Question

Describe geometrically the set of points in the complex plane satisfying the following equations. Re \(z>2\)

Short answer

The set of points is the half-plane to the right of the vertical line \( x = 2 \) in the complex plane.

Step-by-step solution

Understand the Complex Number Representation

A complex number is generally represented as \( z = x + yi \) where \( x \) is the real part and \( y \) is the imaginary part.

Interpret the Given Condition

The given condition is \( \text{Re}(z) > 2 \). This means we are interested in the set of complex numbers whose real part is greater than 2.

Express the Condition in Terms of Real and Imaginary Parts

For the complex number \( z = x + yi \), the real part is \( x \). Therefore, the condition \( \text{Re}(z) > 2 \) can be rewritten as \( x > 2 \).

Describe the Set of Points Geometrically

The inequality \( x > 2 \) represents a vertical line in the complex plane at \( x = 2 \). The set of points satisfying \( x > 2 \) are all the points to the right of this vertical line.

Interpret the Geometric Representation

Thus, geometrically, the set of points satisfying \( \text{Re}(z) > 2 \) is the half-plane to the right of the vertical line \( x = 2 \) in the complex plane.

Key concepts

complex plane

The complex plane is a fundamental tool for visualizing complex numbers. Imagine a standard Cartesian coordinate system. The horizontal axis, commonly known as the x-axis, represents the real part of a complex number. The vertical axis, or y-axis, represents the imaginary part. In this setup, each point in the plane corresponds to a complex number. For instance, the point \( (3, 4) \) represents the complex number \( 3 + 4i \). By using this plane, we can easily understand and manipulate complex numbers geometrically.

Complex numbers are plotted as points in this plane, which makes it easier to apply geometric interpretations to problems involving them. This introduces a strong visual element into the study of complex numbers, aiding in comprehension and solution of problems.

real part of complex number

Each complex number \( z = x + yi \) has a real part (x) and an imaginary part (y). The real part, represented as \( x \), is what we find on the horizontal axis (or x-axis) of the complex plane. In the context of the exercise, we focused on numbers where the real part is greater than 2, so we described it as \( x > 2 \).

In regular math, we might see inequalities involving real numbers, and we handle them by finding values that satisfy the condition. For complex numbers it's somewhat similar. The real part of a complex number determines its position along the horizontal axis. This allows us to define regions of the complex plane satisfying certain conditions related only to their real parts.

So, when we say \( \text{Re}(z) > 2 \), we mean we are only interested in points (complex numbers) where part of the value is strictly greater than 2 on the horizontal axis.

geometric interpretation

Geometric interpretation involves visualizing complex numbers on the complex plane. Let's follow the interpretation for the problem in the exercise:

• The condition given was \( \text{Re}(z) > 2 \).
• This tells us that we're looking for points (complex numbers) where the real part is more than 2.
• On the complex plane, the real part corresponds to the x-axis. Drawing the line \( x = 2 \) allows us to visualize which points have real parts greater than 2.
• The area to the right side of this line (going infinitely up and down) represents all points where \( \text {Re}(z) > 2 \).

Hence, geometrically, this means that our solution is a half-plane, specifically the region to the right of the vertical line where \( x = 2 \). Complex numbers satisfying this condition hence lie within this half-plane.
A visual approach can make such problems easier and more intuitive to grasp.