Problem 12 - Complex Analysis for Mathematics and Engineering

Question

Show that if \(z \neq 0\), the four points \(z, i z,-z\), and \(-i z\) are the vertices of a square with its center at the origin.

Step-by-Step Solution

Answer

The points form a square with its center at the origin, as all sides and diagonals are equal.

Step 1: Define the Points

Let's express the given complex numbers as points in the complex plane. The points are: \( z \), \( i z \), \( -z \), and \( -i z \). These correspond to the complex numbers: \((x, y)\), \((-y, x)\), \((-x, -y)\), and \((y, -x)\), assuming \(z = x + iy\).

Step 2: Calculate Distances

To verify the points form a square, we need to check that all sides have equal length and the diagonals are equal. The distance between any two complex numbers \(a = (x_1, y_1)\) and \(b = (x_2, y_2)\) is given by \(\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}\).Calculate the distance between consecutive points: - \(z \to iz\): \(\sqrt{(-y-x)^2 + (x-y)^2} = \sqrt{x^2 + y^2}\)- \(iz \to -z\): \(\sqrt{(-x+y)^2 + (x+y)^2} = \sqrt{x^2 + y^2}\)- \(-z \to -iz\): \(\sqrt{(-y-x)^2 + (x-y)^2} = \sqrt{x^2 + y^2}\)- \(-iz \to z\): \(\sqrt{(-x+y)^2 + (x+y)^2} = \sqrt{x^2 + y^2}\)

Step 3: Check Diagonals

Next, verify that the diagonals are equal. Compute the distances:- \(z \to -z\): Distance is \(\sqrt{(x+x)^2 + (y+y)^2} = \sqrt{4x^2 + 4y^2} = 2\sqrt{x^2 + y^2}\)- \(iz \to -iz\): Distance is \(\sqrt{(-y+y)^2 + (x-x)^2} = \sqrt{4x^2 + 4y^2} = 2\sqrt{x^2 + y^2}\)Since the diagonals are equal and twice the length of the sides, the quadrilateral is a square.

Step 4: Verify Center at Origin

The diagonals of a square intersect at their midpoints, which is the center of the square. The midpoint of each diagonal is obtained as:- Midpoint of \(z\) and \(-z\) is \(\left(\frac{x-x}{2}, \frac{y-y}{2}\right) = (0, 0)\).- Midpoint of \(iz\) and \(-iz\) is \(\left(\frac{-y+y}{2}, \frac{x-x}{2}\right) = (0, 0)\).Thus, the center of the square is at the origin.

Key Concepts

Complex Plane

The complex plane is a two-dimensional plane used to graphically represent complex numbers. It combines a real axis (horizontal) and an imaginary axis (vertical). Each complex number corresponds to a unique point on this plane.
Imagine the real part of a complex number as its position along the x-axis and the imaginary part as its position along the y-axis.

• The real component of a complex number is shown horizontally.
• The imaginary component is shown vertically.

When given a complex number, such as \( z = x + iy \), the point \( (x, y) \) on the complex plane represents this number. Moving or rotating these points, like using multiplication by \( i \) (which rotates points 90 degrees counterclockwise), can significantly simplify operations within complex number computations. By visualizing complex numbers on this plane, mathematical problems often become easier to understand and solve.

Vertices of a Square

To understand why the points \( z, iz, -z, -iz \) create the vertices of a square, imagine these points as existing on the complex plane. A square in geometrical terms is a shape with:

• Four sides of equal length.
• Four right angles.

What makes \( z, iz, -z, -iz \) form a square is their position in relation to each other:

• Each point \( z, iz, -z, -iz \) is an equal distance \( \sqrt{x^2 + y^2} \) from its neighboring point.
• The rotation operation (multiplying by \( i \)) effectively ensures right angles, rotating any vector by 90 degrees.
• If the first point \( z \) is aligned with the x-axis, multiplying by \( i \) rotates it to align with the y-axis, effectively forming 90-degree angles between each side.

This symmetry and equal spacing support the fact that these points are indeed vertices of a square, neatly organized around the origin.

Distance Formula

The distance formula is crucial for verifying the structure of squares, among many other shapes in geometry. On the complex plane, the distance between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:\[\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]
For a set of points to form a square, two conditions must be fulfilled using this formula:

• All sides between consecutive points must be of equal length.
• The diagonals must be equal and longer than the sides, ideally \( \sqrt{2} \) times a side's length.

In the exercise with the vertices \( z, iz, -z, -iz \), these conditions are met because each pair of consecutive vertices has the same computed distance, \( \sqrt{x^2 + y^2} \). Meanwhile, the diagonals \( z \) to \( -z \), and \( iz \) to \( -iz \), are \( 2\sqrt{x^2 + y^2} \).
Understanding the distance formula enables verification of the geometric properties necessary for verifying squares and other similar shapes on the complex plane.