April Fool's Day, and the Pythagorean Theorem

      Here is a post in honor of April Fools' Day, which unlike March 14, is celebrated by countries that use either the month-day format or the day-month format.

      Not everyone is a math person, but nearly everyone remembers the Pythagorean Theorem.

      Well, not everyone. The Scarecrow got it wrong in the 1939 movie, "The Wizard of Oz". It is arguable whether the writers accidentally or deliberately got it wrong, or whether the actor Ray Bolger flubbed the line. (See also Singh, "The Simpsons and Their Mathematical Secrets", pp. 119-121.)

      So here is a little Pythagorean Theorem problem. In the following right triangle, calculate C.   A² + B² = C², and then take the square root of C. How hard can it be?

      Of course you have to know if A = (√-1), then A² = (√-1)² = -1, but that seems reasonable.

      But does something seem wrong with the resulting value of C?

      I plotted the triangle in a complex number plane, not a real number plane.   (√-1) is not a "real" number, and perhaps it should be thought of as a non-real number. (I am intentionally avoiding the dreaded "i" word.)   Perhaps what is happening is that in the complex plane, the hypotenuse C is really a line segment of magnitude zero, and hence a point? No, not "really".

      So April Fool's. You calculated a hypotenuse of a right triangle and concluded it has length zero.

      A better answer is that a point (x, y) in the complex plane is represented as x + yi, and the Euclidean distance between two complex points (x, y) and (u, v) equals √( (u - x)² + (v - y)² ) which here is the expected √2.

      Here is the R code for the plot, followed by the Python code for the plot. Note that in R, the shading is done with geom_polygon, while in Python it is done with plt.fill_between:

library(ggplot2)

# Define the vertices of the triangle
triangle_data <- data.frame(
    x = c(0, 0, 1),    # x-coordinates of vertices
    y = c(0, 1, 0)     # y-coordinates of vertices
)

ggplot(data = NULL) +
    geom_polygon(data = triangle_data, aes(x = x, y = y), fill = "#9B111E", color = "black") +
    geom_segment(aes(x = 0, y = 0, xend = 1, yend = 0), color = "black", size = 2) +
    geom_segment(aes(x = 0, y = 0, xend = 0, yend = 1), color = "black", size = 2) +
    geom_segment(aes(x = 0, y = 1, xend = 1, yend = 0), color = "black", size = 2) +
    ggtitle("Use the Pythagorean Theorem to calculate C") +
    xlab(expression(bold(A == sqrt(-1)))) +  # Square root symbol for x-axis label
    ylab("B = 1") +     # Label for y-axis
    geom_text(aes(x = 0.55, y = 0.55, label = "C = ?"), fontface = "bold", size = 6) +
    theme_classic() +
    theme(
        legend.position = "none",
        axis.line = element_blank(),
        axis.text = element_blank(),
        axis.ticks = element_blank(),
        axis.title.x = element_text(size = 20, face = "bold"),
        axis.title.y = element_text(size = 20, face = "bold"),  
        plot.title = element_text(size = 20, face = "bold")
    )       

---

import numpy as np
import matplotlib.pyplot as plt

# Create figure and axes
fig, ax = plt.subplots()
ax.set_title('Use the Pythagorean Theorem to calculate C', fontsize=15, fontweight='bold')

# Define the points for the line
x = [0, 1]
y = [1, 0]

# Plot the horizontal and vertical lines
ax.axhline(y=0, color='black')  # x-axis
ax.axvline(x=0, color='black')  # y-axis

# Plot the diagonal line
ax.plot(x, y, color='black', linewidth=1.5)

# Fill the region enclosed by the axes and the line
plt.fill_between(x, y, 0, color='#9B111E', alpha=0.5)

# Add axis labels with bold and enlarged fonts
ax.text(0.5, -0.05, r"$\mathbf{A = \sqrt{-1}}$", ha='center', va='center', fontsize=14, transform=ax.transAxes)
ax.text(-0.05, 0.5, r"$\mathbf{B = 1}$", ha='center', va='center', fontsize=14, rotation=90, transform=ax.transAxes)

# Add diagonal line label with bold text, enlarged font, and moved farther off the line
ax.text(.6, .5, r"$\mathbf{C}$", ha='center', va='center', fontsize=14, color='black')

# Turn off the axes' tick marks and spines
ax.spines['top'].set_color('none')
ax.spines['right'].set_color('none')
ax.spines['bottom'].set_color('none')
ax.spines['left'].set_color('none')
ax.tick_params(left=False, bottom=False, labelleft=False, labelbottom=False)

# Display the plot
plt.show()

# end