Problem 92 - Algebra and Trigonometry

Question

Solve the equation. $$ z^{8}-i=0 $$

Step-by-Step Solution

Answer

The equation has 8 roots: \(e^{i\frac{(4n+1)\pi}{16}}\) for \(n = 0, 1,...,7\).

Step 1: Isolate the variable

The given equation is \(z^{8} - i = 0\). Add \(i\) to both sides to isolate \(z^{8}\): \(z^{8} = i\).

Step 2: Express in exponential form

Recall that the imaginary unit \(i\) can be expressed in exponential form as \(e^{i\frac{\pi}{2} + 2\pi n i}\) (since \(i\) lies at \(90\degree\) or \(\pi/2\) on the unit circle).

Step 3: Use the exponential form to find roots

Since \(z^{8} = i = e^{i(\frac{\pi}{2} + 2n\pi)}\), take the 8th root of both sides in exponential form:\(z = e^{i(\frac{\pi}{16} + \frac{2\pi n}{8})}\) for \(n = 0, 1, 2, 3, 4, 5, 6, 7\).

Step 4: Calculate all 8 roots

Now calculate the roots by plugging in values for \(n\):- For \(n = 0\), \(z = e^{i\frac{\pi}{16}}\)- For \(n = 1\), \(z = e^{i\frac{5\pi}{16}}\)- For \(n = 2\), \(z = e^{i\frac{9\pi}{16}}\)- For \(n = 3\), \(z = e^{i\frac{13\pi}{16}}\)- For \(n = 4\), \(z = e^{i\frac{17\pi}{16}}\)- For \(n = 5\), \(z = e^{i\frac{21\pi}{16}}\)- For \(n = 6\), \(z = e^{i\frac{25\pi}{16}}\)- For \(n = 7\), \(z = e^{i\frac{29\pi}{16}}\)These are the eight roots of the original equation.

Key Concepts

Roots of Complex Numbers

Understanding the concept of roots in complex numbers is crucial to solving equations like \(z^8 - i = 0\). Complex roots are solutions to polynomial equations where the numbers have both real and imaginary parts. When dealing with complex numbers, we extend the idea of square roots and higher roots to include solutions that are not strictly real numbers.

• To find the roots of a complex number like \(z^8 = i\), we need to determine all the possible solutions \(z\) that, when raised to the chosen power, give the original number.
• Complex roots occur in conjugate pairs when derived from real coefficients.

The number of roots corresponds to the degree of the polynomial. So with \(z^8\), there are 8 distinct roots. These roots are evenly spaced around a circle in the complex plane since they form a geometric progression. This circle is commonly referred to as the unit circle due to its radius of one.

Polar Form

Complex numbers can be expressed in different forms, one of which is the polar form. The polar form represents a complex number in terms of its magnitude and angle, rather than its horizontal (real) and vertical (imaginary) components.

• The magnitude is the distance from the origin to the point in the complex plane, often denoted as \( r \).
• The angle, often denoted \( \theta \), is the angle formed with the positive real axis.

Expressing a complex number such as \( i \) (which is at a 90-degree angle on the unit circle with a magnitude of 1) in polar form is straightforward. It is \( (1, \frac{\pi}{2}) \). Polar form is advantageous because it simplifies the process of multiplying and dividing complex numbers. Additionally, it is highly useful for finding roots, as manipulation in polar form can be easier than in rectangular form. Instead of directly dealing with both real and imaginary parts, you deal with scaling and rotating vectors.

Exponential Notation

Exponential notation for complex numbers is a way of expressing numbers in the polar coordinate system using Euler's formula. This formula shows a profound relationship between trigonometry and the exponential function: \[ e^{i\theta} = \cos\theta + i\sin\theta \] This is why any complex number can be expressed as \( e^{i\theta} \), where \( \theta \) is the angle.

• For example, the number \( i \) in exponential form is \( e^{i\frac{\pi}{2}} \) because it equals zero on the real axis and one on the imaginary axis.
• Using this notation is efficient, especially in solving equations involving powers of complex numbers, like finding all 8 roots of \( i \).

In fairness to roots calculation, for any \( n^{th} \) root of a complex number, use \( z = re^{i\theta/n} \). This simplifies calculations significantly, hence why exponential notation is often preferred.
This notation benefits from the clean algebraic manipulations allowed through trigonometric identities and properties, which makes finding powers and roots of complex numbers more manageable.