![Imaginary Numbers Are Real [Part 10: Complex Functions] @WelchLabs](https://i.ytimg.com/vi/pNp8Qf20-sA/hqdefault.jpg)
Uploaded June 2016 | Updated September 2026, 2 weeks ago
Supporting Code: github.com/stephencwelch/Imaginary-Numbers-Are-Real
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.
Supporting Code: github.com/stephencwelch/Imaginary-Numbers-Are-Real
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 10: Complex Functions]](https://i.ytimg.com/vi/pNp8Qf20-sA/mqdefault.jpg "Imaginary Numbers Are Real [Part 10: Complex Functions]
Supporting Code: https://github.com/stephencwelch/Imaginary-Numbers-Are-Real
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.")
![Why Deep Learning Works Unreasonably Well [How Models Learn Part 3]](https://i.ytimg.com/vi/qx7hirqgfuU/mqdefault.jpg "Why Deep Learning Works Unreasonably Well [How Models Learn Part 3]
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Sections
0:00 - Intro
4:49 - How Incogni Saves Me Time
6:32 - Part 2 Recap
8:10 - Moving to Two Layers
9:15 - How Activation Functions Fold Space
11:45 - Numerical Walkthrough
13:42 - Universal Approximation Theorem
15:45 - The Geometry of Backpropagation
19:52 - The Geometry of Depth
24:27 - Exponentially Better?
30:23 - Neural Networks Demystifed
31:50 - The Time I Quit YouTube
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References
Simon Prince, Understanding Deep Learning. https://udlbook.github.io/udlbook/
Liang, Shiyu, and Rayadurgam Srikant. Why deep neural networks for function approximation?. arXiv preprint arXiv:1610.04161 (2016).
Hanin, Boris, and David Rolnick. Deep relu networks have surprisingly few activation patterns. *Advances in neural information processing systems* 32 (2019).
Hanin, Boris, and David Rolnick. Complexity of linear regions in deep networks. *International Conference on Machine Learning*. PMLR, 2019.
Fan, Feng-Lei, et al. Deep relu networks have surprisingly simple polytopes. *arXiv preprint arXiv:2305.09145* (2023).
All Code:
https://github.com/stephencwelch/manim_videos
100k neuron wide example training code: https://github.com/stephencwelch/manim_videos/blob/master/_2025/backprop_3/notebooks/Wide%20Training%20Example.ipynb
Code from viewer Hugo Brouwer that achieves 99.5%+ with less than 100 neurons using Fourier Features!
https://github.com/AgntBrwr/baarle-hertog-fourier-features
Code from viewer Nico Waser that uses 100 neurons:
https://github.com/Waser2004/Illustrated Guide_to_AI/tree/main/Chapter%204%20-%20Deep%20Learning
Written by: Stephen Welch
Produced by: Stephen Welch, Sam Baskin, and Pranav Gundu
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![How to Science [Part 5: Mathematics]](https://i.ytimg.com/vi/sTSNTWKGkbw/mqdefault.jpg "How to Science [Part 5: Mathematics]
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![Learning to See [Part 4: Machine Learning]](https://i.ytimg.com/vi/sarVw-iVWgc/mqdefault.jpg "Learning to See [Part 4: Machine Learning]
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 14: Better Heuristics]](https://i.ytimg.com/vi/tPHImr2sFBM/mqdefault.jpg "Learning To See [Part 14: Better 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")
