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Chapter 8: Problem 16

Graph each complex number along with its opposite and conjugate.\(.5\)

Short Answer

Expert verified
Graph \(0.5\), its opposite \(-0.5\), and conjugate \(0.5\) on the real line at (0.5, 0), (-0.5, 0), and (0.5, 0) respectively.

Step by step solution

01

Understand the Complex Number

The complex number given is \(0.5\). Since it has no imaginary part, it can also be written as \(0.5 + 0i\).
02

Identify the Opposite

The opposite of a complex number \(a + bi\) is \(-a - bi\). For our number \(0.5 + 0i\), the opposite is \(-0.5 + 0i\), which is simply \(-0.5\).
03

Find the Conjugate

The conjugate of a complex number \(a + bi\) is \(a - bi\). For \(0.5 + 0i\), the conjugate is \(0.5 - 0i\), which simplifies to \(0.5\).
04

Graph the Points

On the complex plane, the x-axis represents the real part and the y-axis represents the imaginary part. Plot the points: \(0.5\) at (0.5, 0), its opposite \(-0.5\) at (-0.5, 0), and its conjugate \(0.5\) also at (0.5, 0).
05

Interpret the Graph

Since the original number and its conjugate are both on the real line and identical, they appear as the same point at (0.5, 0). The opposite appears at (-0.5, 0), distinct from the original and conjugate.

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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 a tool used to visually represent complex numbers, making them easier to comprehend. Imagine a two-dimensional grid where the x-axis corresponds to the real part of complex numbers and the y-axis correlates to the imaginary part. Each point on this plane represents a unique complex number.
  • For instance, for the complex number \(0.5 + 0i\), which has no imaginary part, you will plot it at the point \((0.5, 0)\).
  • This means it lies directly on the real number line, since the imaginary component is zero.
  • On the other hand, if a complex number had an imaginary part, such as \(a + bi\), it would appear at \((a, b)\).
The complex plane is an essential concept because it allows us to see relationships and operations involving complex numbers graphically, aiding in our understanding of how these numbers behave.
Conjugate
The conjugate of a complex number is a simple yet powerful concept. It involves mirroring the imaginary part of the number, which can be very useful in various mathematical operations.
  • If you have a complex number \(a + bi\), its conjugate is \(a - bi\).
  • In the case of our example number \(0.5 + 0i\), the conjugate is \(0.5 - 0i\), simplifying back to \(0.5\).
  • Note that when there is no imaginary part, like in our example, the conjugate is identical to the original number.
Finding the conjugate helps in simplifying complex division and is vital in multiplying complex numbers to deal with denominators, ensuring real results.
Opposite of a Complex Number
The opposite of a complex number is another fundamental operation performed on complex numbers. It involves changing the signs of both the real and imaginary components.
  • For any complex number \(a + bi\), the opposite is represented as \(-a - bi\).
  • In our case, \(0.5 + 0i\) has its opposite as \(-0.5 + 0i\) or simply \(-0.5\).
  • Visualizing this on the complex plane, the opposite number is plotted symmetrically opposite to the original number with respect to the origin \((0, 0)\).
Understanding the concept of the opposite is crucial because it is frequently used in calculating and understanding vector differences, making it an invaluable tool in geometry and algebra.

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Most popular questions from this chapter

Graph each complex number along with its opposite and conjugate.\(2-i\)

Use a calculator to help write each complex number in standard form. Round the numbers in your answers to the nearest hundredth.\(1 \operatorname{cis} 261^{\circ}\)

Graph each complex number. In each case, give the absolute value of the number.\(3+4 i\)

Use a calculator to help write each complex number in standard form. Round the numbers in your answers to the nearest hundredth.\(1 \operatorname{cis} 205^{\circ}\)

Write each complex number in standard form.\(4\left(\cos \frac{2 \pi}{3}+i \sin \frac{2 \pi}{3}\right)\)
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