Question

Show that the sum of the \(n\) nth roots of any complex number is zero.

Short answer

The sum of the nth roots of any complex number is zero because the geometric series formed by these roots sums to 0.

Step-by-step solution

Representing the Complex Number

Consider a complex number in its polar form: \[ z = re^{i\theta} \]where \( r \) is the magnitude and \( \theta \) is the argument (angle) of the complex number.

Finding the nth Roots

The nth roots of the complex number are given by: \[ z_k = r^{1/n} e^{i(\theta + 2k\pi)/n} \]for \( k = 0, 1, 2, \ldots, n-1 \).

Summing the nth Roots

To find the sum of all nth roots, we need to sum the expression from Step 2 over all values of \( k \): \[ \sum_{k=0}^{n-1} z_k = \sum_{k=0}^{n-1} r^{1/n} e^{i(\theta + 2k\pi)/n} \]

Setting Up a Geometric Series

Notice that the terms form a geometric series with \( n \) terms. The common ratio of this series is \( e^{i 2\pi /n} \). Recall the sum of a geometric series \( 1 + x + x^2 + \cdots + x^{n-1} \) is given by \[ S = \frac{1 - x^n}{1 - x} \]

Calculating the Sum of the Geometric Series

For our series, \( x = e^{i 2\pi /n} \) and \( x^n = e^{i 2\pi} = 1 \). Thus, the sum of the series is: \[ S = \frac{1 - e^{i 2\pi}}{1 - e^{i 2\pi /n}} = \frac{1 - 1}{1 - e^{i 2\pi /n}} = 0 \]

Concluding the Proof

Since the sum of the nth roots of any complex number forms a geometric series whose sum is 0, it follows that \[ \sum_{k=0}^{n-1} z_k = 0 \]. Therefore, the sum of the nth roots of any complex number is zero.

Key concepts

complex numbers

Complex numbers are numbers that have both a real part and an imaginary part. They are usually written in the form:

• Real Part: The real component. For example, in 3 + 4i, the number 3 is the real part.
• Imaginary Part: The imaginary component, which is a multiple of the imaginary unit 'i'. In 3 + 4i, the number 4i is the imaginary part.

Complex numbers are very powerful in mathematics and engineering, as they allow for the description of rotations and oscillations. They can be represented in Cartesian form or, as seen in this exercise, in polar form. In polar form, a complex number is expressed as: \[ z = re^{i\theta} \]Where 'r' is the magnitude (distance from the origin) and 'θ' is the argument (angle from the positive real axis).Understanding complex numbers and their polar representation is crucial for grasping advanced topics such as nth roots.

nth roots

Finding the nth roots of a complex number involves determining 'n' distinct complex numbers that, when raised to the 'nth' power, return the original number. This task is essential in many fields, including signal processing and quantum mechanics.When a complex number is in polar form, its nth roots can be found using the formula: \[ z_k = r^{1/n} e^{i(\theta + 2k\text{π})/n} \]Here, 'k' runs from 0 to n-1, providing 'n' distinct roots.

• The term \( r^{1/n} \) gives the magnitude of each root.
• The expression itself represents the angle for each root.

Finding nth roots in this way helps break down complex problems and is useful for understanding periodic functions and waves.

geometric series

To understand the sum of the nth roots of a complex number, we need to recognize that the roots form a geometric series. A geometric series is a sequence of terms where each term is a constant multiple of the previous one. For example, in sequence \( 1, x, x^2, x^3, \ldots, x^{n-1} \), the common ratio is 'x'. For our nth roots problem, the common ratio is \( e^{i 2π /n} \).The sum of the first 'n' terms of a geometric series can be calculated using the formula: \[ S = \frac{1 - x^n}{1 - x} \]When applied to our problem, with 'n' nth roots, this formula simplifies to: \[ S = \frac{1 - e^{i 2π}}{1 - e^{i 2π /n}} = 0 \] Because \( e^{i 2π} \) is 1, the numerator becomes 0, proving that the sum of the nth roots is zero.

polar form

The polar form of a complex number is a way of expressing it using its magnitude and angle. This representation is particularly useful for multiplication and finding roots. A complex number in polar form looks like: \[ z = re^{i\theta} \]Where:

• 'r' is the magnitude and represents the distance from the origin.
• 'θ' is the argument and represents the angle formed with the positive real axis.

Polar form simplifies many operations that are cumbersome in Cartesian form. For instance, multiplying two complex numbers in polar form is straightforward: \[ z_1 z_2 = r_1 r_2 e^{i(\theta_1 + \theta_2)} \]By understanding and utilizing polar form, finding roots becomes a systematic process, paving the way for solving advanced mathematical problems.