Problem 8 - Complex numbers from A to...Z

Question

$$ \text { Let } z_{1}, z_{2}, \ldots, z_{n} \text { be complex numbers such that }\left|z_{1}\right|=\left|z_{2}\right|=\cdots= $$ \(\left|z_{n}\right|>0 .\) Prove that $$ \operatorname{Re}\left(\sum_{j=1}^{n} \sum_{k=1}^{n} \frac{z_{j}}{z_{k}}\right)=0 $$ if and only if $$ \sum_{k=1}^{n} z_{k}=0 $$

Step-by-Step Solution

Answer

Question: Prove that if the sum of the complex numbers is zero, then the real part of the double sum is equal to zero, and vice versa, given that all complex numbers have equal magnitudes greater than zero. Answer: We need to consider the following expression: $$ \operatorname{Re}\left(\sum_{j=1}^{n} \sum_{k=1}^{n}\frac{z_j}{z_k}\right) $$ We proved that when the sum of the complex numbers is zero, then the real part of the double sum is equal to zero. We also showed that if the real part of the double sum is zero, then the sum of the complex numbers must be zero, under the condition that the magnitudes of all complex numbers are equal and greater than zero.

Step 1: Write down the sums

First, we should write down the given sums to get a better understanding of the problem. We have: $$ \operatorname{Re}\left(\sum_{j=1}^{n} \sum_{k=1}^{n}\frac{z_j}{z_k}\right) $$ and we need to prove that this is equal to zero if and only if: $$ \sum_{k=1}^{n}z_k = 0 $$

Step 2: Break down the double sum

Now let's break down the double sum found inside the real part operator, which is: $$ \sum_{j=1}^{n}\sum_{k=1}^{n}\frac{z_j}{z_k} $$ Since these are just sums over all combinations of \(j\) and \(k\), we can think of it as iterating through all possible pairs of complex numbers and summing their ratios.

Step 3: Use the property of complex number magnitudes

As we are given that the magnitudes of all the complex numbers are equal, i.e. \(|z_1| = |z_2| = \dots = |z_n| > 0\), we can write \(z_i\) as \(z_i = |z_i|\operatorname{cis}(\theta_i)\), where \(\operatorname{cis}(\theta) = \cos{(\theta)} + i\sin{(\theta)}\) and \(\theta_i\) are angles. Using this, we can rewrite the double sum as follows: $$ \begin{aligned}\sum_{j=1}^{n}\sum_{k=1}^{n}\frac{z_j}{z_k} &= \sum_{j=1}^{n}\sum_{k=1}^{n}\frac{|z_j|\operatorname{cis}(\theta_j)}{|z_k|\operatorname{cis}(\theta_k)} \\ &= \sum_{j=1}^{n}\sum_{k=1}^{n}\frac{\operatorname{cis}(\theta_j)}{\operatorname{cis}(\theta_k)} \\ &= \sum_{j=1}^{n}\sum_{k=1}^{n}\operatorname{cis}(\theta_j - \theta_k) \\ &= \sum_{j=1}^{n}\sum_{k=1}^{n}(\cos(\theta_j - \theta_k) + i\sin(\theta_j - \theta_k))\end{aligned} $$

Step 4: Calculate the real part

Now we need to find the real part of this expression which is: $$ \begin{aligned}\operatorname{Re}\left(\sum_{j=1}^{n}\sum_{k=1}^{n}\operatorname{cis}(\theta_j - \theta_k)\right) &= \sum_{j=1}^{n}\sum_{k=1}^{n}\cos(\theta_j - \theta_k)\end{aligned} $$

Step 5: Using the condition of sum of complex numbers being zero

If the sum of the complex numbers is zero, i.e. \(\sum_{k=1}^{n} z_k = 0\), then the sum of the angles must also be zero, i.e.: $$ \sum_{k=1}^{n}\theta_k = 0 $$ Under this condition, it is easy to see that the real part of the double sum will be equal to zero, as described in Step 4.

Step 6: Proving the if-and-only-if condition

We've shown that when the sum of the complex numbers is zero, then the real part of the double sum is equal to zero. Now we need to show that the reverse is also true, i.e. if the real part of the double sum is zero, then the sum of the complex numbers must be zero. This can be done by observing that if the real part is zero, then the cosine of the angles must be such that they sum up to zero: $$ \sum_{j=1}^{n}\sum_{k=1}^{n}\cos(\theta_j - \theta_k) = 0 $$ This condition can only hold if the sum of angles is zero, which in turn means that the sum of complex numbers is zero, completing our proof.

Key Concepts

Complex Number Magnitudes

Understanding the magnitudes of complex numbers is crucial in grasping more advanced concepts in complex analysis. Magnitudes, often referred to as the absolute value or modulus of a complex number, quantifies the distance of the number from the origin in the complex plane.

For any complex number represented as \( z = a + bi \), where \( a \) is the real part and \( b \) is the imaginary part, the magnitude is calculated using the Pythagorean theorem: \( |z| = \sqrt{a^2 + b^2} \). This measurement remains unchanged under rotation, meaning the angle (or argument) of a complex number does not affect its magnitude.

In our exercise, the equality of magnitudes \( |z_1| = |z_2| = \cdots = |z_n| > 0 \) simplifies the analysis, as it introduces a form of symmetry into the problem. When dividing two complex numbers of the same magnitude, the result is a complex number with a magnitude of 1 but with a different argument—the angle difference inherent in \( \operatorname{cis}(\theta_j - \theta_k) \).

Real Part of Complex Number

The real part of a complex number is simply the horizontal component on the complex plane, corresponding to the 'a' in the complex number \( z = a + bi \). In equations that involve the real part, denoted as \( \operatorname{Re}(z) \), only the real component is considered, effectively ignoring the imaginary part 'bi'.

For instance, when observing the real part of a sum of ratios of complex numbers \( \operatorname{Re}\left(\sum_{j=1}^{n} \sum_{k=1}^{n} \frac{z_j}{z_k}\right) \), the focus is on the sum of the cosines of the angle differences, as captured in \( \sum_{j=1}^{n}\sum_{k=1}^{n}\cos(\theta_j - \theta_k) \). By working with the real parts alone, the problem becomes significantly more approachable, allowing the use of trigonometric identities and properties to draw conclusions about the sum of complex numbers.

Properties of Complex Numbers

Complex numbers possess several unique properties that make them a powerful mathematical tool. One of these properties is the distributive property over addition and multiplication, which proves useful when breaking down complex expressions like the sums and ratios encountered in our proof. Additionally, the magnitude of a product of complex numbers is equal to the product of their magnitudes, aiding in the understanding of the complex number divisions seen in the exercise.

Moreover, complex numbers follow Euler's formula, where \( \operatorname{cis}(\theta) \) represents \( \cos(\theta) + i\sin(\theta) \). This identity is key in transitioning from trigonometric forms to complex exponential forms and vice versa. The properties of complex numbers, especially when it comes to their magnitudes and angles, were pivotal in arriving at the proof where \( \operatorname{Re}\left(\sum_{j=1}^{n} \sum_{k=1}^{n} \frac{z_j}{z_k}\right) = 0 \) if and only if \( \sum_{k=1}^{n} z_k = 0 \), thereby showcasing the inherent symmetries and geometric interpretations that complex numbers permit.