Problem 2 - Algebra and Geometry

Question

Show that if \(w=r e^{i \theta}\) and \(w \neq 0\), then \(z^{2}=w\) if and only if \(|z|=\sqrt{r}\) and \(\arg z\) is \(\theta / 2\) or \(\theta / 2+\pi\)

Step-by-Step Solution

Answer

If \( z^2 = w \), then \(|z| = \sqrt{r}\) and \(\arg z = \theta/2\) or \(\arg z = \theta/2+\pi\).

Step 1: Understanding the Given Equation

We are given that if \( w = re^{i\theta} \) and \( w eq 0 \), we want to find \( z \) such that \( z^2 = w \). This equation tells us that \( z \) is a complex number whose square equals \( w \).

Step 2: Expressing \( z \) in Polar Form

Let \( z = ho e^{i\phi} \) where \( \rho = |z| \) is the modulus of \( z \), and \( \phi = \arg(z) \) is the argument of \( z \).

Step 3: Substitute and Equate Moduli and Arguments

Substitute \( z \) into the equation \( z^2 = w \):\[z^2 = (\rho e^{i\phi})^2 = \rho^2 e^{i(2\phi)} = re^{i\theta}\]From this, equate the moduli: \( \rho^2 = r \). Therefore, \( \rho = \sqrt{r} \).

Step 4: Equating Arguments

Since \( e^{i(2\phi)} = e^{i\theta} \), we can say:\[2\phi = \theta + 2k\pi\quad \text{for some integer } k\]Solving for \( \phi \):\[\phi = \frac{\theta}{2} + k\pi\]

Step 5: Identifying the Possible Values for \( z \)

Considering the periodicity of the argument, \( k \) can be 0 or 1 for distinct values within a full rotation:1. If \( k = 0 \), then \( \phi = \frac{\theta}{2} \).2. If \( k = 1 \), then \( \phi = \frac{\theta}{2} + \pi \).

Step 6: Conclusion

Thus, \( |z| = \sqrt{r} \) and \( \arg(z) = \frac{\theta}{2} \) or \( \arg(z) = \frac{\theta}{2} + \pi \). This confirms that for \( z^2 = w \), these are the necessary conditions for \( z \).

Key Concepts

Polar Form

Expressing complex numbers in polar form is an elegant way to show their properties of magnitude and direction. When we talk about polar form, we're referring to writing a complex number using its modulus and argument. If we have a complex number written as \( z = x + yi \), it can also be expressed as \( r e^{i\theta} \). Here, \( r \) is the modulus of the complex number, representing its distance from the origin in the complex plane, and \( \theta \) is the argument, which is the angle it makes with the positive real axis.

The transformation to polar form from the standard form \( x + yi \) is done using the relationships:

• \( r = \sqrt{x^2 + y^2} \)
• \( \theta = \tan^{-1}(\frac{y}{x}) \)

This representation is particularly useful in multiplication and division operations of complex numbers, as it simplifies the process by converting multiplication into a simple addition of angles and multiplication of magnitudes.

Modulus

The modulus of a complex number is like the measure of its size or length. It represents the distance a complex number lies from the origin of the complex plane, where the real and imaginary axes meet. Given a complex number \( z = x + yi \), its modulus is calculated as \( |z| = \sqrt{x^2 + y^2} \).

For example, if \( z = 3 + 4i \), then its modulus is \( \sqrt{3^2 + 4^2} = \sqrt{25} = 5 \). This modulus is critical, especially when transforming complex numbers into their polar forms.

In the context of solving complex equations like \( z^2 = w \), the modulus helps in understanding the magnitude-related aspect, meaning the lengths from the origin should also correspond when squared, leading to matching the modulus from \( z^2 \) with \( w \).

Argument

The argument of a complex number tells us the direction of the number from the origin, measured in terms of an angle with respect to the positive real axis. It represents the 'heading' or the angle \( \theta \) a complex number makes.

The argument is found through the formula \( \theta = \tan^{-1}(\frac{y}{x}) \), where \( x \) and \( y \) are the real and imaginary components of a complex number \( z = x + yi \).

Knowing the argument is vital when working with polar coordinates because it directly influences how the complex number behaves in terms of directionality. When we take the square of a complex number, its argument doubles, impacting the resulting direction in polar form. For example, if a complex number has an argument of \( \theta \), then its square will have an argument of \( 2\theta \). This property becomes essential in solving equations like \( z^2 = w \) because one must consider how arguments interact and equate when squared.

Complex Equation

A complex equation involves expressions that deal with complex numbers. These equations often appear intimidating due to the imaginary component, but by breaking them down into polar forms, we can simplify the task.

An example of a complex equation is \( z^2 = w \), where \( w \) is a non-zero complex number. The challenge here is to find values of \( z \) that, when squared, yield \( w \). By expressing \( z \) in polar form as \( \rho e^{i\phi} \) and \( w \) as \( re^{i\theta} \), we can compare both moduli and arguments to find solutions.

This approach converts a multi-variable problem into more straightforward modulus and argument comparisons, making calculations more manageable and insightful. It essentially involves solving equations for modulus \( \rho = \sqrt{r} \) and argument \( \phi = \frac{\theta}{2} \) or \( \phi = \frac{\theta}{2} + \pi \), which suggest specific orientations of \( z \) that satisfy the initial condition.

Square Root of Complex Number

Finding the square root of a complex number involves a bit more work than dealing with real numbers. This process comes into play in solving equations like \( z^2 = w \), where determining \( z \) requires extracting the square root of \( w \).

In polar form, if \( w = re^{i\theta} \), then its square root \( z \) is \( \sqrt{r}e^{i(\theta/2)} \) or \( \sqrt{r}e^{i(\theta/2 + \pi)} \). Here, \( \sqrt{r} \) is the modulus of \( z \), and the argument becomes half of that of \( w \) due to the properties of exponents and how angles double. This means there are usually two possible values for the argument, reflecting the periodic nature of trigonometric functions.

Always remember, square roots provide two values (since \( z^2 = w \) yields two solutions for \( z \)), emphasizing the importance of considering the periodic nature of angles in complex numbers.