Problem 1 - Advanced Engineering Mathematics

Question

In Problems 1-8, sketch the graph of the given equation. $$ \operatorname{Re}(z)=5 $$

Step-by-Step Solution

Answer

The graph is a vertical line at \( x = 5 \) on the complex plane.

Step 1: Understanding the Equation

The equation given is \( \operatorname{Re}(z) = 5 \). Here, \( z \) is a complex number expressed as \( z = x + yi \), where \( x \) is the real part and \( y \) is the imaginary part. The function \( \operatorname{Re}(z) \) indicates the real part of the complex number, which, according to the equation, is equal to 5.

Step 2: Identifying the Components of the Complex Number

Since \( \operatorname{Re}(z) = 5 \), we know that the real part \( x \) of the complex number must be 5, irrespective of the value of the imaginary part \( y \). Thus, the complex numbers can be expressed as \( z = 5 + yi \).

Step 3: Sketching the Graph

On the complex plane, each complex number corresponds to a point. The real part of the complex number is graphed on the horizontal axis, and the imaginary part is on the vertical axis. Our equation \( \operatorname{Re}(z) = 5 \) suggests that \( x = 5 \) for all values, and \( y \) can be any number. Hence, the graph of this equation is a vertical line passing through \( x = 5 \) on the real axis.

Key Concepts

Complex Plane

The complex plane is a two-dimensional plane used to represent complex numbers visually. Each complex number corresponds to a unique point on this plane, which helps in understanding and visualizing the nature of complex numbers.

A complex number is usually expressed in the form \( z = x + yi \), where \( x \) is the real part and \( y \) is the imaginary part. On the complex plane:

• The horizontal axis is called the real axis, representing the real parts of complex numbers.
• The vertical axis is the imaginary axis, representing the imaginary components of complex numbers.

The complex plane is an essential tool for complex number graphing, making it easier to interpret operations and relationships like addition, multiplication, and, as in the exercise, the real part of a complex number.

Real and Imaginary Parts

Every complex number \( z = x + yi \) is composed of a real part \( x \) and an imaginary part \( y \). Understanding these components is crucial when working with complex numbers.

• The real part \( x \) is a standard coordinate on the real axis. It's what you would expect from a typical number line.
• The imaginary part \( y \) involves the imaginary unit \( i \), where \( i \) is defined such that \( i^2 = -1 \).

In this exercise, the given equation \( \operatorname{Re}(z) = 5 \) specifies that the real part \( x = 5 \). This means no matter what value \( y \) takes, as long as the equation \( z = 5 + yi \) holds, all such numbers are valid solutions. This understanding is key to interpreting the graphical representation on the complex plane.

Vertical Line Graph

The graphical representation of the equation \( \operatorname{Re}(z) = 5 \) is a vertical line on the complex plane. This is because the real part is fixed and does not change with different values of the imaginary part.

On the complex plane:

• This vertical line crosses the real axis at the point where \( x = 5 \).
• The line extends infinitely in both directions parallel to the imaginary axis, recognizing that \( y \) can be any real number.

Graphing it helps visualize that although the imaginary part \( y \) can vary, the real part remains consistently at 5. Thus, the vertical line effectively represents all numbers of the form \( 5 + yi \), showcasing the unchanging nature of the real part amidst varying imaginary parts.