That makes no difference insofar as the premise of the article is concerned.
927 karma · joined December 5, 2024
The entire point of the article was that human language isn't suitable for instructing computers, and you somehow make the leap to "humans instructing humans must be the same thing".
Reading comprehension must be a thing of the past at this point.
The lack of intelligence in the loop is the foolishness the title refers to.
But since you seem to be into that kind of, um, content, just cut to the chase and spare us all from listening to another angry man for an hour:
https://en.wikipedia.org/wiki/Zionist_Occupation_Government_...
I purchased a domain name from them yesterday, in addition to a few I already had.
This will be the last purchase unless I see some significant traction in the case we're discussing in particular, and company wide in general.
Hope others will do the same. Not punishing this behavior is encouraging such abuse in the entire industry (if Namecheap gets away with that, everyone else will follow).
And as a former Google Maps employee... I'm loving it.
Such security, very rules, wow.
Because an ZIP file with the encrypted key and the password to it in plain text is sooooo much more secure (but it doesn't break the rules, right?).
If you want to see how Scott Adams has gone crazy, watch e.g. Behind the Bastards episode(s) on him:
Never ever seen it in my life though. It was a big "WAS" even then.
The i386-i486 machines we had in middle school were on the Ethernet bus (I remember the BNC cable going along the classroom, with T-joints to the network cards).
IIRC Token Ring wasn't a thing in practice for 100Mbit speeds and beyond, and went the way of the dodo faster than DOS did.
Oh, and hear me out: we have now enough bytes to allow unlimited number of syllables in haiku poems!
Isn't it neat? Now this comment can also be a haiku, and MS Paint can be Photoshop.
Yay progress!
11/10, does exactly what I need it to do.
Anyway I just use Everything by Voidtools[1] for file search because it works instantly and, well, actually works.
Yes you do, if you want to claim it's easy.
Otherwise you're just bullshitting.
>I've skimmed over announcements of the feature becoming available in a product or two
See above. That's not equivalent to "it's easy". You don't have the basis to say that.
>I've never actually done it
You can't be serious.
If it's easy, go do it, find out, and come back here to tell us about it.
And the solution was: make it the user's problem.
Yay.
Can you help me figure out some things?
1. One time when I was abroad in Poland, a thief walked into my room at night and stole my wallet, phone, and laptop.
(The thief was eventually caught 6 months later, but that's another story).
Question: what to do in that case if I were using passkeys?
Is there a solution that doesn't require trusting a cloud provider?
2. Currently passkeys are offered as an alternative to passwords. What are the mechanisms that require passwords to be available as well?
I.e. what stops my bank from simply giving me a passkey at a branch and offering no other option?
3. Clouds are currently the easiest way to use passkeys. Last time I checked, you were locked into that cloud, but things move fast.
What ensures the ability to move the keys out of the cloud?
And what do I need to do to have my own "cloud" like that?
4. What protection does the passkey ecosystem offer for physical threat to the user?
With passwords, there's no way for anyone to know which Hacker News accounts are mine, or whether I still have access to them. I don't remember has plausible deniability (who hasn't ever forgotten a password?).
With passkeys, getting detained with your device means surrendering all your passwords, to my understanding.
What am I missing?
Thanks!
The spec requires passwords to still be an option, they say, so there's nothing to see here, move along — and avert their eyes should an inconvenient argument come their way.
Because, see, we all know that specs are ironclad, never change, and are always fully implemented.
Also, putting all your eggs, ahem, passkeys into one cloud basket (as this article recommends) to "move around" is surely the most best solution better than passwords.
Because we can always trust the cloud, right? Not like it's somebody else's computer or anything, that they can instruct to not do what you tell it.
Anyway, none of the passkey lovers ever answered me what to do if my computer and cell phone ever get stolen again when I'm traveling abroad in the Brave New Passkey World.
The best answer I got was carry a yubikey with me at all times.
One more thing to worry about losing, yay. Much convenience, very security, wow.
If there are any new developments on that front, would love to hear them.
>But when you represent a complex number as a+bi, what's the meaning of '+' there?
Here, treat a as the notation for "move a steps to the right", and bi as "move b steps up". Think about it like moving a pawn on a chess board.
These operations are translations: taking something and moving it. You can execute them as instructions.
"+" means composition of these two operations; that is doing them one after another.
In which order? Doesn't matter, the result will be the same.
Since we talk about "+", let's talk about "-".
The operations we are invertible, you can undo each of them (or any combination).
If "3" stands for the command "Move a pawn 3 steps right", you can introduce a notation "-3" meaning "move it back" (i.e., move it 3 steps left).
You have then: a + -a = 0, and to save yourself some ink, you just write a - b instead of a + (-b).
Same with bi; -bi means move b steps down.
>In Argand diagram, the numbers are coordinates. Why do you sum up coordinates?
Say, you move a pawn 3 steps right and 2 cells up.
Then you move it again by 1 cell right and 4 cells up.
That's the same as moving it (3 + 1) = 4 cells right and (2 + 4) = 6 cells up.
In our previous notation: (3 + 2i) + (1 + 4i) = (3 + 1) + (2 + 4)i = 4 + 6i.
If you know linear algebra: it works this way because the operations 1 (moving one step right) and i (moving one step right) form a basis[1] for the vector space of translations in a plane.
In simpler terms, it works this way because you can combine the move operations in any order (we can understand that easily), and so we can:
1. Write each move operation as "move ___ left/right and __ up/down"
2. Add up left/right and up/down components separately
>More fundamentally what's addition and multiplication?
We are treating numbers here as translation operators, i.e. commands: "move by this much".
Addition is composition, i.e. doing commands in order (which, turns out, doesn't matter).
Multiplication - well, that's where the salt of the article is. Or rather, should have been.
First, let's look at multiplication of "real" numbers.
Geometrically, the operation is proportional scaling. You can think of it as redefining the unit.
Say, we agreed that "3" means "make 3 steps to the right". But for that to make sense, we need to know what a step is.
We can define, by decree, that a step on a chess board is one cell (1). So "3" by itself really means "3 * 1" (3 of 1 step to the right).
We can scale a number by changing the step size; defined geometrically by proportion:
(a * b) is a distance such that (a * b) : b = a : 1.
This might seem weird, but that's how Euclid would've done it, because it allows us to think of a and b as quantities of equal kind (geometric lengths), avoiding distinguishing geometric quantities, like lengths and translation operators (aka vectors), and scalars, i.e. quantities you can scale by.
Side note: vector literally means "carriers", because the vector operation (translation) moves something from one place to another. A mosquito is a malaria vector.
scaling (multiplication) and adding (composition) interact in a familiar way:
c(a + b) = ca + cb
This is fine and dandy as long as we have only one dimension to deal with.
When we have two, we can first restrict ourselves to multiplying by "real" numbers, because we can say that multiplying by a is simply making everything a times bigger or smaller, and it doesn't break anything.
You still get the same interplay of addition and multiplication.
If we want to keep all the other niceties that we had from real numbers, we will be forced to have the following meaning of multiplication of couples of them (i.e. 2D vectors, translations in a plane):
To get a product of two 2D vectors, u and v, you ADD the angles that u and v make with the x-axis, and you MULTIPLY their lengths.
If you denote this operation as u * v, then you have all the rules for * and + that worked for real numbers still working.
It turns out, essentially, that there's no other way to cook up a product of 2D vectors that has these properties.
And that for 3D vectors, there is no such way at all, a consequence of the Hairy Ball Theorem[5].
And that for 4D vectors, you can define a product rule of that sort only if you give up on the order not mattering. For real and complex numbers, a * b = b * a, but in 4D, the order matters.
And for other dimensions, you really don't get anything nice.
Dimensions 1, 2, and 4 are special; but 2 more so than others.
> Is there a continuity between them, like an operator that represents both addition and multiplication meeting them half-way?
There is a continuity between addition and multiplication.
You can define multiplication of integers by treating one as "number of times to do a thing" and the other as "distance to translate by":
3 * 5 = (move by 3 steps to the right 5 times)
= (move by 5 steps to the right 3 times)
So real number multiplication flows from real number addition by continuity.
Another connection is that multiplication of powers is addition of exponents:
a^n * a^m = a^(n + m).
a^n = a * a * a ... * a (n times)
This is how you can define exponentiation. There are other ways. But that's to say that addition and multiplication are inherently inseparable when it comes to real numbers because an integer can be thought of as either a length or a count (that's to say, in 1D vector are scalars), and keeping that puts a lot of structure on what you do.
I think the true spirit of your question deserves this answer though:
Complex multiplication exactly "meets multiplication and addition half-way", because you add the angles and multiply lengths.
>What's the significance of square and square root? Aren't they just some exponent values (2, 1/2) which have no particular significance on the real line? Why should imaginary numbers have a special place if they are tied to square roots (exponent of 0.5)?
This, my friend, is known as the Fundamental Theorem of Algebra, and boy was it not easy to figure out. There was about a handful of failed attempts to prove it (i.e. proofs that people universally agreed were correct, but later realized they weren't).
In short: there is no significance, on the surface at least. You could have picked a root of any polynomial equation that has no real solution as your "imaginary unit", and got the same set of complex numbers in the end - and nothing more.
But you'll end up with sqrt(-something) anyway.
This comes from the following (non-obvious) fact. Any real polynomial in x can be written as a product of the form c(x - r_1)(x - r_2) ... (x - r_n), where c is a real number and r_i are the roots. For the coefficients to end up being real numbers, the roots must come in conjugate pairs (if (a + bi) is a root, so is (a - bi)), so when you multiply (x - (a - bi))(x - (a + bi)) out, you get a quadratic with real coefficients: x² - 2ax + a² + b²
So you can decompose any real polynomial into a product of real polynomials of degree 2 (you might have one term (x - [real root]) left over if the degree is odd - which is another reason why odd-degree polynomials always have a real root).
If you extend your real number field[4] by adjoining a real number, you get nothing new.
And if you get something new, you have adjoined a root of a quadratic equation.
Why sqrt(-1) instead of the root of x^2 + 5 = 0? There is convenience.
(1, sqrt(-1)) is convenient pair to define the complex plane because they form an orthonormal basis [3], i.e. a neat coordinate grid, with easy multiplication rule.
Nobody is forcing you to use i=sqrt(-1). You can define j as a non-real root of x^3 - 7 = 0. Throwing it into the bucket will get you the same "complex" numbers.
But now they'll be really annoying to work with. E.g.: if j = sqrt(-3), then
(a + bj)(c + dj) = (ac - 3bd) + (ad + bc)j
This extra 3 is annoying. In the end, you can still write your j in the form a + b * sqrt(-1), which is a nicer form, and you no longer have "extra" constants popping up in your formulas.
So you do that.
[1] https://en.wikipedia.org/wiki/Basis_(linear_algebra)
[2] https://en.wikipedia.org/wiki/Fundamental_theorem_of_algebra
[3] https://en.wikipedia.org/wiki/Orthonormal_basis
[4] https://en.wikipedia.org/wiki/Field_extension
I would love to see the AI-unassisted version.
AI assistance is like adding vomit flavor to whatever you're cooking. You don't need much before it starts reeking.
I have written a decent amount of math content. On the internet, and off the internet.
See: https://romankogan.net/math
Do you want to get into a sentence-by-sentence analysis?
> I’m not sure I quite agree, so I’d like to hear why you think it’s LLM slop?
I have pointed out one reason: because the text titled "visual introduction" lacks visuals, and the introduction of complex number in it is purely algebraic without a single picture.
The numbers are introduced algebraically, and then explained visually, after the reader was already confused. You only get anything out of this if you already know complex numbers well enough; in which case it's not an introduction anyway.
The author says, quote "I did use AI assistance for some of the components", so I hope this settles the discussion as to whether it's AI slop or not.
...and it shows.
All I can say is that it made the article substantially worse.
A "visual intro" that doesn't even mention Mobius transformations[1] or Riemann surfaces [2] or any of the beautiful, colorful ways to visualize complex functions [3], or..
..well, it's just lazy LLM slop.
Is it better than the average high school introduction to the subject?
Sadly, yes. But it's such a low bar.
[1] https://www.youtube.com/watch?v=0z1fIsUNhO4
For starters, this "visual intro" introduces complex numbers algebraically.
The text has a visual explanation, but the intro had zero pictures.
Good question!
It does work with dividing by zero :D
It doesn't just appear to be substantially written by an LLM; it was written by an LLM.
It lacks the punch. Anyone who didn't know the geometric definition would just be bored to hell by going over the history of how they were discovered which has nothing to do with geometry.
The article says "Visual Intro", but the introduction is literally not visual. The complex numbers are introduced through historical/algebraic lens, and then reintroduced visually, by which time the audience has been lost.
Of course this article will rise to the top here.
You can do them purely geometrically, so lack of preparation in arithmetic and algebra isn't a pre-requisite.
It all feels so farcical to the extent that it feels like it was either staged somehow, or they took enough of his family hostage to leave him no other choice.
Because you can't be that stupid and live to be that high and that old in Putin's regime.
Wait, no, that's ACTUAL HUMAN pensions, because plenty of AI companies are still "investment grade"
https://www.forbes.com/sites/brandonkochkodin/2026/10/12/ai-...
This tracks with the actual revenue impact of AI adoption:
https://www.bcg.com/publications/2026/the-formula-for-agenti...
Their very cheerful "high" revenue increase from AI adoption (that few companies in the index manage to achieve) sits at 4.5%. Most of those in the "value generating" minority are seeing sub-1% returns.
Which, of course, is non-zero, real, and all that, and matters when we're talking about a 4-5% increase of a big company's revenue, assuming the expenses are not catching up with that.
Which they absolutely are. The AI is subsidized by venture capital, and the subsidies are drying up (hence OpenAI upcoming funding round). None of the AI companies are profitable (much less self-sustaining), and AI prices are inevitably increasing.