Problem 111 - Algebra and Trigonometry

Question

Plot each of the complex fourth roots of 1

Step-by-Step Solution

Answer

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 1: 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.

Step 2: 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: \(x_1 = \sqrt[4]{1}(\cos(\frac{0 + 2*0*\pi}{4})+i\sin(\frac{0 + 2*0*\pi}{4})) = 1\), \(x_2 = \sqrt[4]{1}(\cos(\frac{0 + 2*1*\pi}{4})+i\sin(\frac{0 + 2*1*\pi}{4})) = i\), \(x_3 = \sqrt[4]{1}(\cos(\frac{0 + 2*2*\pi}{4})+i\sin(\frac{0 + 2*2*\pi}{4})) = -1\), \(x_4 = \sqrt[4]{1}(\cos(\frac{0 + 2*3*\pi}{4})+i\sin(\frac{0 + 2*3*\pi}{4})) = -i\).

Step 3: 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.