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Chapter 1: Problem 8

Sketch the graph of the given equation in the complex plane. $$ \operatorname{Im}(z-i)=\operatorname{Re}(z+4-3 i) $$

Short Answer

Expert verified
The graph is a line with equation \(y = x + 5\).

Step by step solution

01

Interpret the Complex Numbers

The given equation is \(\operatorname{Im}(z-i) = \operatorname{Re}(z+4-3i)\). Split the complex number \(z\) into its real and imaginary parts: let \(z = x + yi\), where \(x\) is the real part and \(y\) is the imaginary part.
02

Substitute and Simplify

Substitute \(z = x + yi\) into \(\operatorname{Im}(z-i)\). We have \(z - i = x + (y-1)i\). Therefore, \(\operatorname{Im}(z-i) = y-1\).
03

Compute the Other Side

Substitute \(z = x + yi\) into \(\operatorname{Re}(z+4-3i)\). We have \(z + 4 - 3i = (x + 4) + (y-3)i\). Therefore, \(\operatorname{Re}(z+4-3i) = x+4\).
04

Set Up the Equation

Equating both sides, we get \(y - 1 = x + 4\). Simplify this to get the equation of the line: \(y = x + 5\).
05

Sketch the Graph

The equation \(y = x + 5\) represents a line in the complex plane. This line has a slope of 1 and intersects the y-axis at 5. Sketch this line on the Argand plane, where the horizontal axis is \(x\) (real part) and the vertical axis is \(y\) (imaginary part).

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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 a fundamental concept in mathematics, representing numbers in the form of \(z = x + yi\). Here, \(x\) is the real part and \(y\) is the imaginary part, multiplied by the imaginary unit \(i\), where \(i^2 = -1\).
  • The real part of a complex number can be visualized on the horizontal axis of the complex plane, also known as the Argand plane.
  • The imaginary part is visualized along the vertical axis.
  • Complex numbers allow us to solve equations that involve the square roots of negative numbers, which are not possible with real numbers alone.
Complex numbers are essential in various fields such as engineering, physics, and mathematics for modeling oscillations, waves, and rotations.
Imaginary Part
The imaginary part of a complex number \(z = x + yi\) is the coefficient \(y\) of the imaginary unit \(i\). It represents the value plotted on the vertical axis in the complex plane.
  • To extract the imaginary part, we simply look at the coefficient attached to \(i\).
  • For example, if \(z - i = x + (y-1)i\), the imaginary part is \(y-1\).
The concept of imaginary parts allows us to separate and analyze the dual components of complex numbers and is crucial for mathematical operations involving complex conjugates and complex arithmetic.
Real Part
The real part of a complex number, in the form \(z = x + yi\), is simply the value \(x\) without any imaginary component. This is the component that is usually plotted on the horizontal axis of the complex plane.
  • The real part helps to retain context in complex numbers, where it represents the real-world measurable quantity.
  • In the case of the transformation \(z + 4 - 3i = (x+4) + (y-3)i\), the real part we focus on is \(x+4\).
Being able to distinguish the real part allows analysts to perform tasks like filtering real signals from complex functions, making it crucial in various computational applications.
Graph Sketching
Graph sketching in the complex plane involves plotting complex numbers on a two-dimensional plane, known as the Argand plane. The horizontal axis represents the real part, while the vertical axis represents the imaginary part.
For the given equation \(y = x + 5\), this represents a straight line on the Argand plane.
  • The line has a slope of 1, indicating a 45-degree angle relative to both axes.
  • It intercepts the imaginary part axis (y-axis) at the point where the imaginary part equals 5.
Sketching this graph helps visualize the relationship between complex numbers and their components, providing a geometric perspective on complex equations.

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Most popular questions from this chapter

Sketch the set \(S\) of points in the complex plane satisfying the given inequality. Determine whether the set is (a) open, (b) closed, (c) a domain, (d) bounded, or (e) connected. $$ \operatorname{Im}(z)<\operatorname{Re}(z) $$

Sketch the set \(S\) of points in the complex plane satisfying the given inequality. Determine whether the set is (a) open, (b) closed, (c) a domain, (d) bounded, or (e) connected. $$ |\operatorname{Re}(z)|>2 $$

Use de Moivre's formula (10) with \(n=3\) to find trigonometric identities for \(\cos 3 \theta\) and \(\sin 3 \theta\).

Find solutions of the given homogeneous differential equation. $$ y^{\prime \prime}+2 y^{\prime}+4 y=0 $$

Use a \(\mathrm{CAS}^{\S}\) to first find \(z^{n}=w\) for the given complex number and the indicated value of \(n\). Then, using the output and the same value of \(n\), determine whether \(w^{1 / n}=\left(z^{n}\right)^{1 / n}=z .\) If not, explain why not. $$ 2+2 i ; n=12 $$
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