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Chapter 8: Problem 364

Plot the complex number in the complex plane. $$6-2 i$$

Short Answer

Expert verified
Plot the point (6, -2) on the complex plane.

Step by step solution

01

Identify the Real and Imaginary Parts

The complex number given is \(6 - 2i\). The real part of this complex number is 6. The imaginary part is \(-2\).
02

Determine the Coordinates

In the complex plane, complex numbers are plotted similar to coordinates in a Cartesian plane. The real part (6) is plotted on the x-axis (horizontal), and the imaginary part (-2) is plotted on the y-axis (vertical). Thus, the complex number \(6 - 2i\) corresponds to the point (6, -2) in the complex plane.
03

Plot the Point on the Complex Plane

To plot the point, start at the origin (0, 0) of the complex plane. Move 6 units to the right (positive x-direction) and 2 units down (since the imaginary part is negative). Mark the point at these coordinates, which is (6, -2). You have now successfully plotted the complex number \(6 - 2i\) on the complex plane.

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Key Concepts

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

complex plane
The complex plane is an incredibly useful tool for visualizing complex numbers. It works similarly to a standard Cartesian plane, with two perpendicular axes. The horizontal axis represents the real part of a complex number, while the vertical axis represents the imaginary part.
  • The x-axis: This is where you plot the real component. For the complex number \(6 - 2i\), the real part is 6.
  • The y-axis: This vertical line is where the imaginary component is plotted. Here, for \(6 - 2i\), the imaginary part is \(-2\).
By utilizing these two axes, you can represent any complex number as a point in the plane, making it much easier to understand their relationships and operations.
real and imaginary parts
Understanding the real and imaginary parts of a complex number is crucial. Each complex number is composed of these two parts, often written in the form \( a + bi \), where \( a \) is the real part and \( b \) is the imaginary part.
  • Real Part: This is a regular number without an imaginary unit \(i\). For \(6 - 2i\), the real part is 6.
  • Imaginary Part: This component includes the imaginary unit \(i\). The imaginary unit \(i\) is defined as \( \sqrt{-1} \), and for \(6 - 2i\), the imaginary part is \(-2\).
These two parts allow us to manipulate and perform operations on complex numbers effectively, such as addition, subtraction, and plotting on the complex plane.
plotting complex numbers
Once you have identified the real and imaginary parts of a complex number, plotting it on the complex plane becomes a straightforward task. The objective is to correlate these parts to a specific point on the plane.
  • Start at the origin, point \( (0,0) \), on the complex plane.
  • Move horizontally along the x-axis by the value of the real part. For \(6 - 2i\), this means moving 6 units to the right.
  • From this position, adjust vertically along the y-axis by the value of the imaginary part. Here, because the imaginary part is \(-2\), you'll move 2 units down.
This path leads you to the point labeled \((6, -2)\) on the complex plane. Successfully plotting the complex number in this manner can make complex number operations more intuitive and less abstract.

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Most popular questions from this chapter

For the following exercises, convert the given Cartesian equation to a polar equation. $$x^{2}+y^{2}=64$$

Solve the triangle, rounding to the nearest tenth, assuming \(\alpha\) is opposite side \(a, \beta\) is opposite side \(b,\) and \(\gamma\) s opposite sidec : \(a=4, b=6, c=8\) .

For the following exercises, test each equation for symmetry. Sketch a graph of the polar equation \(r=5 \sin (7 \theta)\).

Convert the complex number from polar to rectangular form: \(z=5 \operatorname{cis}\left(\frac{2 \pi}{3}\right)\).

For the following exercises, find the absolute value of each complex number. $$4-3 i$$
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