Problem 11 - Complex Analysis for Mathematics and Engineering

Question

Show that if \(z \neq 0\), the four points \(z\), \(\bar{z},-z\), and \(-\bar{z}\) are the vertices of a rectangle with its center at the otigin.

Step-by-Step Solution

Answer

The points form a rectangle centered at the origin by symmetry and equal diagonal lengths.

Step 1: Understand the Points and their Relations

The points in the complex plane are defined as \(z\), its conjugate \(\bar{z}\), \(-z\), and \(-\bar{z}\). These points are related through reflection across the real and imaginary axes.

Step 2: Calculate Distances Between Points

Calculate the distances between the pairs of points: \(z\) and \(\bar{z}\), \(z\) and \(-z\), \(\bar{z}\) and \(-\bar{z}\), and so on. Note that all diagonal pairs will have equal lengths, as they are reflections across either the real or imaginary axis, implying \(|z - \bar{z}| = |-z + \bar{z}|\) and \(|z + \bar{z}| = |z - (-\bar{z})|\).

Step 3: Check Right Angles

Verify that each angle between consecutive points (e.g., \(\angle z\bar{z}\), \(\angle \bar{z}-z\)) is a right angle. The vectors connecting these points form perpendicular bisectors, confirming the right angles of a rectangle.

Step 4: Verify Center at Origin

The symmetry of these points around the origin can be seen from their equations: \(z\) and \(-z\) are reflections across the origin, as are \(\bar{z}\) and \(-\bar{z}\). The origin \((0,0)\) serves as the midpoint of these reflections, thus is the center of the rectangle.

Key Concepts

Complex Numbers

Complex numbers are fascinating mathematical objects that can represent quantities in two dimensions. A complex number is usually denoted by the letter \(z\) and can be represented as \(z = a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit with the property \(i^2 = -1\).

These numbers are visualized in the complex plane, a coordinate system where the horizontal axis represents the real part \(a\) and the vertical axis represents the imaginary part \(bi\). Each complex number corresponds to a point in this plane. This plane helps in visualizing operations on complex numbers, such as addition, subtraction, or finding magnitudes, called modulus, and can even include complex conjugation.

The complex conjugate of a number \(z = a + bi\) is denoted by \(\bar{z}\) and is defined as \(\bar{z} = a - bi\). The conjugate geometrically reflects the point about the real axis.

Reflection in the Complex Plane

Reflection in the complex plane is a geometric transformation that allows us to understand symmetries and properties of complex numbers more deeply. When you take a complex number \(z = a + bi\) and find its conjugate \(\bar{z} = a - bi\), you are performing a reflection of \(z\) across the real axis. This mirrors the point from above the axis to below it or vice versa.

This concept can be extended further.

• Reflection of \(z\) to obtain \(-z = -a - bi\) mirrors the point through the origin.
• Reflection of \(\bar{z}\) yields \(-\bar{z} = -a + bi\), which reflects across both the real and imaginary axes.

These reflections are integral when analyzing symmetric properties of complex-number shapes, such as the rectangle formed in the given exercise.

Geometry of Complex Numbers

The geometry of complex numbers opens up a world where algebra meets geometry, interacting richly in the complex plane. This can be particularly intriguing when investigating specific shapes, like the rectangle formed from \(z\), \(\bar{z}\), \(-z\), and \(-\bar{z}\).

To visualize these as vertices of a rectangle, note:

• These points are symmetric about the origin.
• Equal distances between diagonal points (e.g., \(|z - \bar{z}|\) and \(|-z + \bar{z}|\)) show that opposite sides are equal and parallel, typical for rectangles.
• The angles formed at these vertices are right angles. When two vectors from the center form a right angle, their dot product is zero, confirming our rectangle has right angles.

The rectangle centered at the origin exhibits perfect symmetry due to the nature of reflections in complex numbers, bridging the study of algebraic properties with geometric representation.