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Chapter 8: Problem 366

Plot the complex number in the complex plane. $$1-4 i$$

Short Answer

Expert verified
Plot the point at (1, -4) in the complex plane.

Step by step solution

01

Identify the real and imaginary parts

The given complex number is \(1 - 4i\). Here, the real part is \(1\) and the imaginary part is \(-4\). This means the complex number can be represented as \((1, -4)\) in the complex plane.
02

Understand the complex plane coordinates

In the complex plane, the horizontal axis (x-axis) represents the real part, and the vertical axis (y-axis) represents the imaginary part. Therefore, the number \((1, -4)\) means going \(1\) unit to the right and \(4\) units down on the complex plane.
03

Plot the point

To plot \(1 - 4i\) in the complex plane, mark a point at the intersection of \(x = 1\) (rightward from the origin) and \(y = -4\) (downward from the origin). This gives you the location of the complex number on the 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 a two-dimensional space used to visualize complex numbers. Unlike the Cartesian plane, where points are represented by \((x, y)\) coordinates, the complex plane uses the format of \((a, b)\). Here, "\(a\)" is the real part and "\(b\)" is the imaginary part of the complex number. In this plane:
  • The horizontal axis is the real axis.
  • The vertical axis is the imaginary axis.
Complex numbers are plotted in a similar manner to Cartesian coordinates, but understanding the terminology and orientation is crucial. The real-axis behaves like the x-axis, while the imaginary-axis functions as the y-axis. This setup allows complex numbers, such as \(1-4i\), to be represented visually and analyzed geometrically.
real and imaginary parts
When dealing with complex numbers, it's essential to understand the significance of real and imaginary parts. A complex number typically takes the form \(a + bi\), where:
  • "\(a\)" is the real part.
  • "\(b\)" is the imaginary part, and is paired with the imaginary unit "\(i\)".
For example, in the complex number \(1 - 4i\), the real part is \(1\) and the imaginary part is \(-4\). This separation of parts allows complex numbers to be easily represented on the complex plane. Additionally, it aids in computations and transformations involving complex numbers, as understanding each component's role simplifies many mathematical processes.
plotting complex numbers
Plotting complex numbers is straightforward once you grasp the real and imaginary components. Consider the number \(1 - 4i\). The process goes as follows:
  • Identify the real part: \(1\).
  • Identify the imaginary part: \(-4\).
On the complex plane, from the origin, move:
  • 1 unit rightward (real part \(1\)).
  • 4 units downward (imaginary part \(-4\)).
Thus, you will mark the point at the intersection of these coordinates. This visual representation helps to understand the behavior and properties of complex numbers by providing a spatial relationship. Plotting aids in further mathematical operations and problem-solving strategies involving complex numbers.

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