Problem 2 - Advanced Engineering Mathematics

Question

Sketch the graph of the given equation. $$ \operatorname{Im}(z)=-2 $$

Step-by-Step Solution

Answer

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

Step 1: Understand the Context

The equation given is \( \operatorname{Im}(z) = -2 \). In complex numbers, the imaginary part is the component multiplied by the imaginary unit \( i \). This tells us that in the complex plane (Argand diagram), any complex number \( z \) that satisfies this has its imaginary part constant at \( -2 \).

Step 2: Identify the Graph Type

For any complex number \( z = a + bi \), the imaginary part is \( b \). The equation \( \operatorname{Im}(z) = -2 \) means that \( b = -2 \). This results in a horizontal line on the Argand diagram at \( y = -2 \).

Step 3: Sketch the Graph

To sketch the graph, draw a horizontal line on the complex plane (Argand diagram) that intersects the imaginary axis at \( y = -2 \). This line runs parallel to the real axis, representing all complex numbers where the imaginary part is \( -2 \).

Step 4: Describe the Graph Characteristics

The graph of \( \operatorname{Im}(z) = -2 \) is a horizontal line across all possible real numbers. It includes points like \( 1 - 2i \), \( 3 - 2i \), \( -5.5 - 2i \), etc., implying that any real part is permissible as long as the imaginary part remains \( -2 \).

Key Concepts

Imaginary Part of a Complex Number

Complex numbers often puzzle students at first glance, as they incorporate both real and imaginary components. In mathematics, a complex number is typically expressed in the form \( z = a + bi \). Here, \( a \) is the real part, while \( b \), the coefficient of the imaginary unit \( i \), represents the imaginary part. The imaginary unit \( i \) is defined as \( \sqrt{-1} \), which is a key element in extending the number system beyond real numbers. Understanding the imaginary part is crucial when handling expressions and equations involving complex numbers. In the equation \( \operatorname{Im}(z) = -2 \), the notation \( \operatorname{Im}(z) \) explicitly refers to the imaginary part \( b \). It essentially states that for all applicable complex numbers, their imaginary component is fixed at \(-2\). This consistency across various numbers is what the equation represents and revolves strongly around our understanding of complex numbers and their components.

Argand Diagram

The Argand diagram is a valuable tool for visualizing complex numbers. Named after the mathematician Jean-Robert Argand, it provides a two-dimensional plane, resembling a Cartesian coordinate system, where each complex number is represented as a point. In this plane:

• The horizontal axis is called the real axis, representing the real part \( a \) of the complex number.
• The vertical axis is referred to as the imaginary axis, where the imaginary part \( b \) is plotted.

Visualizing complex numbers on an Argand diagram simplifies understanding various properties and operations such as addition, multiplication, and notably, graphical interpretations of equations. For example, plotting the imaginary part of \(-2\) results in a precise line on this plane, providing immediate visual comprehension of entire sets of complex numbers with this particular characteristic. Thus, the Argand diagram serves not only as a representation tool but also as a method to explore and interpret complex mathematical concepts effectively.

Horizontal Line on Complex Plane

Drawing a horizontal line on the complex plane (or Argand diagram) signifies a set of complex numbers where the imaginary part remains constant while the real part is variable and unrestricted. For the equation \( \operatorname{Im}(z) = -2 \), this translates to a horizontal line at \( y = -2 \) on the imaginary axis. Here are some important points about this line:

• Every point on this line shares the same imaginary value \(-2\).
• This line spans infinitely along the real axis, covering every possible real number.
• Examples of points on the line include any complex number such as \( 1 - 2i \), \( 0 - 2i \), or \( -3.5 - 2i \).

By recognizing this horizontal line, we better grasp how specific constraints on imaginary components can transform into clear and broad graphical depictions on the complex plane, demonstrating the flexibility and power of visualizing complex number equations.