Question

17\. Find the natural logarithms of the three cube roots of 1 and of the five fifth roots of 1 .

Short answer

Answer: The natural logarithms of the cube roots of 1 are: ln(1) = 0 ln(e^{i(2\pi/3)}) = i(2\pi/3) ln(e^{i(4\pi/3)}) = i(4\pi/3) The natural logarithms of the fifth roots of 1 are: ln(1) = 0 ln(e^{i(2\pi/5)}) = i(2\pi/5) ln(e^{i(4\pi/5)}) = i(4\pi/5) ln(e^{i(6\pi/5)}) = i(6\pi/5) ln(e^{i(8\pi/5)}) = i(8\pi/5)

Step-by-step solution

Find the cube roots of 1 using polar form and De Moivre's theorem.

To find the cube roots of 1, we need to solve the equation z^3 = 1, where z is a complex number. We can write z in its polar form as z = 1 * e^{i\theta}. Using De Moivre's theorem, we have: (z^3)^{1/3} = 1^{1/3} * e^{i\theta/3} (e^{3i\theta})^{1/3} = 1 * e^{i(2k\pi/3)} z = e^{i(\frac{2k\pi}{3}+\frac{2m\pi}{3})} where k = 0, 1, 2 and m is an integer such that 0 <= m < 3. The cube roots of 1 are: z_1 = e^{i(0)} z_2 = e^{i(2\pi/3)} z_3 = e^{i(4\pi/3)}

Calculate the natural logarithm of the cube roots.

Now, we will calculate the natural logarithm of each cube root using the identity ln(e^z) = z: ln(z_1) = ln(e^{i(0)}) = 0 ln(z_2) = ln(e^{i(2\pi/3)}) = i(2\pi/3) ln(z_3) = ln(e^{i(4\pi/3)}) = i(4\pi/3) The natural logarithms of the cube roots of 1 are: ln(1) = 0 ln(e^{i(2\pi/3)}) = i(2\pi/3) ln(e^{i(4\pi/3)}) = i(4\pi/3)

Find the fifth roots of 1 using polar form and De Moivre's theorem.

To find the fifth roots of 1, we need to solve the equation z^5 = 1, where z is a complex number. We can write z in its polar form as z = 1 * e^{i\theta}. Using De Moivre's theorem, we have: (z^5)^{1/5} = 1^{1/5} * e^{i\theta/5} (e^{5i\theta})^{1/5} = 1 * e^{i(2k\pi/5)} z = e^{i(\frac{2k\pi}{5}+\frac{2m\pi}{5})} where k = 0, 1, 2, 3, 4 and m is an integer such that 0 <= m < 5. The fifth roots of 1 are: z_1 = e^{i(0)} z_2 = e^{i(2\pi/5)} z_3 = e^{i(4\pi/5)} z_4 = e^{i(6\pi/5)} z_5 = e^{i(8\pi/5)}

Calculate the natural logarithm of the fifth roots.

Now, we will calculate the natural logarithm of each fifth root using the identity ln(e^z) = z: ln(z_1) = ln(e^{i(0)}) = 0 ln(z_2) = ln(e^{i(2\pi/5)}) = i(2\pi/5) ln(z_3) = ln(e^{i(4\pi/5)}) = i(4\pi/5) ln(z_4) = ln(e^{i(6\pi/5)}) = i(6\pi/5) ln(z_5) = ln(e^{i(8\pi/5)}) = i(8\pi/5) The natural logarithms of the fifth roots of 1 are: ln(1) = 0 ln(e^{i(2\pi/5)}) = i(2\pi/5) ln(e^{i(4\pi/5)}) = i(4\pi/5) ln(e^{i(6\pi/5)}) = i(6\pi/5) ln(e^{i(8\pi/5)}) = i(8\pi/5)

Key concepts

De Moivre's Theorem

When exploring the fascinating world of complex numbers, De Moivre's Theorem emerges as a powerful tool for raising complex numbers to integer powers. In essence, this theorem establishes a relationship between complex number powers and trigonometric functions. Formulated by Abraham de Moivre, it states that for any complex number given in polar form as \(z = r(\text{cos} \theta + i\text{sin} \theta)\), and an integer \(n\), the power \(z^n\) can be expressed as:\[z^n = r^n (\text{cos}(n\theta) + i\text{sin}(n\theta))\]
For students working with natural logarithms of roots of unity, De Moivre's Theorem simplifies the process of finding the powers and roots of complex numbers. It's particularly handy when dealing with the roots of unity, as it describes how to efficiently calculate these roots by strategically using angles that are fractions of \(2\text{pi}\).

Complex Numbers

Complex numbers, written as \(a + bi\) where \(a\) is the real part, \(b\) is the imaginary part, and \(i\) is the imaginary unit defined by \(i^2 = -1\), are the backbone of many advanced mathematical concepts. They extend the idea of one-dimensional numbers to a two-dimensional number line by incorporating the notion of direction. This direction is represented by the imaginary part of the number. Complex numbers can be visualized on the complex plane, with the horizontal axis representing the real part and the vertical axis representing the imaginary part.

When tackling natural logarithms of roots of unity, knowing the structure of complex numbers is crucial. Each root can be expressed as a point on the complex plane, with their position revealing valuable information about their magnitude and direction.

Polar Form Representation

Transitioning to the polar form of complex numbers, it becomes evident why it is so useful for multiplicative and exponential operations. In this form, a complex number \(z\) is not represented by its \(x\) and \(y\) coordinates (or the real and imaginary parts), but rather by its magnitude (or modulus) \(r\) and the angle \(\theta\) it makes with the positive real axis, known as the argument. The polar form is generally written as \(z = r(\text{cos} \theta + i\text{sin} \theta)\) or its equivalent exponential notation \(z = re^{i\theta}\) thanks to Euler's formula.

When finding natural logarithms of roots of unity, polar form proves invaluable. It simplifies multiplication and finding powers by converting these operations into adding angles and multiplying magnitudes, making the calculations more straightforward than in rectangular form.

Roots of Unity

Roots of unity refer to the complex numbers that satisfy the equation \(z^n = 1\) for a given positive integer \(n\). These roots have a significant trait: they are evenly distributed on the unit circle in the complex plane and have an angle of \(2k\text{pi}/n\), where \(k\) is an integer ranging from 0 to \(n-1\). Because of their symmetrical placement, roots of unity often emerge in areas like Fourier analysis and are fundamental in understanding complex polynomial equations.

Let's take an example: the \(n\text{th}\) roots of unity can be represented as \(e^{i(2k\text{pi}/n)}\), where \(k\) is between 0 and \(n-1\). For natural logarithms of these roots, you simply apply the logarithmic identity \(ln(e^z) = z\), which makes it manageable to calculate their logarithms as they have clear expressions in exponential form.