Problem 2 - Advanced Engineering Mathematics

Question

In Problems 1-8, sketch the graph of the given equation. $$ \operatorname{Im}(z)=-2 $$

Step-by-Step Solution

Answer

The graph is a horizontal line at \( y = -2 \) in the complex plane.

Step 1: Understand the Equation

The equation given is \( \operatorname{Im}(z) = -2 \), which describes points in the complex plane. In terms of complex numbers \( z = x + yi \), it means the imaginary part, \( y \), is always \(-2\).

Step 2: Interpret in Complex Plane

The complex number \( z \) is expressed as \( x + yi \), where \( x \) is the real part and \( y \) is the imaginary part. The equation \( \operatorname{Im}(z) = -2 \) indicates that all points have their imaginary component fixed at \(-2\).

Step 3: Plot Horizontal Line

To graph this equation, plot a horizontal line across the complex plane. Since the imaginary part \( y = -2 \), the line is parallel to the real axis (\( x \)-axis) and passes through the point \( (0, -2) \) on the imaginary axis.

Step 4: Sketch the Graph

Draw the horizontal line which cuts through all points with the imaginary part \(-2\). For all real numbers \( x \), corresponding complex numbers can be \( z = x - 2i \).

Key Concepts

Imaginary Part

When dealing with complex numbers, understanding the imaginary part is crucial. A complex number is represented as \( z = x + yi \), where \( x \) is the real part and \( y \) is the imaginary part, attached with the imaginary unit \( i \). The imaginary unit \( i \) is defined by the property \( i^2 = -1 \).

In the context of the given problem, \( \operatorname{Im}(z) = -2 \) specifies that the imaginary part \( y = -2 \). This essentially means that every point we're dealing with in the complex plane will have a vertical position corresponding to \( y = -2 \). Any variation will only occur in the real part of the complex number, indicating a horizontal arrangement.

Complex Numbers

Complex numbers can be visualized as points or vectors in a two-dimensional plane known as the complex plane. This plane is structured much like the Cartesian coordinate system with one axis for the real parts and another perpendicular axis for the imaginary parts.

Each complex number can be expressed as a combination of a real component and an imaginary component: \( z = x + yi \). The real number \( x \) dictates the position on the horizontal axis, while \( y \), the imaginary part, decides the vertical position along the imaginary axis.

• **Real part**: Located on the horizontal axis, denoted by \( x \).
• **Imaginary part**: Values along the vertical axis, denoted by \( y \).

In our example, all the complex numbers will have the form \( z = x - 2i \), implying that no matter what real value \( x \) takes, the imaginary part remains fixed at \(-2\). This forms a horizontal linear pattern on the complex plane.

Horizontal Line in Complex Plane

A horizontal line in the complex plane signifies that the imaginary part for all complex numbers on the line remains constant. Since the equation is \( \operatorname{Im}(z) = -2 \), the line is horizontal and parallel to the real axis and crosses the imaginary axis at \( -2 \).

This results in:

• A straight line where every point has exactly the same imaginary component \( y = -2 \).
• The line is not restricted by the real part \( x \), meaning it stretches infinitely left and right.

The concept of horizontal lines helps in simplifying the understanding of how complex numbers interact on a visual level. It separates the influence of the imaginary component from that of the real component, allowing us to see complex relationships clearly in a static view. This visualization becomes particularly relevant when trying to understand operations such as addition or multiplication on complex numbers, where it’s important to manage the impacts on both the real and imaginary plane.