4m\) for any integer \(m\)). Let's take the simplest case: \[ \frac{n}{2} = 2 \] \[ n = 4 \] 4. Check the options: The value \(n=4\) is one of the given options. For n=2, we have \((-i)^{2/2} = (-i)^1 = -i \neq -1\). For n=6, we have \((-i)^{6/2} = (-i)^3 = i \neq -1\). For n=8, we have \((-i)^{8/2} = (-i)^4 = 1 \neq -1\). Therefore, the only possible value among the options is \(n=4\). Step 4: Final Answer:
By simplifying the complex fraction to \(-i\) and solving the equation \((-i)^{n/2} = -1\), we find that a possible value for \(n\) is 4. This corresponds to option (B).