Problem 25 - Algebra and Trigonometry

Question

Sketch the set in the complex plane. $$ \\{z| | z |<2\\} $$

Step-by-Step Solution

Answer

The set is the open disk centered at the origin with radius 2, excluding the boundary.

Step 1: Identify the Set Description

The set given is \( \{ z | |z| < 2 \} \), meaning we are interested in all complex numbers \( z \) with magnitude less than 2.

Step 2: Understand Magnitude in the Complex Plane

The magnitude \(|z|\) of a complex number \(z = x + yi\) is given by \(|z| = \sqrt{x^2 + y^2} \). This represents the distance from \(z\) to the origin \((0,0)\) in the complex plane.

Step 3: Interpret the Set Condition

The inequality \(|z| < 2\) implies that the distance from the origin to \(z\) is less than 2, meaning \( \sqrt{x^2 + y^2} < 2 \).

Step 4: Sketch the Region in the Complex Plane

The set \( \{ z | |z| < 2 \} \) represents a circle centered at the origin \( (0,0) \) with radius 2, but as it's a strict inequality, we exclude the boundary of the circle. Shade the interior of the circle without the edge to represent this set.

Key Concepts

Complex Plane

The complex plane is a two-dimensional plane where all complex numbers are represented geometrically. Imagine it as a flat sheet where every point corresponds to a complex number. The horizontal axis is called the real axis and represents the real part of the complex numbers. The vertical axis, known as the imaginary axis, represents the imaginary part. Each complex number can be plotted on this plane using its real and imaginary parts, much like plotting points on a Cartesian coordinate system.

To find where a complex number belongs in the complex plane, you simply take its real part, move along the real axis, and take its imaginary part, moving along the imaginary axis. These movements from the origin (0,0) lead you to the specific point that represents the complex number.

Visualizing the complex plane is helpful for understanding operations on complex numbers such as addition, subtraction, and finding magnitudes and angles.

Magnitude of Complex Numbers

The magnitude of a complex number is a measure of its size or length. For a complex number represented as \( z = x + yi \), its magnitude is the distance from this point to the origin \((0,0)\) of the complex plane. This can be visualized as the length of the line segment connecting the origin and the point \(z\) on the complex plane.

To calculate this magnitude, use the formula

• \(|z| = \sqrt{x^2 + y^2}\)

This comes from applying the Pythagorean theorem, where \(x\) and \(y\) are legs of a right triangle, and the magnitude \(|z|\) is the hypotenuse.

Understanding the magnitude is crucial as it helps in determining how far the number is from the origin. It also plays a vital role in complex number operations and transformations, including rotations and scaling.

Inequality in Complex Numbers

Inequalities involving complex numbers can sometimes be tricky, as they often relate to their magnitude rather than their real or imaginary parts separately. For example, the inequality \(|z| < 2\) defines a set of all complex numbers whose magnitudes are less than 2.

In this case, the inequality \(|z| < 2\) implies that in the complex plane, these numbers lie strictly inside a circle of radius 2 centered at the origin. This means every point inside the circle satisfies the inequality, but points exactly on the perimeter do not because the inequality is strict (< rather than ≤).

Inequalities like these can be visualized as specific regions in the complex plane. Understanding how they define areas helps in sketching sets and analyzing complex number conditions. They are an important tool in complex analysis, helping to articulate and solve problems involving limits, convergence, and other advanced concepts.