Problem 12 - Complex numbers from A to...Z

Question

Let \(a, b, c, d\) be distinct complex numbers with \(|a|=|b|=|c|=|d|\) and \(a+b+c+d=0\). Then the geometric images of \(a, b, c, d\) are the vertices of a rectangle.

Step-by-Step Solution

Answer

Question: Prove that when four distinct complex numbers of the same magnitude are summed and equal to zero, their representations in the complex plane form the vertices of a rectangle. Solution: We have shown that when the sum of four distinct complex numbers of the same magnitude is zero, the differences between alternate vertices are either purely real or purely imaginary. This implies that the sides of the polygon formed by these complex numbers are parallel to either the real or imaginary axis, which confirms that the resulting polygon is a rectangle.

Step 1: Represent the given information in the problem

First, let's represent the given information. We know that \(|a|=|b|=|c|=|d|\) and \(a+b+c+d=0\). We can represent this geometrically as follows: 1. Plot the complex numbers \(a, b, c\), and \(d\) in the complex plane. 2. Connect consecutive points to form a polygon. We must now prove that this polygon is a rectangle.

Step 2: Consider the difference between two complex numbers

Let's consider the difference between two complex numbers \(a-c\) and \(b-d\). Taking the difference will help us determine the relationship between the pairs of points we will need to prove our conclusion. We know that \(a+b+c+d=0\). If we subtract \(c\) and \(d\) from both sides of the equation, we get: \(a - c = -(b - d)\) It will be helpful to consider the magnitudes of \(a - c\) and \(b - d\).

Step 3: Calculate the magnitudes of the differences

To calculate the magnitudes, we can simply multiply the complex number by its conjugate. We have: \(|(a-c)(a-c)^*| = |(b-d)(b-d)^*|\) Since \(|z_1z_2| = |z_1||z_2|\) for all complex numbers \(\displaystyle z_1\) and \(\displaystyle z_2\), we can also write: \(|(a - c)||a - c| = |b - d||b - d|\) Since \(|a|=|c|\) and \(|b|=|d|\), we can write: \(|a - c|^2 = |b - d|^2\)

Step 4: Prove that the differences are either purely real or purely imaginary

At this point, we have shown that the square of the magnitudes of the differences are equal. Now we must prove that either \(\displaystyle a-c\) and \(\displaystyle b-d\) are purely real or purely imaginary. Recall that \(\displaystyle a-c\) and \(\displaystyle b-d\) are negatives of each other. Thus, the products ``(a-c)(b-d)`` and ``(a-c)(d-b)`` are equal, so \((a-c)(b-d) = (a-c)(d-b)\). Expanding this expression, we obtain: \(ab - (a+d) c + cd = ad - (a+c)b + cd\) This simplifies to: \((a+b)(c+d)=0\) Since \(a+b+c+d=0\), we know that \(a+b=-(c+d)\), which means either \(\displaystyle a-c\) and \(\displaystyle b-d\) are purely real, or they are purely imaginary.

Step 5: Conclude that the polygon is a rectangle

We have shown that the differences between alternate vertices are either purely real or purely imaginary, implying that the sides of the polygon are either parallel to the real or imaginary axis. Therefore, we can conclude that the polygon formed by the complex numbers \(a, b, c\), and \(d\) is a rectangle.

Key Concepts

Complex Plane

The complex plane, also known as the Argand plane, is a two-dimensional plane used to visually represent complex numbers. The horizontal axis, typically called the real axis, represents the real part of a complex number, while the vertical axis, known as the imaginary axis, represents the imaginary part. Complex numbers are plotted much like coordinates in a Cartesian system, with the position \(x, y\) corresponding to the complex number \(x + yi\).

Through the complex plane, various geometric concepts apply to complex numbers, much like they do to real numbers in the Cartesian plane. For example, consider the problem of showing that the complex numbers \(a, b, c, d\), given certain conditions, are vertices of a rectangle. By meticulously plotting these points on the complex plane and examining their relationship through complex arithmetic, one can conclude the geometrical shape that these points form. In the given exercise, since the sum of them is zero and their magnitudes are equal, this implies a symmetry that hints towards a rectangle in the complex plane.

Magnitude of Complex Numbers

The magnitude (or absolute value) of a complex number \(z\) represented as \(z = x + yi\) is a measure of its distance from the origin in the complex plane, akin to the length of a vector in physics. It is calculated as \(\sqrt{x^2 + y^2}\) and is denoted as \(|z|\). An essential property of magnitudes is that for any two complex numbers \(z_1\) and \(z_2\), the magnitude of their product is the product of their magnitudes: \(|z_1z_2| = |z_1||z_2|\).

In the context of our exercise, recognizing that all given complex numbers have the same magnitude allows us to understand that they are equidistant from the origin. This equidistance is a strong geometric clue that supports the conclusion of them being vertices of a rectangle when composed with additional arithmetic properties like the sum equating to zero.

Conjugate of Complex Numbers

The conjugate of a complex number is crucial in various complex number operations. For a complex number \(z = x + yi\), the conjugate is \(z^* = x - yi\). Notably, multiplying a complex number by its conjugate gives a real number \(z z^* = (x + yi)(x - yi) = x^2 + y^2\), which is the square of its magnitude: \(|z|^2\).

This property is of particular interest in the exercise at hand. By multiplying the differences of complex numbers by their conjugates and comparing their magnitudes, it facilitates the proof that the vertices form a rectangle. Knowing that \(|(a-c)(a-c)^*|\) is equal to \(|(b-d)(b-d)^*|\) emphasizes that opposite sides of the polygon have equal lengths. Since conjugation preserves magnitude, it helps verify the equality of the relationships between points, ultimately contributing to the geometric proof.