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Chapter 20: Problem 62

Image of \(|z|=2\) under \(w=z+3+2 i\) is

Short Answer

Expert verified
Image of |z|=2 is a circle of radius 2 centered at (3, 2).

Step by step solution

01

Understand the geometric representation of |z|=2

The equation \(|z| = 2\) represents a circle in the complex plane with center at the origin \((0,0)\) and radius 2. All points \(z\) on this circle have a magnitude of 2.
02

Apply the transformation w = z + 3 + 2i

The transformation \(w = z + 3 + 2i\) shifts each point \(z\) in the complex plane by \(3\) units along the real axis and \(2\) units upward along the imaginary axis. This is like adding the complex number \(3 + 2i\) to \(z\).
03

Determine the new center and radius after the transformation

Adding \(3 + 2i\) to every point on the circle \(|z|=2\) results in a new circle, but the radius remains unchanged. The new center is at the point \((3, 2)\) in the complex plane, while the radius remains 2.

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Key Concepts

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

Geometric Representation
The concept of geometric representation in complex numbers is fascinating because it provides a visual way of understanding these numbers. In this context, complex numbers can be seen as points or vectors in a plane known as the complex plane. When dealing with \( |z| = 2 \), the geometric representation is a circle centered at the origin \( (0,0) \) with a radius of 2. This means that any complex number \( z \) on this circle is exactly 2 units away from the origin.
Think of this as drawing a perfect circle on a piece of graph paper, where each point on the circle maintains a constant distance, in this case, 2, from the center located at the origin of the axes.
Complex Plane
Understanding the complex plane is crucial for visualizing complex numbers. The complex plane is much like a regular coordinate plane but adjusted for complex numbers.
  • The horizontal axis, or real axis, represents the real part of a complex number.
  • The vertical axis, or imaginary axis, represents the imaginary part.
Therefore, any complex number \( z = a + bi \) can be represented by the point \( (a,b) \) in the complex plane. This makes it easier to visualize operations involving complex numbers, like addition or multiplication. In our scenario, the circle defined by \( |z|=2 \) is centered at \( (0,0) \) with a radius that stretches 2 units in all directions, capturing the transcendental beauty of complex numbers as both abstract and tangible.
Transformation
Transformation in the context of complex numbers introduces a dynamic shift. When we say \( w = z + 3 + 2i \), we are talking about moving every point \( z \) on our original circle by a specific vector in the complex plane. This vector \( 3 + 2i \) effectively shifts each point 3 units right and 2 units up.
  • The horizontal shift (3 units) adds to the real part of each \( z \).
  • The vertical shift (2 units) adds to the imaginary part of each \( z \).
The transformation results in a new circle that maintains the same size or radius but is now centered at \( (3,2) \). Thus, transformations allow us to understand how complex numbers and their geometric representations move and interact within the complex plane, emphasizing the interconnectedness of algebra and geometry.

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