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Magazine Chapter 7: Problem 47
Find all the complex solutions of the equations. $$i z^{3}=1$$
Short Answer
Expert verified
The complex solutions for the given equation are \(z_1= e^{i\pi/6}, z_2=e^{i5\pi/6}\) and \(z_3=e^{i3\pi/2}\).
Step by step solution
01
Convert the given equation to the exponential form of the complex number
We know that \(e^{i\theta} = \cos\theta + i\sin\theta\). We can write the equation \(i z^{3}=1\) as \(i\cdot(|z|^3\cdot e^{i\theta})=e^{0}\), where \(|z|^3\) indicates the modulus of the \(z^3\) and \(\theta\) the argument.
02
Compare the two forms
Set \(|z|^3 = 1\) and \(3\theta = \arg(i)\). Solving the first equation gives us |z| = 1. The argument of \(i\) is \(\pi/2\) (or \(2k\pi + \pi/2\) for multiples of \(2\pi\)). Therefore, \(3\theta = 2k\pi + \pi/2\).
03
Find the values for \(\theta\)
Solve the equation \(3\theta = 2k\pi + \pi/2\) for \(\theta\), knowing that \(k\) can be 0, 1, or 2 (since it's a cubic equation and thus has at most three roots). This leads to three distinct solutions: \(\theta_1= \pi/6, \theta_2=5\pi/6\) and \(\theta_3=3\pi/2\).
04
Use the values to determine the solutions
Now calculate the solutions using \(z_k = |z|\cdot e^{i\theta_k}\). As |z|=1, the solutions are: \(z_1= e^{i\pi/6}, z_2=e^{i5\pi/6}\) and \(z_3=e^{i3\pi/2}\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Complex Solutions
Complex solutions of an equation like \(iz^3 = 1\) involve not only a magnitude but also a direction, represented by an angle on the complex plane. The complex plane, also called the Argand plane, is a two-dimensional plane where the x-axis represents real numbers, and the y-axis represents imaginary numbers. When we solve for complex solutions, we are essentially finding points in this plane that satisfy the given equation.
- The equation \(iz^3 = 1\) implies that z must be a complex number such that its cube, when multiplied by \(i\), results in the real number 1.
- A cubic equation like this generally has three distinct solutions, owing to the fact that a polynomial of degree \(n\) can have up to \(n\) roots.
- Each solution represents a unique point in the complex plane that, when cubed and multiplied by \(i\), equals 1.
Exponential Form
The exponential form of complex numbers is a very elegant way to express numbers through Euler's formula: \(e^{i\theta} = \cos\theta + i\sin\theta\). This representation is particularly useful when dealing with powers and roots of complex numbers because it simplifies multiplication and division operations.
- To convert the initial equation \(iz^3 = 1\) into exponential form, we express both sides using \(e^{i\theta}\). Here, we write \(z^3\) as \(|z|^3 \cdot e^{i\theta}\) and the right-hand side \(1\) as \(e^0\).
- Matching the magnitudes (|z|^3 = 1) and the arguments (3\theta = \arg(i)) allows us to solve for the necessary variables independently. This way, we find values for \(|z|\) and \(\theta\) that satisfy the equation.
- With the exponential form, the sometimes cumbersome trigonometric calculations become much simpler, turning the solution of the equation into a straightforward process of finding the roots in an argument.
Modulus and Argument
Understanding the modulus and argument of complex numbers is crucial when solving equations like \(iz^3 = 1\). These terms refer to a complex number's magnitude and its direction in the complex plane, respectively.
- The **modulus**, denoted as \(|z|\), is the distance from the origin to the point \(z\) in the complex plane. For our solutions, we determine that \(|z| = 1\), which means each solution is on the unit circle.
- The **argument**, or \(\arg(z)\), is the angle the radius to the point \(z\) makes with the positive real axis. For the given equation, the argument of \(i\) (which is the right-hand side of the equation \(iz^3 = 1\)) is \(\pi/2\), leading us to solve for \(\theta\).
- By writing \(z\) in its exponential form, this allows the distinct comfort of linearizing what would otherwise be a daunting trigonometric task to a simple arithmetic procedure.
