Problem 24 - Advanced Engineering Mathematics

Question

Describe the set of points in the complex plane that satisfies \(|\operatorname{Re}(z)| \leq|z| .\)

Step-by-Step Solution

Answer

All points in the complex plane satisfy the condition.

Step 1: Understand the Problem

We are asked to describe the set of points in the complex plane that satisfy the inequality \(|\operatorname{Re}(z)| \leq|z|\). Here, \(z\) is a complex number written commonly as \(z = x + yi\), where \(x\) is the real part and \(y\) is the imaginary part.

Step 2: Interpret the Modulus and Real Part

The modulus of \(z\), which is \(|z|\), is calculated as \(|z| = \sqrt{x^2 + y^2}\). The real part of \(z\), \(\operatorname{Re}(z)\), is equal to \(x\). The absolute value of the real part is \(|x|\).

Step 3: Set up the Inequality

The inequality \(|\operatorname{Re}(z)| \leq |z|\) translates to \(|x| \leq \sqrt{x^2 + y^2}\). This is the condition we need to analyze.

Step 4: Square Both Sides of the Inequality

To simplify \(|x| \leq \sqrt{x^2 + y^2}\), square both sides to obtain \(x^2 \leq x^2 + y^2\).

Step 5: Simplify the Inequality

From squaring, the simplified inequality is \(0 \leq y^2\). This is always true because \(y^2\) represents a square of a real number, which is always non-negative.

Step 6: Determine the Set of Points

Since the inequality \(0 \leq y^2\) is always satisfied for all real numbers \(y\), the condition \(|x| \leq \sqrt{x^2 + y^2}\) holds for all points in the complex plane.

Key Concepts

Complex Numbers

Complex numbers are an essential part of mathematics, often appearing in a form like \( z = x + yi \). The two components of a complex number are the real part \( x \) and the imaginary part \( y \), where \( i \) is the imaginary unit with the property \( i^2 = -1 \).

Complex numbers offer a more comprehensive number system that includes not only real numbers but also solutions to equations that have no real solutions.

In the context of the complex plane, each complex number corresponds to a unique point. The horizontal axis represents the real part, while the vertical axis represents the imaginary part.

This representation allows us to treat complex numbers geometrically, making it easier to see relationships, perform operations, and understand constraints like inequalities.

Inequalities in Complex Analysis

Inequalities in complex analysis often involve determining specific regions within the complex plane. These regions represent sets of complex numbers that satisfy a given inequality.

For example, consider the inequality \(|\operatorname{Re}(z)| \leq |z|\). Translating this into components, it means that the magnitude of the real part of \( z \) should not exceed the magnitude of the complex number itself. This condition helps in visualizing a broad region since it does not impose a strict limitation, allowing any point in the complex plane.

Solving such inequalities often involves techniques like expressing the inequality using Cartesian coordinates (\( x \) and \( y \)), simplifying, and interpreting the regions that these expressions describe.

Modulus and Real Part

The modulus and real part of a complex number are key concepts that help in understanding and analyzing complex numbers.

The modulus \( |z| \) of a complex number \( z = x + yi \) is calculated as \( |z| = \sqrt{x^2 + y^2} \). It represents the distance of the point \( z \) from the origin in the complex plane.

Meanwhile, the real part \( \operatorname{Re}(z) \) is simply \( x \), the horizontal coordinate when visualizing the number on the complex plane.

Understanding these components is crucial when analyzing inequalities. For instance, in the original exercise, comparing \(|\operatorname{Re}(z)|\) and \(|z|\) helps determine spatial relationships and constraints on \( z \) within the plane.