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Chapter 8: Problem 17

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

Short Answer

Expert verified
Shade the upper-left quadrant, including the origin and the boundaries along the negative real and positive imaginary axes.

Step by step solution

01

Understand the Set's Components

The set \( \{ z = a + bi | a \leq 0, b \geq 0 \} \) describes all 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.
02

Identify the Complex Plane Quadrants

In the complex plane, the horizontal axis represents the real part \( a \), and the vertical axis represents the imaginary part \( b \). Quadrant II is where \( a \leq 0 \) and \( b \geq 0 \).
03

Recognize the Boundaries

The condition \( a \leq 0 \) defines the left half of the plane, starting from the imaginary axis to the left. The condition \( b \geq 0 \) includes the upper half of the plane, starting from the real axis upward.
04

Combine the Conditions

The set \( \{ z = a + bi | a \leq 0, b \geq 0 \} \) includes all points that satisfy both conditions simultaneously, forming the upper-left (second) quadrant and including the boundaries where \( a = 0 \) and \( b = 0 \). Thus, the entire boundary on the imaginary axis and the real axis up to the origin is included.
05

Sketch the Set

On the complex plane, shade the upper-left quadrant (Quadrant II). This includes the entire region to the left of and above the origin, as well as the boundary lines along the negative real axis (\( b = 0 \)) and positive imaginary axis (\( a = 0 \)). Make sure to include the origin as well.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Complex Numbers
Complex numbers are numbers that consist of two parts: a real part and an imaginary part. These numbers have the general form of \( z = a + bi \), where \( a \) is the real part, and \( bi \) is the imaginary part, with \( i \) being the imaginary unit. The imaginary unit \( i \) is defined as \( \sqrt{-1} \), which means \( i^2 = -1 \). Complex numbers are used extensively in mathematics and engineering to solve equations that do not have real solutions.

Here are some important points about complex numbers:
  • If \( b = 0 \), the number is purely real, reducing to \( a \).
  • If \( a = 0 \), the number is purely imaginary, expressed as \( bi \).
  • Complex numbers are crucial for performing transformations and mathematical operations that are not possible with real numbers alone.
Complex Plane Quadrants
The complex plane is a two-dimensional plane similar to the Cartesian coordinate system but designed to represent complex numbers. In this plane:
  • The horizontal axis is the real axis, representing the real component \( a \).
  • The vertical axis is the imaginary axis, representing the imaginary component \( b \).

The complex plane is divided into four quadrants, each with distinct conditions for \( a \) and \( b \):
  • Quadrant I: both \( a \) and \( b \) are positive.
  • Quadrant II: \( a \leq 0 \) and \( b \geq 0 \), meaning here \( a \) is non-positive and \( b \) is non-negative.
  • Quadrant III: both \( a \) and \( b \) are negative.
  • Quadrant IV: \( a \geq 0 \) and \( b \leq 0 \), with \( a \) non-negative and \( b \) non-positive.
Understanding these quadrants helps in visualizing and graphing complex numbers according to their real and imaginary parts.
Graphing in the Complex Plane
Graphing complex numbers in the complex plane involves plotting their points based on the values of \( a \) and \( b \). Each complex number \( z = a + bi \) corresponds to a unique point \( (a, b) \) in the plane. To graph the given set \( \{ z = a + bi | a \leq 0, b \geq 0 \} \):
  • Identify the conditions: Here, \( a \leq 0 \) and \( b \geq 0 \) means you are looking to fill the upper-left quarter of the plane, which is Quadrant II.
  • Consider the boundaries: Set includes the entire negative side and upper side of the plane up to the origin, embracing the sections where \( a = 0 \) (imaginary axis) and \( b = 0 \) (real axis).
  • Shade the region: Shade the entire Quadrant II along with the boundaries to indicate inclusion of all points \( (a, b) \) meeting the criteria.
Proper graphing ensures clear visual representation, making it easier to grasp the spatial distribution of the complex numbers in the set.

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