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Chapter 8: Problem 14

Graph each complex number along with its opposite and conjugate.\(-3 i\)

Short Answer

Expert verified
The complex number -3i is plotted at (0, -3), its opposite at (0, 3), and its conjugate at (0, 3) in the complex plane.

Step by step solution

01

Understanding the Complex Number

The given complex number is -3i. It has no real part and only an imaginary part, which is -3.
02

Plotting the Complex Number

In the complex plane, the horizontal axis represents the real part and the vertical axis represents the imaginary part. Here, since the real part is 0, plot the point at 0 on the real axis and -3 on the imaginary axis for the complex number -3i.
03

Finding the Opposite

The opposite of a complex number a + bi is -a - bi. For our number -3i, the opposite is 3i, since there is no real part to negate.
04

Plotting the Opposite

Plot the opposite, 3i, on the complex plane. Again, the real part is still 0, but now plot the point at 3 on the imaginary axis.
05

Finding the Conjugate

The conjugate of a complex number a + bi is a - bi. For the complex number -3i, the conjugate is 3i.
06

Plotting the Conjugate

Plot the conjugate, which is also 3i, on the complex plane. It will coincide with the opposite in this case.

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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 two-dimensional plane used to graph complex numbers. Unlike the typical X-Y plane, the complex plane has a unique feature: it consists of a real axis and an imaginary axis.

  • The horizontal axis is the real axis. Points on this axis have no imaginary part.
  • The vertical axis is the imaginary axis. Points on this axis have no real part.

The complex plane is a helpful tool to visualize and comprehend complex numbers. Each complex number is represented as a point or a vector on this plane. For example, the complex number a + bi is plotted where a on the real axis and b on the imaginary axis intersect. In the exercise, the complex number -3i is plotted directly on the imaginary axis because the real part is 0 .
Imaginary Axis
The imaginary axis is the vertical line on the complex plane. It is analogous to the Y-axis in the regular Cartesian coordinate system, but it specifically represents the imaginary part of complex numbers.

  • Points on the imaginary axis have a form of 0 + bi.
  • The position along this axis is determined by the coefficient b.

When plotting a complex number like -3i, find its position on the imaginary axis where -3 is the y-coordinate. To graph it, mark a point at this location. The real part, being 0, means the point lies on the imaginary axis directly.
Conjugate of a Complex Number
The conjugate of a complex number is a fundamental concept in complex number operations. For a complex number of the form a + bi, the conjugate is a - bi. This operation essentially reflects the complex number across the real axis in the complex plane.

  • The real part remains unchanged.
  • The sign of the imaginary part is reversed.

This process is particularly straightforward when dealing with numbers like -3i, which have no real part. The conjugate of -3i is 3i. In this case, the conjugate and the opposite are the same due to the absence of a real component. This makes plotting simpler since both lie on the imaginary axis, just on opposite sides.

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Most popular questions from this chapter

Simplify each expression to a single trigonometric function.\(\sin 30^{\circ} \cos 90^{\circ}+\cos 30^{\circ} \sin 90^{\circ}\)

\(x^{2}+y^{2}=4 x\)

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

Graph each complex number. In each case, give the absolute value of the number.\(1+i\)

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