Problem 25 - Precalculus Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

A circle centered at the origin with radius 2, excluding the boundary, represents the set.

Step 1: Understand the Set Notation

The given set \( \{z||z|<2\} \) describes all complex numbers \( z \) such that the modulus (or magnitude) of \( z \) is less than 2.

Step 2: Interpret the Magnitude Condition

The magnitude of a complex number \( z = x + yi \) is given by \(|z| = \sqrt{x^2 + y^2}\). The condition \(|z| < 2\) implies that these numbers lie within a circle centered at the origin (0,0) with a radius of 2.

Step 3: Sketch the Set

On the complex plane, draw a circle centered at the origin with a radius of 2. The set consists of all points inside this circle but not on it, representing all complex numbers whose distance from the origin is less than 2.

Key Concepts

Complex Plane

The complex plane is a two-dimensional plane used to graphically represent complex numbers. Complex numbers have two components: a real part and an imaginary part. This plane, therefore, allows every complex number to be represented as a point. For a complex number of the form \( z = x + yi \), where \( x \) is the real part and \( y \) is the imaginary part, \( x \) is plotted along the horizontal axis (the real axis), and \( y \) is plotted along the vertical axis (the imaginary axis).
The position of a complex number on this plane is unique, hence making it a convenient tool for visualizing operations like addition, subtraction, and more that involve complex numbers. This plane also allows us to convey more intricate concepts such as the set of complex numbers that satisfy certain conditions, like those seen in equations or inequalities.
This tool can be likened to the Cartesian coordinate system from geometry, providing an infinite space to aptly exhibit any complex number or expressions containing them.

Modulus of a Complex Number

The modulus of a complex number is essentially its distance from the origin in the complex plane. For any complex number \( z = x + yi \), its modulus is calculated using the formula:

• \(|z| = \sqrt{x^2 + y^2}\)

This expression draws direct inspiration from the Pythagorean theorem, considering the real component \( x \) and the imaginary component \( y \) as two sides of a right-angled triangle. The hypotenuse, formed by connecting these points to the origin, gives us the modulus upon solving.
The modulus conveys information not only about the magnitude of a complex number but also about its position relative to others when plotted. It tells us "how far," in terms of absolute value, a complex number is from the origin.
Understanding the modulus is crucial, as it aids in evaluating inequalities, comparing complex numbers, and transforming complex operations within the complex plane effectively.

Inequalities in Complex Numbers

Inequalities involving complex numbers can sometimes appear perplexing but are quite manageable once broken down. Complex inequalities involve conditions set on the modulus. In our exercise, the inequality \(|z| < 2\) mandates that we consider only the complex numbers with a modulus, or distance from the origin, less than 2.
One should visualize this as all points residing within a circle of radius 2 centered at the origin on the complex plane. Importantly, this circle includes points within its perimeter but does not incorporate those lying on the circle itself, due to the strict inequality ('less than' rather than 'less than or equal to').
Such inequalities are fundamental in complex analysis, as they describe regions in the complex plane that meet specific criteria. They provide a way to delimit areas and solve problems that necessitate an understanding of complex entities in a bounded realm.