![How to Science [Part 3: Experiments] @WelchLabs](https://i.ytimg.com/vi/b-CPQdWU-sI/maxresdefault.jpg)
Uploaded December 2017 | Updated September 2026, 3 weeks ago
PDF: welchlabs.com/guides
Support Welch Labs: patreon.com/welchlabs
Music:
premiumbeat.com/royalty_free_music/songs/jazz-manouche-forever
premiumbeat.com/royalty_free_music/songs/razer-trap
PDF: welchlabs.com/guides
Support Welch Labs: patreon.com/welchlabs
Music:
premiumbeat.com/royalty_free_music/songs/jazz-manouche-forever
premiumbeat.com/royalty_free_music/songs/razer-trap
![How to Science [Part 3: Experiments]](https://i.ytimg.com/vi/b-CPQdWU-sI/mqdefault.jpg "How to Science [Part 3: Experiments]
PDF: http://www.welchlabs.com/guides/
Support Welch Labs: https://www.patreon.com/welchlabs
Music:
https://premiumbeat.com/royalty_free_music/songs/jazz-manouche-forever
https://premiumbeat.com/royalty_free_music/songs/razer-trap")
![Learning to See [Part 11: Haystacks on Haystacks]](https://i.ytimg.com/vi/biy2yU3Auc4/mqdefault.jpg "Learning to See [Part 11: Haystacks on Haystacks]
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")
![Neural Networks Demystified [Part 1: Data and Architecture]](https://i.ytimg.com/vi/bxe2T-V8XRs/mqdefault.jpg "Neural Networks Demystified [Part 1: Data and Architecture]
Neural Networks Demystified
Part 1: Data and Architecture
@stephencwelch
Supporting Code:
https://github.com/stephencwelch/Neural-Networks-Demystified
In this short series, we will build and train a complete Artificial Neural Network in python. New videos every other friday.
Part 1: Data + Architecture
Part 2: Forward Propagation
Part 3: Gradient Descent
Part 4: Backpropagation
Part 5: Numerical Gradient Checking
Part 6: Training
Part 7: Overfitting, Testing, and Regularization")
![Self Driving Cars [S1E1: The ALV]](https://i.ytimg.com/vi/cExJbbwOfcw/mqdefault.jpg "Self Driving Cars [S1E1: The ALV]
PATREON: https://www.patreon.com/welchlabs
TWITTER: @welchlabs
MORE: http://www.welchlabs.com
CODE: https://github.com/stephencwelch/self_driving_cars
FURTHER READING
Original ALV Paper: http://www.cs.ucsb.edu/~mturk/Papers/ALV.pdf
Great Book on SCI: https://www.amazon.com/Strategic-Computing-Machine-Intelligence-1983-1993/dp/0262529262/ref=sr_1_1?ie=UTF8&qid=1547071120&sr=8-1&keywords=strategic+computing
Great book on the history of AI: https://www.amazon.com/Ai-Daniel-Crevier/dp/0465001041/ref=sr_1_1?ie=UTF8&qid=1547071150&sr=8-1&keywords=ai+the+tumultuous+history+of+the+search+for+artificial+intelligence
VIDEO REFERENCES
Elon Musk at TED: https://www.youtube.com/watch?v=NcqI76Z4t1A
Cadillac Super Cruise: https://www.youtube.com/watch?v rxW68ADldI
Tesla Perception: https://www.youtube.com/watch?v=VG68SKoG7vE
Waymo Perception: https://www.youtube.com/watch?v=B8R148hFxPw
Chris Urmson TED talk: https://www.youtube.com/watch?v=tiwVMrTLUWg
MUSIC
https://www.premiumbeat.com/royalty-free-tracks/girl-power
https://www.premiumbeat.com/royalty-free-tracks/out-of-the-woods
https://www.premiumbeat.com/royalty-free-tracks/language-2
SPECIAL THANKS TO
Tony Fast
Krish Ravindranath
Karthik Naga
Charles Young
Chang Lee
Mathew Turk
- And -
Vin Soma
Raphael J Vasquez
Nate Fuller")
![How to Science [Part 1: Music]](https://i.ytimg.com/vi/d3mHfqd0VZY/mqdefault.jpg "How to Science [Part 1: Music]
pdf:
http://www.welchlabs.com/guides
music: https://www.premiumbeat.com/royalty_free_music/songs/storytelling-piano
patreon:
https://www.patreon.com/welchlabs
correction:
length table should be in cm, to mm!")
![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")
