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 Real [Part 1: I...  

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*, https://doi.org/10.2307/2973509. 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*, https://doi.org/10.2307/2974078. 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*, https://doi.org/10.2307/2973441. 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*, https://doi.org/10.2307/2972960. Accessed 22 July 2024.

Nice History of Euler’s Formula
Sandifer, Ed. e, pi and i: Why is “Euler” in the Euler identity? http://eulerarchive.maa.org/hedi/HEDI...

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: https://scholarlycommons.pacific.edu/...

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
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