In mathematics and physics, complex numbers aren't just "imaginary" values—they are the secret language of 2D rotation. While real numbers live on a 1D line, complex numbers inhabit a 2D plane, and multiplying them acts as a bridge between dimensions.
1. The Geometry of i
To understand how we switch dimensions, look at the imaginary unit i. In a standard real-number system, you only move left or right. Adding i introduces a vertical axis.
* The 90-degree turn: Multiplying a real number by i is geometrically equivalent to a 90° counter-clockwise rotation.
* The Dimension Switch: If you start at 1 (on the x-axis) and multiply by i, you land at i (on the y-axis). You have effectively "switched" your direction from horizontal to vertical.
2. Rotation via Euler’s Formula
The most elegant link between complex numbers and rotation is Euler’s Formula:
This formula places any complex number on a unit circle in the complex plane. When you multiply a vector by e^{i\theta}, you aren't changing its length; you are simply rotating it by the angle \theta.
Why this matters:
* Algebraic Simplicity: Instead of using messy rotation matrices (which involve four separate multiplications and additions), you can rotate a point by simply multiplying two complex numbers.
* Phase in Physics: This is why complex numbers are used in electrical engineering and quantum mechanics. A "phase shift" in a wave is just a rotation in the complex plane.
3. Beyond 2D: Quaternions
If complex numbers (a + bi) handle 2D rotations by adding one imaginary dimension, what happens if we want to rotate in 3D?
To handle 3D space without hitting "Gimbal Lock" (where two axes align and you lose a degree of freedom), mathematicians use Quaternions. These extend the concept to three imaginary units: i, j, and k.
> The Rule of Four: Interestingly, to rotate smoothly in three dimensions, you actually need a four-dimensional number system.
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Summary Table
| Number System | Dimensions | Primary Use in Rotation |
|---|---|---|
| Real Numbers | 1D | Scaling (stretching/shrinking) |
| Complex Numbers | 2D | Planar rotation, oscillations, AC circuits |
| Quaternions | 4D | 3D computer graphics, aerospace navigation |
They can be treated as vectors, but they have "superpowers" that standard vectors do not.
1. The Similarities (The 2D Map)
In a purely visual or structural sense, a complex number z = a + bi behaves exactly like a 2D vector \vec{v} = (a, b).
* Addition: Adding two complex numbers is identical to "tip-to-tail" vector addition.
* Magnitude: The "absolute value" (modulus) of a complex number |z| = \sqrt{a^2 + b^2} is the same as the length of a vector.
* Coordinates: Both represent a point on a 2D plane.
2. The Difference: Multiplication
This is where complex numbers leave standard 2D vectors in the dust.
In standard vector algebra (like what you'd use in an introductory physics class), there isn't a single, clean way to "multiply" two 2D vectors to get another 2D vector. You have the Dot Product (which gives you a single number/scalar) and the Cross Product (which actually points out of the 2D plane into the 3D world).
Complex numbers, however, can be multiplied together to produce another complex number.
The "Rotation" Secret
When you multiply two complex numbers, the math automatically handles two things at once:
* Scaling: The lengths are multiplied.
* Rotation: The angles are added.
Standard vectors cannot do this on their own; you would need to bring in a "Rotation Matrix" to force a vector to turn. A complex number just "knows" how to turn naturally through its imaginary component.
3. When to use which?
Mathematically, complex numbers form a Field, while vectors form a Vector Space.
* Use Vectors when you are dealing with forces, velocities, or any dimension higher than 2 (like 3D space).
* Use Complex Numbers when you are dealing with things that rotate, vibrate, or oscillate (like radio waves, electricity, or quantum particles).
> The Peer-to-Peer Truth: Think of a complex number as a vector with an attitude. It lives in the same 2D house, but it knows how to spin and transform itself algebraically in ways a simple (x, y) coordinate cannot.
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You would be surprised about what the 4.5 models can already do in these ways of thinking. I think that one can unlock this power with the right set of prompts. It's impressive, truly.
It has already understood so much, we just need to reap the fruits.
I'm really looking forward to trying the new version.
How is generating a continuation prompt materially different from compaction? Do you manually scrutinize the context handoff prompt? I've done that before but if not I do not see how it is very different from compaction.
I wonder if it's just: compact earlier, so there's less to compact, and more remaining context that can be used to create a more effective continuation
The first thing I would do in case of depression is to make sure that the patient's energy levels are good and that mitochondria and other energy-related biochem phenomena work as expected.
I know first-hand that low energy-levels and lacking energy production mechanically lead to depression.
Also, look at how people (children also) experience the world and their relationships and their stresses when they are tired (or even just hungry) compared to when they are fit...
Fix those, and the depression might be gone.
This is not bashing against anti-depressants, they play their role to. But in some cases, energy-management is key.
Thank you for the comment. This is not in my area of expertise, so I hope you can clarify - how does one test that "mitochondria and other energy-related biochem phenomena work as expected"?
Chris Masterjohn is a noted quack. He takes bits of actual science and research and weaves them together into narratives that make it sound like he has everything figured out with his unique protocols, but it doesn’t hold up to actual scrutiny. People spend years following his ever changing protocols without getting anywhere (beyond placebo effect and a large bill for supplements)
I know I won’t convince the parent commenter but hopefully I can convince other readers not to go down this road or invest any money in anything related to him.
Pure ad hominem FUD. “This guy sometimes disagrees with scientists employed by the government, don’t listen to him!”.
The technical details are beyond my understanding but I’ve heard from a PhD in the field that Masterjohn’s understanding of metabolism is second to none. Whether his protocols work or not is certainly a case by case matter (like any health protocol), but he always appears to substantiate it with well-cited lines of argument, and is willing to engage with interlocutors.
As for spending years with changing protocols without getting anywhere besides spending lots of money, well that can be said for people with complex issues who go the institutionally approved route as well. It isn’t discrediting in its own right that a protocol didn’t work for some.
I have spent the last week with Claude Code, instructing it to find all it can about Myalgic Encephalomyelitis / Chronic
Fatigue Syndrome (ME/CFS).
This paper is the result _so far_.
Specifically, I asked the AI to propose treatment protocols for people with severe ME/CFS, which should yield significant improvements quickly. With the hope to give them back quickly a decent quality of life and the emotional and physical means to fight against this disease in the long term.
Disclaimer: I am not a doctor.
This work is the result of careful instructions given to an AI, including requests for the AI to formulate novel hypotheses, make links to other diseases with similar symptoms or underlying mechanisms.
So, while a part of the paper is generated in a brainstorming mode, the protocols of chapters 16 and 17 are not, and most of the paper has been generated and documented with explicit request of honesty, strictest math. and logic rigor and correctness, rigorous citations of sources.
It also proposes several research directions, sorted from most promising to least promising.
This paper provides additional information and insights (not medical advice - that, you might get from your doctor, maybe sharing the paper with them), I wish it will help those out there who need help managing ME/CFS or who face a continuous lack of energy, wherever it may come from.
Incapable: that happens when the acceptance of an idea implies that their perception of their identity is flawed and has, logically to change in order to adapt for the new reality where the idea has its place.
Denial is a protection mechanism, and it is very effective when the reality is too difficult to support as it is.
Identity is so essential in our beliefs, attitudes and behaviours that most of us won't accept anything that requires it to change.
Unless we accept that failure is part of our identity and that this means that our identity sometimes has to evolve. But that has to be done willingly, explicitly (in our minds).
You can swap out the motherboard, so yeah. Not sure how long they support specific Laptop bodies (or frames) for but I would imagine some of their frames are good for a long minute.
Probably because they include the CPU if I am not mistaken, but probably makes sense and is worth the hassle if your concern is your CPU / main hardware is falling behind.
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