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In 1747, Euler spent 133 pages proving that logarithms have infinitely many solutions - and in doing so accidentally revealed why every LLM, every signal processor, and every Fourier transform works the way it does. The imaginary unit I doesn't break exponential functions. It rotates them, turning exponential growth...

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Deliriumm

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118,141 Aufrufe • vor 1 Tag •via X (Twitter)

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The AI Therapistvor 1 Tag

Euler’s complex logs map to rotation. LLMs use similar math for attention vectors, rotating meaning in high-D space so context sticks. It is basically magic that works because the angles align perfectly with human logic. Kind of elegant, don’t you think?

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Ricardo Alvarengavor 1 Tag

Links e créditos por gentileza!

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Deepvor 1 Tag

133 pages for that is wild. i use complex numbers in audio code every week and never once thought about euler proving why they work.

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Most people first see Euler’s Formula as a strange equation in a textbook. Then years later, they realize it quietly powers the modern world. Leonhard Euler discovered that these seemingly unrelated mathematical ideas: → exponential growth → imaginary numbers → sine waves → cosine waves → rotation … are all deeply connected through one identity: e^(iθ) = cos(θ) + i·sin(θ) At first glance it looks impossible. How can an exponential function suddenly produce circles and waves? The key insight is that multiplying by a complex exponential creates rotation. As the angle θ changes: - the cosine term tracks horizontal motion - the sine term tracks vertical motion - together they trace a perfect circle in the complex plane Euler showed that waves and rotation are mathematically the same phenomenon viewed differently. That single idea changed science and engineering forever. Today, Euler’s Formula sits underneath: → Fourier Transforms → signal processing → wireless communication → MRI scanners → quantum mechanics → electrical engineering → audio compression → radar systems → GPS → neural network frequency analysis Even modern AI systems indirectly rely on mathematics built on top of these foundations. The famous special case is Euler’s Identity: e^(iπ) + 1 = 0 Richard Feynman reportedly called it: "our jewel." Because it revealed that mathematics is not a collection of separate topics. It is one connected language describing reality itself.
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Most people first see Euler’s Formula as a strange equation in a textbook. Then years later, they realize it quietly powers the modern world. Leonhard Euler discovered that these seemingly unrelated mathematical ideas: → exponential growth → imaginary numbers → sine waves → cosine waves → rotation … are all deeply connected through one identity: e^(iθ) = cos(θ) + i·sin(θ) At first glance it looks impossible. How can an exponential function suddenly produce circles and waves? The key insight is that multiplying by a complex exponential creates rotation. As the angle θ changes: - the cosine term tracks horizontal motion - the sine term tracks vertical motion - together they trace a perfect circle in the complex plane Euler showed that waves and rotation are mathematically the same phenomenon viewed differently. That single idea changed science and engineering forever. Today, Euler’s Formula sits underneath: → Fourier Transforms → signal processing → wireless communication → MRI scanners → quantum mechanics → electrical engineering → audio compression → radar systems → GPS → neural network frequency analysis Even modern AI systems indirectly rely on mathematics built on top of these foundations. The famous special case is Euler’s Identity: e^(iπ) + 1 = 0 Richard Feynman reportedly called it: "our jewel." Because it revealed that mathematics is not a collection of separate topics. It is one connected language describing reality itself.

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Fourier wrapped a signal around a circle in 1822. Every Spotify equalizer, every noise filter in your AirPods, and every frequency layer inside ChatGPT runs on what he found. The trick is simple. Spin the signal at different speeds. When the spin matches a hidden frequency, the center of mass shifts. That shift is the answer. Claude and GPT use this same idea to separate relevant tokens from noise. 20 minutes. Bookmark & watch today. The math is older than the telephone.
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Fourier wrapped a signal around a circle in 1822. Every Spotify equalizer, every noise filter in your AirPods, and every frequency layer inside ChatGPT runs on what he found. The trick is simple. Spin the signal at different speeds. When the spin matches a hidden frequency, the center of mass shifts. That shift is the answer. Claude and GPT use this same idea to separate relevant tokens from noise. 20 minutes. Bookmark & watch today. The math is older than the telephone.

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Complex numbers aren't imaginary. They're the only way to describe a wave that has both amplitude and phase at the same time - and every LLM processes language as exactly that kind of wave through its attention layers. The same math that describes quantum wave interference is what makes Fourier transforms work, which is what lets GPT process tokens in parallel instead of one at a time. Descartes named them "imaginary" before anyone understood what they were. 20 minutes. Bookmark & watch today. The confusion clears up faster than any textbook manages.
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Complex numbers aren't imaginary. They're the only way to describe a wave that has both amplitude and phase at the same time - and every LLM processes language as exactly that kind of wave through its attention layers. The same math that describes quantum wave interference is what makes Fourier transforms work, which is what lets GPT process tokens in parallel instead of one at a time. Descartes named them "imaginary" before anyone understood what they were. 20 minutes. Bookmark & watch today. The confusion clears up faster than any textbook manages.

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If you're an entrepreneur and you're taking a lot of risk, you wanna get five million bucks in T-bills sitting in account. That is what I call the f*** you money that gives you the power to do whatever you want. That protects you and you have to have the discipline to park that. You get to look at it every day and every month you get a big check off it, and leave it alone. Not everyone can get there, but if you get your first pass, take five and just put it into T-bills. Really boringo.
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If you're an entrepreneur and you're taking a lot of risk, you wanna get five million bucks in T-bills sitting in account. That is what I call the f*** you money that gives you the power to do whatever you want. That protects you and you have to have the discipline to park that. You get to look at it every day and every month you get a big check off it, and leave it alone. Not everyone can get there, but if you get your first pass, take five and just put it into T-bills. Really boringo.

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how beautiful it must be to have someone in your life that supports you every step of the way and does it all just because he loves you and is proud of you
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