The most beautiful equation in math, explained visually [Euler’s Formula]

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References & Notes
Welch Labs Imaginary Numbers Series: • Imaginary Numbers Are ...
Excellent History of Logarithms by Florian Cajori
Cajori, Florian. “History of the Exponential and Logarithmic Concepts.” *The American Mathematical Monthly*, vol. 20, no. 1, 1913, pp. 5-14. *JSTOR*, doi.org/10.230.... Accessed 22 July 2024.
Cajori, Florian. “History of the Exponential and Logarithmic Concepts.” *The American Mathematical Monthly*, vol. 20, no. 2, 1913, pp. 35-47. *JSTOR*, doi.org/10.230.... Accessed 22 July 2024.
Cajori, Florian. “History of the Exponential and Logarithmic Concepts:” *The American Mathematical Monthly*, vol. 20, no. 3, 1913, pp. 75-84. *JSTOR*, doi.org/10.230.... Accessed 22 July 2024.
Cajori, Florian. “History of the Exponential and Logarithmic Concepts.” *The American Mathematical Monthly*, vol. 20, no. 4, 1913, pp. 107-17. *JSTOR*, doi.org/10.230.... Accessed 22 July 2024.
Nice History of Euler’s Formula
Sandifer, Ed. e, pi and i: Why is “Euler” in the Euler identity? eulerarchive.ma...
Much of the visual approach presented here comes from Needham’s incredible book:
Needham, T. (1997). Visual Complex Analysis. United Kingdom: Clarendon Press.
Other books referenced
Maor, E. (2011). E: The Story of a Number. Ukraine: Princeton University Press.
Penrose, R. (2021). The Road to Reality: A Complete Guide to the Laws of the Universe. United Kingdom: Knopf Doubleday Publishing Group.
Dunham, W. (2022). Euler: The Master of Us All. United States: AMM Press.
Wilson, R. (2019). Euler's Pioneering Equation: The Most Beautiful Theorem in Mathematics. United Kingdom: Oxford University Press.
Nahin, P. J. (2010). An Imaginary Tale: The Story of √-1. Ukraine: Princeton University Press.
Stillwell, J. (2013). Mathematics and Its History. United Kingdom: Springer New York.
Euler’s Amazing 1747 Paper
Euler, Leonard. "Sur les logarithmes des nombres négatifs et imaginaires” Written in 1747, but not published until 1862. Euler did publish a similar paper in 1749. See Cajori #3 above.
English Translation: scholarlycommo...
Note on Benroulli’s area of sectors
Euler’s counterexample using Bernoulli’s sector area comes in a couple flavors. The one presented in his 1747 paper "Sur les logarithmes des nombres négatifs et imaginaires” is a bit different than an earlier example in a letter to Bernoulli. I chose the earlier example for clarity. See Cajori vol 2 and Sandifer.
Feynman Lectures - Algebra
www.feynmanlec...