Question

Write a function that will accept a complex number \(c\) and plot that point oa a Cartesian coordinate system with a circular marker. The plot should include both the \(x\) and \(y\) axes, plus a vector drawn from the origin to the location of \(\mathrm{c}\).

Short answer

To write a function that accepts a complex number, \(c\), and plots it on a Cartesian coordinate system, you can use Python and the Matplotlib library. First, find the real (\(a\)) and imaginary (\(b\)) parts of the complex number, and then use a plotting function like this: ```python import numpy as np import matplotlib.pyplot as plt def plot_complex_number(c: complex): a, b = c.real, c.imag fig, ax = plt.subplots() ax.set_aspect("equal") ax.axhline(0, color='black', linewidth=1) ax.axvline(0, color='black', linewidth=1) ax.set_xlim(-10, 10) ax.set_ylim(-10, 10) ax.plot(a, b, 'bo', markersize=8) ax.quiver(0, 0, a, b, angles='xy', scale_units='xy', scale=1, color='r') plt.xlabel('Real') plt.ylabel('Imaginary') plt.title(f'Complex number: {c}') plt.grid(True) plt.show() plot_complex_number(3 + 4j) ``` This code will create a plot with x and y axes, a circular marker at the location of the complex number, and a vector from the origin to the point.

Step-by-step solution

Find the real and imaginary parts of the complex number

A complex number is of the form \(c = a + bi\), where \(a\) is the real part and \(b\) is the imaginary part. You can split \(c\) into these two parts by simply identifying the values of \(a\) and \(b\).

Plot the point on a Cartesian coordinate system

Using the real part, \(a\), as the x-coordinate, and the imaginary part, \(b\), as the y-coordinate, you can plot the complex number on a 2D plane. Now that we know how to convert the complex number into Cartesian coordinates, we can write a function in Python using a plotting library like Matplotlib to draw the axes, the point, and the vector. ```python import numpy as np import matplotlib.pyplot as plt def plot_complex_number(c: complex): # Step 1: Find the real and imaginary parts of the complex number a, b = c.real, c.imag # Step 2: Plot the point on a Cartesian coordinate system fig, ax = plt.subplots() ax.set_aspect("equal") ax.axhline(0, color='black', linewidth=1) ax.axvline(0, color='black', linewidth=1) ax.set_xlim(-10, 10) ax.set_ylim(-10, 10) # Plot circular marker at the complex number's location ax.plot(a, b, 'bo', markersize=8) # Plot vector from the origin to the complex number's location ax.quiver(0, 0, a, b, angles='xy', scale_units='xy', scale=1, color='r') plt.xlabel('Real') plt.ylabel('Imaginary') plt.title(f'Complex number: {c}') plt.grid(True) plt.show() # Example usage plot_complex_number(3 + 4j) ``` This Python function takes a complex number as input and follows the steps mentioned earlier to plot the point on a Cartesian coordinate system, including a vector from the origin to the point. The output is a visual representation of the input complex number.

Key concepts

MATLAB Programming

MATLAB is an advanced programming language and interactive environment commonly used for mathematical computing, algorithm development, data visualization, and numerical analysis. When it comes to complex numbers, MATLAB simplifies these tasks with built-in functions and powerful visualization tools.

To create a plot similar to the one described using MATLAB, one would typically employ functions such as plot for points and quiver for vectors. MATLAB's ability to handle complex numbers directly in its calculations and its user-friendly interface for plotting makes it a preferred tool for engineers and scientists to represent complex data visually.

Importantly, understanding how to translate complex numbers to graphical representations in MATLAB enhances comprehension of mathematical concepts and can further aid in the development of simulations or more advanced data analysis.

Cartesian Coordinate System

The Cartesian coordinate system is a two-dimensional plane defined by a pair of perpendicular axes. In this system, every point in the plane is identified by an ordered pair of numbers (x, y), which represent the positions on the horizontal (x-axis) and vertical (y-axis) axes.

This coordinate system is particularly useful for visualizing complex numbers, as every complex number can be represented as a point in the plane. The real part of the complex number corresponds to the x-coordinate, while the imaginary part corresponds to the y-coordinate. Learning how to plot complex numbers on the Cartesian coordinate system can significantly aid a student's understanding of the spatial relationships between numbers and foster the development of more intuitive problem-solving skills.

Python Matplotlib Visualization

Python is a versatile programming language that, when combined with the Matplotlib library, offers extensive capabilities for data visualization. Matplotlib is especially well-suited for creating static, interactive, and animate plots in Python, making it a useful tool for representing mathematical concepts like complex numbers graphically.

When we use Matplotlib to visualize complex numbers, as demonstrated in the provided Python example, we are creating a visual aid that can help illuminate abstract concepts. This practical visualization technique, wielded correctly, is an invaluable resource that can significantly enhance learning by providing a clear visual representation of complex data.

Complex Number Representation

Complex numbers are a fundamental concept in mathematics and engineering, consisting of a real part and an imaginary part. They are typically written in the form a + bi, where a is the real part, and b represents the imaginary part, with i being the imaginary unit.

In geometric terms, complex numbers can be represented as points or vectors on a two-dimensional plane, often referred to as the complex plane or Argand diagram. This representation is potent for both educational and practical purposes as it provides a way to interpret operations on complex numbers, such as addition, subtraction, and multiplication, as geometric transformations. By visualizing these numbers graphically, students can more easily comprehend the behavior of complex numbers and develop a more robust understanding of complex number operations and properties.