Complex Plane Introduction

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Complex numbers are, well, complex. But fear not, they're not as intimidating as they seem. By the end of this article, you'll have a solid understanding of the complex plane and how to plot complex numbers on it.

Complex Numbers

Before we dive into the complex plane, let's quickly recap complex numbers. A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, and i is the imaginary unit, which satisfies the equation i² = -1. In other words, i is the square root of -1. Spooky, right?

The Complex Plane

Now that we're familiar with complex numbers, let's talk about the complex plane. It's similar to the Cartesian plane, but instead of plotting real numbers, we plot complex numbers. In the complex plane, the horizontal axis represents the real part of a complex number, and the vertical axis represents the imaginary part.

Let's say we have a complex number z = 3 + 4i. To plot this number on the complex plane, we find the point where the real part (3) intersects the imaginary part (4). The result is a point with coordinates (3, 4).

Check out this example:

import matplotlib.pyplot as plt
import numpy as np

# Define the complex number
z = 3 + 4j

# Plot the complex number on the complex plane
plt.plot(np.real(z), np.imag(z), 'ro') # 'ro' means red circle
plt.axhline(y=0, color='k') # Imaginary axis
plt.axvline(x=0, color='k') # Real axis
plt.xlabel('Real')
plt.ylabel('Imaginary')
plt.title('Complex Plane')
plt.grid()
plt.show()

This code snippet plots the complex number 3 + 4i on the complex plane using Python's matplotlib library. It displays a graph with a red circle at the point (3, 4).

Operations on the Complex Plane

The complex plane isn't just for plotting complex numbers. We can also perform operations, such as addition, subtraction, multiplication, and division, on complex numbers in the complex plane. The results of these operations can be visualized by connecting the complex numbers using vectors. See complex-number-operations for a deeper dive into these operations and their visualizations.

Conclusion

We've just scratched the surface of the fascinating world of the complex plane. By understanding the basics of the complex plane and how complex numbers are plotted, you're one step closer to mastering complex numbers and their applications in mathematics, science, and engineering. Keep exploring, and don't let imaginary numbers spook you!

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