Problem 21 - Precalculus Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

The set is the second quadrant of the complex plane, including the negative real and positive imaginary axes.

Step 1: Understand the Set Representation

The given set \( \{ z = a + b i \mid a \leq 0, b \geq 0 \} \) represents complex numbers where the real part \( a \) is less than or equal to zero, and the imaginary part \( b \) is greater than or equal to zero. This set indicates that the complex number \( a+bi \) is located in a specific region of the complex plane.

Step 2: Identify the Region in the Complex Plane

In the complex plane, the real part corresponds to the x-axis, and the imaginary part corresponds to the y-axis. For the given set, since \( a \leq 0 \), we are considering points to the left of the y-axis (including the y-axis), and since \( b \geq 0 \), the points are above the x-axis (including the x-axis itself). Thus, we are dealing with the region that is in the second quadrant of the complex plane, including the negative real axis and positive imaginary axis.

Step 3: Sketch the Region

To sketch the region, first draw the coordinate axes, labeling the real axis (horizontal) and the imaginary axis (vertical). Shade the entire area in the second quadrant, including the boundary on the imaginary axis and real axis. This shaded region represents all the complex numbers \( z = a + bi \) where \( a \leq 0 \) and \( b \geq 0 \).

Key Concepts

Complex Numbers

Complex numbers are unique numbers in mathematics that consist of two parts: a real part and an imaginary part. The standard format for a complex number is given by

• \( z = a + bi \)

Here, \( a \) illustrates the real part, while \( b \) represents the imaginary part, and \( i \) is the imaginary unit satisfying \( i^2 = -1 \). This dual nature allows complex numbers to be represented graphically in a 2-dimensional plane known as the complex plane. Unlike real numbers that lie on a single line, complex numbers can describe more complex phenomena due to their two-dimensional nature.
Consider a complex number, \( z = 3 + 4i \). This number has a real part \( a = 3 \), and an imaginary part \( b = 4 \). It can be plotted on the complex plane, providing a spatial way to understand its value. Exploring complex numbers enables solving equations that do not have solutions among the real numbers, like \( x^2 + 1 = 0 \), and offers deeper insights into various fields such as engineering and physics.

Quadrants in the Complex Plane

The complex plane is divided into four quadrants, much like the Cartesian coordinate system. Each quadrant is determined by the signs of the real and imaginary parts of complex numbers. Imagine this plane with the horizontal x-axis as the real axis and the vertical y-axis as the imaginary axis.

• **First Quadrant**: Both real and imaginary parts are positive \( (a > 0, b > 0) \)
• **Second Quadrant**: Real part is negative, imaginary part is positive \( (a < 0, b > 0) \)
• **Third Quadrant**: Both real and imaginary parts are negative \( (a < 0, b < 0) \)
• **Fourth Quadrant**: Real part is positive, imaginary part is negative \( (a > 0, b < 0) \)

In this exercise, the region defined by \( a \leq 0 \) and \( b \geq 0 \) places the complex numbers in the second quadrant. This quadrant includes numbers with a negative real component and a positive imaginary component. Recognizing these quadrants is essential in graphically distinguishing and solving problems involving complex numbers.

Real and Imaginary Parts

Every complex number can be decomposed into two parts: the real part and the imaginary part. The real part is the component \( a \) in \( a + bi \), whereas the imaginary part is \( b \), multiplied by the imaginary unit \( i \). Recognizing and understanding these components is crucial for mathematically manipulating complex numbers.

• The real part influences where the number lies horizontally on the complex plane.
• The imaginary part determines the vertical position of the number.

For instance, a complex number with \( a = -2 \) and \( b = 3 \), like \( -2 + 3i \), will be placed in the second quadrant due to its negative real part and positive imaginary part. Recognizing the influence of the real and imaginary parts aids in visualizing how complex numbers behave and allows more straightforward solutions to complex equations. By considering their signs and values, one can determine not only the quadrant of the complex plane in which they lie, but also their application in real-world problems and mathematical conjectures.