![Imaginary Numbers Are Real [Part 9: Closure] @WelchLabs](https://i.ytimg.com/vi/dLn5H69lS0w/hqdefault.jpg)
Uploaded November 2015 | Updated September 2026, 2 weeks ago
More information and resources: welchlabs.com
Imaginary numbers are not some wild invention, they are the deep and natural result of extending our number system. Imaginary numbers are all about the discovery of numbers existing not in one dimension along the number line, but in full two dimensional space. Accepting this not only gives us more rich and complete mathematics, but also unlocks a ridiculous amount of very real, very tangible problems in science and engineering.
Part 1: Introduction
Part 2: A Little History
Part 3: Cardan's Problem
Part 4: Bombelli's Solution
Part 5: Numbers are Two Dimensional
Part 6: The Complex Plane
Part 7: Complex Multiplication
Part 8: Math Wizardry
Part 9: Closure
Part 10: Complex Functions
Part 11: Wandering in Four Dimensions
Part 12: Riemann's Solution
Part 13: Riemann Surfaces
Want to learn more or teach this series? Check out the Imaginary Numbers are Real Workbook: welchlabs.com/resources.
More information and resources: welchlabs.com
Imaginary numbers are not some wild invention, they are the deep and natural result of extending our number system. Imaginary numbers are all about the discovery of numbers existing not in one dimension along the number line, but in full two dimensional space. Accepting this not only gives us more rich and complete mathematics, but also unlocks a ridiculous amount of very real, very tangible problems in science and engineering.
Part 1: Introduction
Part 2: A Little History
Part 3: Cardan's Problem
Part 4: Bombelli's Solution
Part 5: Numbers are Two Dimensional
Part 6: The Complex Plane
Part 7: Complex Multiplication
Part 8: Math Wizardry
Part 9: Closure
Part 10: Complex Functions
Part 11: Wandering in Four Dimensions
Part 12: Riemann's Solution
Part 13: Riemann Surfaces
Want to learn more or teach this series? Check out the Imaginary Numbers are Real Workbook: welchlabs.com/resources.
![Imaginary Numbers Are Real [Part 9: Closure]](https://i.ytimg.com/vi/dLn5H69lS0w/mqdefault.jpg "Imaginary Numbers Are Real [Part 9: Closure]
More information and resources: http://www.welchlabs.com
Imaginary numbers are not some wild invention, they are the deep and natural result of extending our number system. Imaginary numbers are all about the discovery of numbers existing not in one dimension along the number line, but in full two dimensional space. Accepting this not only gives us more rich and complete mathematics, but also unlocks a ridiculous amount of very real, very tangible problems in science and engineering.
Part 1: Introduction
Part 2: A Little History
Part 3: Cardans Problem
Part 4: Bombellis Solution
Part 5: Numbers are Two Dimensional
Part 6: The Complex Plane
Part 7: Complex Multiplication
Part 8: Math Wizardry
Part 9: Closure
Part 10: Complex Functions
Part 11: Wandering in Four Dimensions
Part 12: Riemanns Solution
Part 13: Riemann Surfaces
Want to learn more or teach this series? Check out the Imaginary Numbers are Real Workbook: http://www.welchlabs.com/resources.")
![Learning to See [Part 5: To Learn is to Generalize]](https://i.ytimg.com/vi/efR8ybG7Ihs/mqdefault.jpg "Learning to See [Part 5: To Learn is to Generalize]
In this series, well explore the complex landscape of machine learning and artificial intelligence through one example from the field of computer vision: using a decision tree to count the number of fingers in an image. Its gonna be crazy.
Supporting Code: https://github.com/stephencwelch/LearningToSee
welchlabs.com
@welchlabs")
![The most beautiful equation in math, explained visually [Euler’s Formula]](https://i.ytimg.com/vi/f8CXG7dS-D0/mqdefault.jpg "The most beautiful equation in math, explained visually [Euler’s Formula]
Welch Labs Imaginary Numbers Book!
https://www.welchlabs.com/resources/imaginary-numbers-book
Book Digital Version
https://www.welchlabs.com/resources/imaginary-numbers-are-real-book-digital-download
Euler’s Formula Poster!
https://www.welchlabs.com/resources/eulers-formula-poster-13x19
Poster Digital Version
https://www.welchlabs.com/resources/eulers-formula-dark-mode-poster-digital-download
Special thanks to the Patrons:
Juan Benet, Ross Hanson, Yan Babitski, AJ Englehardt, Alvin Khaled, Eduardo Barraza, Hitoshi Yamauchi, Jaewon Jung, Mrgoodlight, Shinichi Hayashi, Sid Sarasvati, Dominic Beaumont, Shannon Prater, Ubiquity Ventures, Matias Forti
Tattoo by @themutemaker - thank you! Check out her awesome work here:
https://www.instagram.com/themutemaker
Welch Labs
Ad free videos and exclusive perks: https://www.patreon.com/welchlabs
Watch on TikTok: https://www.tiktok.com/@welchlabs
Learn More or Contact: https://www.welchlabs.com/
Instagram: https://www.instagram.com/welchlabs
X: https://twitter.com/welchlabs
References & Notes
Welch Labs Imaginary Numbers Series: https://www.youtube.com/watch?v=T647CGsuOVU
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-2007-08.pdf
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). Eulers 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/cgi/viewcontent.cgi?filename=0&article=1806&context=euler-works&type=additional
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
https://www.feynmanlectures.caltech.edu/I_22.html
CFAQJOTYQHT7JYIT")
![Channel Update [April 2017]](https://i.ytimg.com/vi/gZPy7OxAjRM/mqdefault.jpg "Channel Update [April 2017]
Help me make more videos: https://www.patreon.com/welchlabs
YouTube channels referenced: MinutePhysics, Veritaseum, VSauce, Physics Girl, CGP Grey, Mathologer, 3Blue1Brown, and Numberphile.
A special thanks to Charles Young for all his fantastic help with the motion rig. Thanks to Joan and Alison Young for cutting out an absolutely amazing ransom note.
Music: https://www.premiumbeat.com/royalty_free_music/songs/yours-faithfully
https://www.premiumbeat.com/royalty_free_music/songs/it-s-always-funny-in-philadelphia")
![Learning To See [Part 13: Heuristics]](https://i.ytimg.com/vi/g_sA8hYU3b8/mqdefault.jpg "Learning To See [Part 13: Heuristics]
In this series, well explore the complex landscape of machine learning and artificial intelligence through one example from the field of computer vision: using a decision tree to count the number of fingers in an image. Its gonna be crazy.
Supporting Code: https://github.com/stephencwelch/LearningToSee
welchlabs.com
@welchlabs")
![Learning To See [Part 1: Introduction]](https://i.ytimg.com/vi/i8D90DkCLhI/mqdefault.jpg "Learning To See [Part 1: Introduction]
In this series, well explore the complex landscape of machine learning and artificial intelligence through one example from the field of computer vision: using a decision tree to count the number of fingers in an image. Its gonna be crazy.
Become a Patron for exclusive perks: https://www.patreon.com/welchlabs
Supporting Code: https://github.com/stephencwelch/LearningToSee
welchlabs.com
@welchlabs")
