Problem 28 - Precalculus Mathematics for Calculus

Question

Sketch the set in the complex plane. $$\\{z=a+b i | a \geq b\\}$$

Step-by-Step Solution

Answer

Shade the region right of and including the line \( a = b \) in the complex plane.

Step 1: Understand the Set Definition

The set is defined as \( \{ z = a + bi \mid a \geq b \} \), where \( z \) is a complex number represented as \( a + bi \), with \( a \) and \( b \) being real numbers. The condition \( a \geq b \) implies that the real part of \( z \), denoted by \( a \), is greater than or equal to the imaginary part \( b \).

Step 2: Interpret in the Complex Plane

In the complex plane, the real part \( a \) is represented on the horizontal axis, and the imaginary part \( bi \) is represented on the vertical axis. The line \( a = b \) is a diagonal line that extends through the origin at a 45° angle. The region described by \( a \geq b \) includes all points on or to the right of this diagonal line.

Step 3: Graph the Line \( a = b \)

Draw the line given by \( a = b \) on the complex plane, which is a line that starts from the origin \((0,0)\) and extends at a 45° angle, passing through points like \((1, 1)\), \((2, 2)\), etc. This line serves as a boundary separating the regions where \( a \) is greater or less than \( b \).

Step 4: Shade the Region \( a \geq b \)

Shade the region that includes the line \( a = b \) and extends to the right side of this line. This is the set of all points where the horizontal coordinate (real part \( a \)) is greater than or equal to the vertical coordinate (imaginary part \( b \)). Ensure the shading reflects that the points on the line \( a = b \) are included in the set (solid line).

Step 5: Verify the Set Description

Review the graph to ensure that the entire region where \( a \geq b \) is represented. Every point in this region satisfies the condition \( a \geq b \), confirming the set is correctly depicted on the complex plane.

Key Concepts

Complex Numbers

Complex numbers are numbers that consist of two parts: a real part and an imaginary part. They are generally written in the form \( z = a + bi \), where \( a \) is the real part and \( b \) is the imaginary part. The letter \( i \) represents the imaginary unit, which satisfies the property \( i^2 = -1 \). This might seem mysterious at first, but it's a fundamental aspect of complex numbers. Think of complex numbers as coordinates in a two-dimensional space. You can use them to represent points on a plane known as the complex plane.

• The real part \( a \) determines how far left or right the point is from the origin.
• The imaginary part \( b \) determines how far up or down the point is.

This means complex numbers allow us to extend the traditional numbers line into a plane, offering a rich framework for solving various mathematical problems.

Real Part

The real part of a complex number is the component that takes the place of the horizontal axis in the complex plane. In a complex number \( z = a + bi \), \( a \) is the real part.

• The real part indicates the point's position along the horizontal axis.

In the context of the exercise, the condition \( a \geq b \) is important. It means the real part \( a \) must be greater than or equal to the imaginary part \( b \), placing the points either on or to the right of the line \( a = b \).This condition helps in determining the graphical region on the complex plane, ensuring that points lie within the specified set.

Imaginary Part

The imaginary part of a complex number is associated with the vertical axis on the complex plane. For \( z = a + bi \), \( bi \) represents the imaginary part, where \( b \) is a real number and \( i \) is the imaginary unit.

• The imaginary part tells us how high or low the point is along the vertical axis.

In our exercise, the imaginary part \( b \) is a crucial element. When \( a \geq b \) was given, it ensured that we only select points whose real part \( a \) equals or surpasses the imaginary part \( b \). This part forms the boundary for the shading in the graphical representation, ensuring accuracy in depicting complex sets.

Graphical Representation

Graphical representation of complex numbers is done using the complex plane. This plane has two axes:

• Horizontal axis for real parts.
• Vertical axis for imaginary parts.

In this exercise, you learned to plot the set where \( a \geq b \). You first draw the line \( a = b \), which is a diagonal at a 45° angle through points like \((1, 1)\), \((2, 2)\), etc. This line marks the boundary, and we shade the region on or to the right of it. This shading represents all points that have a real part strictly greater than or equal to the imaginary part. Graphical representation is a powerful way to visually understand complex equations and inequalities, providing clarity on concepts that might otherwise appear abstract.