Problem 21 - Algebra and Trigonometry

Question

Sketch the set in the complex plane. $$ \\{z=a+b i | a \leq 0, b \geq 0\\} $$

Step-by-Step Solution

Answer

Shade the upper-left quadrant and the positive imaginary axis in the complex plane.

Step 1: Understand Complex Numbers

Complex numbers are in the form of \( z = a + bi \), where \( a \) is the real part and \( b \) is the imaginary part. The complex plane is a visual representation with the real part on the x-axis and the imaginary part on the y-axis.

Step 2: Identify the Region on the Real Axis

From the set description \( a \leq 0 \), we understand that \( a \) represents the real part of the complex number. This inequality indicates that in the complex plane, we are interested in the half-plane where the real component is non-positive, i.e., left of the imaginary axis (including the imaginary axis itself).

Step 3: Identify the Region on the Imaginary Axis

The condition \( b \geq 0 \) tells us that \( b \), the imaginary part, must be non-negative. Thus, we are considering the upper half-plane including the real axis (which corresponds to \( b = 0 \)).

Step 4: Combine the Regions

Combining both conditions, \( a \leq 0 \) and \( b \geq 0 \), the set consists of all points in the upper-left quadrant of the complex plane and also includes the positive y-axis.

Step 5: Sketch the Set

On a graph with the real number line as the x-axis and the imaginary number line as the y-axis, shade the area to the left of the y-axis (non-positive real), and above or on the x-axis (non-negative imaginary). Highlight the lines \( a = 0, b \geq 0 \) and \( a \leq 0, b = 0 \) as boundary lines because they are part of the set.

Key Concepts

Complex Numbers

Let's start by understanding what complex numbers are. Complex numbers are numbers that include a real part and an imaginary part. They're expressed in the form:

• \( z = a + bi \)

Here, \( a \) represents the real part, and \( b \) is the imaginary part, often followed by the imaginary unit \( i \). The imaginary unit \( i \) is defined such that \( i^2 = -1 \). So, complex numbers are not «just» numbers; they're numbers that help us expand mathematical concepts to new ideas beyond the real line.

In the complex plane, complex numbers are represented as points or vectors. This plane is a vital tool for visualizing these numbers. On this plane, the horizontal axis (x-axis) shows the real part, while the vertical axis (y-axis) shows the imaginary part.

Real and Imaginary Parts

We now seek to dissect the roles of the real and imaginary parts in complex numbers. As mentioned, complex numbers take the form \( z = a + bi \), with:

• \( a \): Real part
• \( b \): Imaginary part

The real part \( a \) determines the position of a complex number along the horizontal axis, while the imaginary part \( b \) dictates the position along the vertical.

When we're asked to consider inequalities like \( a \leq 0 \) and \( b \geq 0 \), it means:

• The real part \( a \) does not go beyond zero; it can be zero or take any negative value.
• The imaginary part \( b \) remains non-negative; it can be zero or any positive value.

In terms of location, this suggests focusing on the upper-left quadrant of the complex plane. This quadrant includes all numbers whose real parts are zero or negative and imaginary parts are zero or positive.

Complex Number Inequalities

Understanding inequalities within complex numbers can be fascinating yet slightly challenging. When dealing with inequalities involving complex numbers, each part is treated individually. So for an inequality set \( \{z = a + bi | a \leq 0, b \geq 0\} \):

• The inequality \( a \leq 0 \) indicates numbers situated on or to the left of the imaginary axis.
• Likewise, the inequality \( b \geq 0 \) conveys numbers positioned on or above the real axis.

Combining these conditions:

• Numbers fitting both criteria occupy the upper-left section of the complex plane.
• This includes points on the positive y-axis and the negative part of the x-axis.

Visualizing this, you’d shade the space in the complex plane where the inequalities overlap, emphasizing the upper-left quadrant.