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Chapter 6: Problem 111

Plot each of the complex fourth roots of \(1 .\)

Short Answer

Expert verified
The fourth roots of 1 are 1, i, -1, and -i. When plotted on the complex plane, they are at points (1,0), (0,1), (-1,0), and (0,-1) respectively.

Step by step solution

01

Recognizing the Roots of 1

The complex fourth roots of 1 are solutions of the equation \(z^4 = 1\). The general formula to find the roots is \(z_k = r(cos \frac{arg(z) + 2k\pi}{n} + isin \frac{arg(z) + 2k\pi}{n})\) where \(r = |z|\), \(arg(z)\) is the argument of z, n is the degree of the root, and k is each integer from 0 to n-1. In this case, \(r = 1\), \(arg(z) = 0\) and \(n = 4\). When we substitute these values into the formula, we get the roots as \(z_0 = cos \frac{0\pi}{4} + i sin \frac{0\pi}{4} = 1\), \(z_1 = cos \frac{2\pi}{4} + i sin \frac{2\pi}{4} = i\), \(z_2 = cos \frac{4\pi}{4} + i sin \frac{4\pi}{4} = -1\) and \(z_3 = cos \frac{6\pi}{4} + i sin \frac{6\pi}{4} = -i\).
02

Plotting the Roots on the Complex Plane

Plot these roots on the complex plane. The complex plane is a two-dimensional number line where the horizontal axis represents real numbers and the vertical axis represents imaginary numbers. Find the points (1,0), (0,1), (-1,0), (0,-1) which correspond to the values 1, i, -1, -i respectively, and mark them on the plane.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Complex Numbers
Complex numbers are an extension of the real numbers and are fundamental in advanced mathematics and engineering. A complex number has a real part and an imaginary part and is usually written in the form of \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit with the property that \(i^2 = -1\).

One can visualize complex numbers by using a two-dimensional plane known as the complex plane鈥攖he horizontal axis (x-axis) represents the real part of the complex number, and the vertical axis (y-axis) represents the imaginary part. This graphical representation not only makes complex numbers more accessible but also allows for the visualization of complex number operations, such as addition, subtraction, and multiplication.
De Moivre's Theorem
De Moivre's theorem is a powerful tool in mathematics that connects complex numbers to trigonometry. The theorem states that for any real number \(\theta\) and any positive integer \(n\), \((\cos \theta + i \sin \theta)^n = \cos(n\theta) + i \sin(n\theta)\).

This formula is especially useful for finding the powers and roots of complex numbers. For example, when determining the \(n^{th}\) roots of a complex number, one would divide the argument (the angle in polar coordinates) by \(n\) and apply the theorem to find each unique root.
  • It simplifies complex calculations,
  • Helps visualize roots in complex plane,

Understanding de Moivre's theorem is essential to tackle problems involving complex number calculations, such as finding roots, as shown in the given exercise.
Complex Plane
The complex plane, also known as the Argand plane, is a geometric representation of complex numbers. It is essentially a two-dimensional graph where every point corresponds to a complex number.

Real and Imaginary Axes

The horizontal axis, called the real axis, represents the real part of complex numbers. The vertical axis represents the imaginary part and is known as the imaginary axis.

Plotting Complex Numbers

Plotting a complex number like \(a + bi\) is done by moving \(a\) units along the real axis and \(b\) units along the imaginary axis. This concept allows us to visualize the fourth roots of 1 in the initial exercise, providing an intuitive grasp of their distribution around the unit circle.

The complex plane isn't just a visualization tool; it's integral for complex analysis and helps in understanding the geometric interpretation of complex number operations.

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