Problem 111 Plot each of the complex fourth ... [FREE SOLUTION]

Chapter 7: Problem 111

Plot each of the complex fourth roots of 1

Short Answer

Expert verified

The fourth roots of the number 1 are 1, i, -1 and -i, corresponding to the points (1,0), (0,1), (-1,0), and (0,-1) on the complex plane.

Step by step solution

Understanding the problem

The question asks for the fourth roots of the number 1. In mathematical terms, we are looking for the solutions of the equation \(x^4 = 1\), where x is a complex number.

Finding the roots

The fourth roots of a complex number can be found by using the formula \(\sqrt[n]{\|z\|}(\cos(\frac{\theta + 2k\pi}{n})+i\sin(\frac{\theta + 2k\pi}{n}))\), where \(k = 0, 1, ..., n-1\) and \(z = \|z\|(\cos\theta+i\sin\theta)\). For our case, the roots are: \(x_1 = \sqrt[4]{1}(\cos(\frac{0 + 2*0*\pi}{4})+i\sin(\frac{0 + 2*0*\pi}{4})) = 1\), \(x_2 = \sqrt[4]{1}(\cos(\frac{0 + 2*1*\pi}{4})+i\sin(\frac{0 + 2*1*\pi}{4})) = i\), \(x_3 = \sqrt[4]{1}(\cos(\frac{0 + 2*2*\pi}{4})+i\sin(\frac{0 + 2*2*\pi}{4})) = -1\), \(x_4 = \sqrt[4]{1}(\cos(\frac{0 + 2*3*\pi}{4})+i\sin(\frac{0 + 2*3*\pi}{4})) = -i\).

Plotting the roots

The roots can now be represented graphically. Complex numbers are plotted on a plane similar to how coordinates are laid out, with the real part of the roots forming the x-axis, and the imaginary part forming the y-axis. Here, (1,0), (0,1), (-1,0), and (0,-1) represent the 4 roots respectively.

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91影视

Plot each of the complex fourth roots of 1

Short Answer

Step by step solution

Understanding the problem

Finding the roots

Plotting the roots

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