/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 8 Plot each point in the complex p... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 13: Problem 8

Plot each point in the complex plane. $$1(=1+0 i)$$

Short Answer

Expert verified
Plot the point at (1, 0) on the complex plane.

Step by step solution

01

Identify Real and Imaginary Components

Examine the complex number \(1\). This can be written in the form \(1 + 0i\), meaning the real part is 1 and the imaginary part is 0.
02

Plot the Real Component

On the complex plane, locate the real axis (horizontal axis). Since the real part is 1, move 1 unit to the right from the origin along this axis.
03

Plot the Imaginary Component

Locate the imaginary axis (vertical axis) on the complex plane. Since the imaginary part is 0, you will stay on the horizontal axis without moving up or down.
04

Mark the Point on the Complex Plane

The point that represents the complex number \(1 + 0i\) is at the coordinates (1, 0). Place a point at this location on the complex plane.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Complex Plane
The complex plane is a visual way to represent complex numbers geometrically. It looks a lot like the regular Cartesian coordinate plane you might be familiar with, but here we work with complex numbers. Instead of having 'x' and 'y' axes, we have a "real" axis and an "imaginary" axis.
  • The horizontal axis represents the real part of complex numbers.
  • The vertical axis represents the imaginary part.
Think of the complex plane as a map for complex numbers. This allows us to "see" complex numbers in a two-dimensional form. Understanding this plane is extremely helpful in grasping how complex numbers are structured and represented.
Real and Imaginary Components
A complex number is usually expressed in the form of \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part. In complex numbers, both parts have key roles.
  • The real component \(a\) determines the position along the real axis.
  • The imaginary component \(b\) determines the position along the imaginary axis.
For example, in the complex number \(1 + 0i\), 1 is the real component, while 0 is the imaginary component. This number is essentially a real number because its imaginary part is zero. Recognizing these parts helps in plotting points and understanding the structure of complex numbers.
Plotting Points
Plotting points on the complex plane involves utilizing both the real and imaginary components. This is similar to how you plot points on a traditional 2D grid, but with a focus on complex numbers.
  • Start by finding the real component. Move horizontally on the real axis for this component.
  • Then locate the imaginary component. Move vertically on the imaginary axis for this component.
  • Once both movements are complete, mark your point on the plane.
Let's take our example, \(1 + 0i\). You would move one unit to the right on the real axis and remain stationary on the imaginary axis, as the imaginary component is zero. This is how you translate numerical values into a visual point on the complex plane.
Coordinates on Complex Plane
Coordinates in the complex plane work similarly to coordinates in a 2D Cartesian plane, but with a distinct purpose. Each complex number has corresponding coordinates \(a, b\) that indicate its position on the plane.
  • The first value, \(a\), represents movement along the real axis.
  • The second value, \(b\), represents movement along the imaginary axis.
For the complex number \(1 + 0i\), its coordinates are \((1, 0)\). This means that on the complex plane, the point is 1 unit from the origin along the real axis (and 0 units along the imaginary axis). Understanding coordinates helps in visualizing and mapping complex numbers accurately.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Carry out the indicated operations. Express your results in rectangular form for those cases in which the trigonometric functions are readily evaluated without tables or a calculator. $$ \left(\cos \frac{2}{5} \pi+i \sin \frac{2}{5} \pi\right) \div\left(\cos \frac{2}{5} \pi+i \sin \frac{2}{5} \pi\right) $$

Find the indicated roots. Express the results in rectangular form. Find the fourth roots of \(8-8 \sqrt{3} i\). Hint: Use the addition formulas or the half-angle formulas.

Use DeMoivre's theorem to find the indicated roots. Express the results in rectangular form. Square roots of \(i\)

(The Ricker model continued) Suppose that the initial size of a population is \(P_{0}=300\) and that the size of the population at the end of year \(t\) is given by $$ P_{t}=10 P_{t-1} e^{-P_{t-1} / 1000} \quad(t \geq 1) $$ (a) Use a graphing utility to compute the population sizes through the end of year \(t=5 .\) (As in Example 5, round the final answers to the nearest integers.) Then use the graphing utility to draw the population scatter plot for \(t=0,1, \ldots, 5 .\) Give a general description (in complete sentences) of how the size of the population changes over this period. (b) Use a graphing utility to compute the population sizes through the end of year \(t=20,\) and draw the scatter plot. To help you see the pattern, use the option on your graphing utility that connects adjacent dots in a scatter plot with line segments. Describe the population trend that emerges over the period \(t=15\) to \(t=20\) (c) For a clearer view of the long-term population behavior, use a graphing utility to compute the population sizes for the period \(t=25\) to \(t=35,\) and draw the scatter plot. As in part (b), use the option on your graphing utility that connects adjacent dots with line segments. Summarize (in complete sentences) what you observe.

Find the coefficient of \(B^{-10}\) in the expansion of \(\left[\left(B^{2} / 2\right)-\left(3 / B^{3}\right)\right]^{10}\).
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Decision Maths

Read Explanation

Probability and Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Pure Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.