Re(z n )=0, where 𝑧

You are asked to find the values of $n$ such that the real part of $z^n$ equals 0, where $z = 3 + i$.

Step 1: Express $z$ in polar form.

First, let's write $z = 3 + i$ in polar form. The modulus $|z|$ and argument $\arg(z)$ are computed as follows:

• Modulus: $|z| = \sqrt{3^2 + 1^2} = \sqrt{9 + 1} = \sqrt{10}$

• Argument: $\arg(z) = \tan^{-1}\left(\frac{1}{3}\right)$

So, the polar form of $z$ is: $z = |z| \left( \cos(\arg(z)) + i \sin(\arg(z)) \right)$ $z = \sqrt{10} \left( \cos(\theta) + i \sin(\theta) \right)$ where $\theta = \arg(z) = \tan^{-1}\left(\frac{1}{3}\right)$.

Step 2: Compute $z^n$.

Using De Moivre’s Theorem, we can express $z^n$ as: $z^n = \left( \sqrt{10} \right)^n \left( \cos(n\theta) + i \sin(n\theta) \right)$ The real part of $z^n$ is given by: $\text{Re}(z^n) = \left( \sqrt{10} \right)^n \cos(n\theta)$

Step 3: Set the real part to 0.

We want the real part to be 0, so we set: $\left( \sqrt{10} \right)^n \cos(n\theta) = 0$ Since $\left( \sqrt{10} \right)^n \neq 0$ for any real $n$, it follows that: $\cos(n\theta) = 0$

The cosine function is 0 when: $n\theta = \frac{\pi}{2} + k\pi \quad \text{for some integer } k$ Thus: $n \cdot \tan^{-1}\left( \frac{1}{3} \right) = \frac{\pi}{2} + k\pi$ Solving for $n$, we get: $n = \frac{\frac{\pi}{2} + k\pi}{\tan^{-1}\left( \frac{1}{3} \right)}$

Step 4: Find the numerical value.

To get a numerical answer, we need the value of $\theta = \tan^{-1}\left( \frac{1}{3} \right)$, which is approximately: $\theta \approx 0.32175 \, \text{radians}$ Therefore: $n = \frac{\frac{\pi}{2} + k\pi}{0.32175}$ $n \approx \frac{1.5708 + k\pi}{0.32175}$ For integer values of $k$, you can compute specific values for $n$.

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Would you like to go further and calculate specific values of $n$ for different $k$ values? Or do you have any questions about this process?

Related Questions:

1. How do you compute the argument of a complex number in polar form?
2. Can De Moivre’s Theorem be applied to any complex number?
3. What are the principal values of the argument for complex numbers?
4. How do you find the real and imaginary parts of powers of complex numbers?
5. What is the geometric interpretation of multiplying complex numbers in polar form?

Tip: When working with complex numbers, converting to polar form can make operations like raising to powers or extracting roots much easier than working directly in rectangular form!