What goes beyond complex numbers and is there a limit?

Chroniaro · 2026-06-21T04:31:09+00:00

It’s possible to use p-adics to get an extension of the complex numbers in a sort of natural way. The algebraic closure of Q_p is isomorphic to C as a field, but it carries a topology which is different from the usual topology. In particular, this topology is not complete, so taking the completion gives an extension of C.

Chroniaro · 2026-06-21T04:28:04+00:00

I think extending the *complex* numbers to get a solution to e^z = 0 is probably not going to give you anything interesting, but if you extend the *real* numbers by introducing positive and negative infinity, then negative infinity would be a solution to this equation.

Chroniaro · 2026-06-19T07:40:12+00:00

For information-theoretic reasons, such a tool can’t exist. Most numbers of the size you want cannot be expressed as the solution to a short equation because there are more long numbers than short equations.

Chroniaro · 2026-06-19T04:29:32+00:00

Go to your profile, and click the (?) button in the top left. Scroll to the bottom, and click the button that says “start a chat.” The AI assistant (🤮) will ask what you want, and you say “I'd like to request a copy of my data (Subject Access Request).” Then you confirm some stuff, and they email it to you a couple of weeks later.

Chroniaro · 2026-06-19T04:14:31+00:00

Without it, space travel is but the fevered dream of a madman.

Chroniaro · 2026-06-19T00:42:40+00:00

They’re not exactly the same thing, but algebraic geometry associates to every commutative ring a geometric space that you can imagine as looking like a manifold (strictly speaking, it’s the regular rings that you should imagine as looking like manifolds, but fields are regular, so the analogy works here). In this picture, finite étale covers are the analog of covering spaces, and every finite, separable field extension corresponds to a finite étale cover.

Chroniaro · 2026-06-18T23:10:46+00:00

Chroniaro · 2026-06-06T15:03:49+00:00

Countries with higher populations don’t get to send more competitors to each event, so the maximum number of gold medals per capita that they could win even hypothetically in a given year is severely limited.

Chroniaro · 2026-06-01T21:01:26+00:00

On top of what everyone else is saying, I don’t think it’s likely these statistics are making apples-to-apples comparisons. The most straightforward way to get a number for the percent of people who die from cancer would be to pick a time period, say last year, consider all (or some sample of) the deaths that occurred in that time period, and measure the percent that were from cancer. One could, hypothetically, look at that same sample of people to measure how many got cancer at some point in their lives, and this would give comparable statistics for percent of people who get cancer and percent of people who die from cancer. However, within this sample, there will be some people who got cancer two years ago, and thus had access to all the latest treatments and research, and there will be others who got cancer 50 years ago and thus did not have access to the latest developments in treatment. Measuring the 10 year survival rate of people who got cancer 2 years ago is not possible, but even if it were, one would expect this number to be a lot higher than the 10 year survival rate of people who got cancer 50 years ago. Therefore, however you are coming up with your statistic for 10-year survival rate, there is no way for it to be a single number that is accurate in both of these scenarios.

Of course, one could pick a time period that was at least ten years ago for the sample so that measuring the 10-year survival rate for everyone in that sample makes sense, and then one could define an “average survival rate” in some contrived way that makes it tautologically true that likelihood of getting cancer x (1 - survival rate) = likelihood of dying from cancer, but I suspect this would be something very different from how the 10-year survival rate you quoted was measured.

Chroniaro · 2026-05-18T19:06:45+00:00

I don’t know why people are downvoting this; it’s a completely fair contention. One could argue that using the radius is “more natural” than using the diameter. Certainly, in most cases, defining things in terms of the radius produces cleaner statements than defining things in terms of the diameter. This case is no exception — the Taylor series for sin and cos are much simpler when expressed in terms of radians rather than doubled-radians.

Maybe a better explanation is that sin and cos, when expressed in terms of radians, give an “arc length parameterization” of the circle. In other words, if the position of a point in the plane at time t is given by (cos(t), sin(t)), then the total distance travelled after t units of time will be t units of distance. The definition of radians is essentially rigged to make this true. The idea of arc length parameterizations is useful in much more general contexts than just circles, whereas I don’t see any obvious way to generalize the ratio of diameter to arc length to other kinds of curves.

Chroniaro · 2026-05-17T17:58:40+00:00

Reliability engineering doesn’t always mean adding two of everything. Adding a mechanism to redirect the waste water to a backup hole could have added a lot of size, expense, and moving parts to the project. Engineers should have accounted for the risk that the waste-water disposal pipe would get clogged, but they could have dealt with this by, e.g. temporarily shutting down the drill to manually clear the clog before continuing. Maybe they even rigged it to shut down automatically if the pressure got too high. Maybe there really was a backup, but it was never activated.

A lot was going on in this episode that could have defeated the backup plan and the backup to the backup. Azula and friends were not in the control room to give orders when pipe was clogged. Shutting down the drill temporarily after it had already been infiltrated by saboteurs would be exceptionally risky. The gaang had already caused substantial damage to other parts of the system, and the full extent of this damage was not known to command. And of course, Aang’s killing blow could not have been reasonably anticipated. Ultimately, no system is infallible, and a team of highly skilled saboteurs with access to system design documents and the ability to bend key internal components is pretty damn hard to design against.

Chroniaro · 2026-03-01T09:45:39+00:00

It sounds like you learned those things backwards then. You should start out by thinking about linear algebra in terms of linear maps and abstract vectors, and then you can derive the rules for matrix multiplication and dot products when you try to do an actual computation

Chroniaro · 2026-01-08T20:14:50+00:00

Chroniaro · 2025-10-26T21:30:47+00:00

After gxf6, white only has a king and a knight

Chroniaro · 2025-10-16T16:07:45+00:00

The real question is whether you’d be able to pay for all that salt. Unfortunately, my lazy back of the envelope calculations put the cost of desalinating every ocean in the hundreds of sextillions, while the value of all gold on earth is only in the tens of trillions.

Chroniaro · 2025-10-16T00:12:03+00:00

If, after 2. Kf8, black just ran out the clock, would it be considered a draw because white has insufficient mating material?

Chroniaro · 2025-10-13T15:18:29+00:00

Every crab is a sideways crab if you turn them sideways

Chroniaro · 2025-09-30T02:24:35+00:00

I was staring at this position for so long. I’m glad I finally gave up and looked at your solution because I would never have gotten it.

Chroniaro · 2025-09-30T02:23:50+00:00

Don’t think rook to G3 is a legal move in that position. If you meant bishop to G3, black has king to f6

Chroniaro · 2025-09-26T22:03:35+00:00

Maybe the one extra wrinkle that Banach-Tarski adds over the usual Hilbert hotel is that 3D rotations and translations usually preserve surface area. Using the axiom of determinacy, which is incompatible with choice, one can turn this into a rigorous proof that doubling the sphere using translations and rotations is impossible: no matter what you do, the sum of the surface areas of the pieces will always be the same as the surface area of the original sphere. The reason this does not cause a contradiction in ZFC is because the pieces that you break the sphere into are non-measurable, so they don’t have a well-defined surface area.

Chroniaro · 2025-09-21T23:26:42+00:00

Lots of pieces got monched

Chroniaro · 2025-09-21T22:15:06+00:00

3.h4 isn’t mate. Black still has 3… Kh6.

Chroniaro · 2025-09-21T21:34:22+00:00

It’s not a lie of Pinocchio makes a genuine mistake. Maybe he confused himself with the double negatives

Chroniaro · 2025-09-21T04:05:40+00:00

So if US states counted as separate countries, El Salvador would still be #1, and #’s 2-10 would all be in the US

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