![Imaginary Numbers Are Real [Part 8: Math Wizardry] @WelchLabs](https://i.ytimg.com/vi/iecUL8_OxrU/sddefault.jpg)
Uploaded October 2015 | Updated September 2026, 1 week 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 8: Math Wizardry]](https://i.ytimg.com/vi/iecUL8_OxrU/mqdefault.jpg "Imaginary Numbers Are Real [Part 8: Math Wizardry]
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.")
![Yann LeCuns $1B Bet Against LLMs [Part 1]](https://i.ytimg.com/vi/kYkIdXwW2AE/mqdefault.jpg "Yann LeCuns $1B Bet Against LLMs [Part 1]
Apply to join Hudson River Trading: https://www.hudsonrivertrading.com/welchlabs
Welch Labs Book: https://www.welchlabs.com/resources/ai-book-ezrzm-msrmc
Patreon: https://www.patreon.com/c/welchlabs
Part 2: https://www.youtube.com/watch?v=v_jDvpEGTIg
Sections
0:00 - Intro
2:28 - The Problem with Deep Learning
4:17 - Intelligence is a Cake
5:15 - The Rise of Generative AI
8:00 - Blurry Images
8:54 - HRT is an awesome place to work
11:16 - But why so Blurry?
13:30 - Do our models need to be generative?
15:16 - Siamese Networks
17:53 - Representation Collapse
19:54 - Yann’s Epiphany & Barlow Twins
27:22 - DINO
28:58 - JEPA & World Models
34:09 - But is JEPA good?
36:19 - Welch Labs Book
Special thanks to: Yann LeCun, Stephane Deny, David Fan, Nicolas Ballas
Clip of Yann from 1989: https://www.youtube.com/watch?v=FwFduRA_L6Q
CNN Paper: http://yann.lecun.com/exdb/publis/pdf/lecun-89e.pdf
LeNet-5 paper: http://vision.stanford.edu/cs598_spring07/papers/Lecun98.pdf
Dashcam video
https://commons.wikimedia.org/wiki/File:Car_Driving_Faadou_4K_HDR Rural_road Canton 327.webm
Image Credits
https://en.wikipedia.org/wiki/File:Dota_2_Gameplay_Aug_2017.jpg
https://commons.wikimedia.org/wiki/File:Felis_catus-cat_on_snow.jpg
https://commons.wikimedia.org/wiki/File:Magnificent_CME_Erupts_on_the_Sun August_31.jpg
https://commons.wikimedia.org/wiki/File:Alcedo_atthis Riserve_naturali_e_aree_contigue_della_fascia_fluviale_del_Po.jpg
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V-JEPA2 Robot Arm Videos
https://ai.meta.com/research/vjepa/
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, Brian Henry, Tim Palade, Petar Vecutin, Nicolas baumann, Jason Singh, Robert Riley, vornska, Barry Silverman, Jake Ehrlich, Mitch Jacobs, Lauren Steely, Jeff Eastman, Rodolfo Ibarra, Clark Barrus, Rob Napier, Andrew White, Richard B Johnston, abhiteja mandava, Burt Humburg, Kevin Mitchell, Daniel Sanchez, Ferdie Wang, Tripp Hill, Richard Harbaugh Jr, Prasad Raje, Kalle Aaltonen, Midori Switch Hound, Zach Wilson, Chris Seltzer, Ven Popov, Hunter Nelson, Amit Bueno, Scott Olsen, Johan Rimez, Shehryar Saroya, Tyler Christensen, Beckett Madden-Woods, Darrell Thomas, Javier Soto, U007D, Caleb Begly, Rick Rubenstein, Brent Hunsaker, Dan Patterson, Tchsurvives, Alex Adai, Walter Reade, Zyansheep, Walter Reade, Duncan Stannett, Reginald Carey, Jean-Manuel Izaret, dh71633, Adrian Rodriguez, Dimitar Stojanovski, Michael Harder, Peter Maldonado, Emily Pesce, David Johnston, Insang Song, FaeTheWolf, Stephen Taylor, KittenKaboodle, EMatter, PATRICKMCCORMACK, John Beahan, Cameron, Cole Jones, Garrett Thornburg, Jeroen W, Rohit Sharma, GlennB, Emmanuel Cortes, Katie Quinn, Karina C, Cakra WW, Mike Ton, Eric Gometz, MacCallister Higgins, Niko Drossos, David Eraso, Tom Zehle, Steve, Brian Lineburg, rjbl, Michael Loh, Perry Vais, Bengal0, Farhad Manjoo, Sara Chipps, Ellis Driscoll, William Taysom, Will Harmon, CK, Abdullah, Peter Cho, Leo Nikora, Griffin Smith, Ash Katnoria, Alex, Markus Hays Nielsen, Catherine H., Vi, David Dobáš, Peter Wang, Sina Sohangir, Danny Thomas, Julian Francis, Hans Adler, Jiayu Peng, Weston M, Youssouf da Silva, John Thomas, Samuel Costello, Sam Adams, Bryan Liles, Malaya Zemlya, Karl, Vahe Andonians, Mike Doughty, Larry Novelo, Jonas Acres, Ludicrum Rex, Robert Blumofe, Anthony Z, Alex Zhao, Dan Babitch, Nikko Patten
Supporting code: https://github.com/WelchLabs/videos
Created by: Sam Baskin, Pranav Gundu, and Stephen Welch
Content ID: CFAQJOTYQHT7JYIT")
![Learning to See [Part 7: There is no f]](https://i.ytimg.com/vi/klWUOO4sHaA/mqdefault.jpg "Learning to See [Part 7: There is no f]
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")
![ChatGPT is made from 100 million of these [The Perceptron]](https://i.ytimg.com/vi/l-9ALe3U-Fg/mqdefault.jpg "ChatGPT is made from 100 million of these [The Perceptron]
Go to https://drinkag1.com/welchlabs to subscribe and save $20 off your first subscription of AG1! Thanks to AG1 for sponsoring todays video.
Imaginary Numbers book is back in stock! Update at 23:11
https://www.welchlabs.com/resources/imaginary-numbers-book
Welch Labs Posters:
https://www.welchlabs.com/resources
Perceptrons designs from viewers:
https://sites.google.com/view/jacklangsdorf-funprojects/home/perceptron?authuser=0
https://github.com/rdb64-hobbies/Perceptron/blob/main/BUILDING.md
Really nice Perceptron simulators built by viewers:
https://github.com/srives/Perceptron
https://priyangsubanerjee.github.io/perceptron-simulator/
Special Thanks to Patrons https://www.patreon.com/welchlabs
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, Brian Henry, Tim Palade, Petar Vecutin, Nicolas baumann, Jason Singh, Robert Riley, vornska, Barry Silverman
References
Rumelhart, D. E., Mcclelland, J. L. (1987). Parallel Distributed Processing, Volume 1: Explorations in the Microstructure of Cognition: Foundations. United Kingdom: Penguin Random House LLC.
Talking Nets: An Oral History of Neural Networks. (2000). United Kingdom: MIT Press.
Prince, S. J. (2023). Understanding Deep Learning. United Kingdom: MIT Press.
Crevier, D. (1993). AI : the tumultuous history of the search for artificial intelligence. New York: Basic Books.
Cat and dog face dataset: https://www.kaggle.com/datasets/andrewmvd/animal-faces?resource=download
Minsky, M., Papert, S. (2017). Perceptrons: An Introduction to Computational Geometry. United Kingdom: MIT Press.
Widrow, Bernard, and Michael A. Lehr. 30 years of adaptive neural networks: perceptron, madaline, and backpropagation. *Proceedings of the IEEE* 78.9 (1990): 1415-1442.
Olazaran, Mikel. A sociological history of the neural network controversy. *Advances in computers*. Vol. 37. Elsevier, 1993. 335-425.
Widrow, Bernard. Generalization and information storage in networks of adaline neurons. *Self-organizing systems* (1962): 435-461.
Widrow, Bernard. Thinking about thinking: the discovery of the LMS algorithm. *IEEE Signal Processing Magazine* 22.1 (2005): 100-106.
Technical Notes
Method for counting neurons in ChatGPT: Starting with GPT-2 implementation here: https://github.com/karpathy/build-nanogpt/blob/master/train_gpt2.py - keys, queries, and values are implemented in Linear layers with n_embd inputs and 3*n_embd outputs, where n_embd is the embedding dimension. Output projection layer has n_embd and n_embd outputs. So a single attention layer will have ~4*n_embd neurons. GPT-3 has an embedding dimension of 12,288, so each attention layer has ~49,152 neurons. Each MLP block has n_embd inputs, 4*n_embd hidden units, and n_embd outputs, so ~5*n_embd total neurons, or ~61,440. Total neuron count for GPT-3 is then 96*(49,152+61,440)=10,616,832, ignoring initial embedding and final unembedding. Finally, GPT-4 reportedly has ~1.8 Trillion parameters (https://semianalysis.com/2023/07/10/gpt-4-architecture-infrastructure/), making it ~10x larger than GPT-3. Note that GPT-4 is reportedly a mixture of experts, and not all experts are used for each inference, so it appears that not all 1.8 trillion parameters are used for a given inference call. Assuming that ~10x the parameters means 10x the neurons, then GPT-4 should have ~100M neurons.
CFAQJOTYQHT7JYIT")
