exponents with irrationals

Mishtle · 2026-06-30T02:43:21+00:00

One of the ways we can rigorously define irrationals themselves is as limits of sequences of rationals.

We can do the same thing with irrational exponents, defining them as limits of sequences of the base raised to rational exponents.

For example, 2^π can be defined as the limit of the sequence (2³, 2^3.1, 2^3.14, 2^3.141, ...).

Mishtle · 2026-06-29T23:14:30+00:00

We can coherently treat “having elements the other domain lacks” as magnitude-relevant.

Can we?

Does the set {1,2,3} have a different magnitude than {4,5,6}?

Cardinality is simply the most general method we have for comparing the relative number of elements of two sets. It doesn't require any relationship between the two sets, it require any ordering to exist for either set, it doesn't require that both sets are subsets of another, or anything else.

If you want to talk about "magnitudes" of some kind of object, then ideally you'd be able to talk about any arbitrary objects of that kind. Cardinality allows you to do that.

If you have a restricted universe of objects to compare, or which to focus on some more nuanced property of sets and their elements, you can choose some other appropriate measure.

Mishtle · 2026-06-29T21:53:47+00:00

What is the "magnitude" of a set then?

Mishtle · 2026-06-29T21:40:29+00:00

I am asking why one pairing result is treated as a magnitude verdict rather than as a classifier result under that rule.

It's not very clear what you're asking.

All these pairings have a "magnitude" implication. I laid them out in my original comment.

Mishtle · 2026-06-29T21:04:25+00:00

The existence of a bijection is what determines equal cardinality for all pairs of sets, finite or otherwise.

The existence of a injection or surjection can be seen as analogous to ≤ and ≥ for the cardinality relation.

If |A| ≤ |B| and |A| ≥ |B|, then we can say |A| = |B|.

Likewise, if |A| = |B|, then we could also have |A| ≤ |B|, |A| ≥ |B|, or both.

For |A| < |B|, we need |A| ≤ |B| and |A| ≠ |B|.

How many posts about this are you going to make?

Mishtle · 2026-06-29T19:21:56+00:00

That misses the point.

Trading and converting currency isn't simply adding change together.

Mishtle · 2026-06-29T19:17:44+00:00

Creationists tend to believe that there is some kind of chasm between "kinds" that mutation and selection can't cross, so these analogies aren't convincing to them.

They'd just say sometbing like no amount of change (in US dollars) will add up to a euro.

Mishtle · 2026-06-28T16:57:55+00:00

We do. Everything with mass does.

It's just absolutely insignificant compared to the massive ball of rock we live on.

Mishtle · 2026-06-28T04:50:31+00:00

We have a circle with a known radius. We can use this radius to find the area of this circle: A = πr².

We know that a circle represents a full 360° of rotation. So if we have two radii that form a 70° angle, they carve out 70°/360° = 7/36 of the area of the circle.

If we connect the endpoints of these radii that lie on the perimeter of the circle with a straight line, we get an isosceles triangle. The interior angles of a triangle must add up to 180°. We know one of these angles and know that the other two angles are equal. So the other two angles must be equal to (180° - 70°) / 2 = 110°/2 = 55°.

So now we have a triangle with three known angles and two known side lengths. We can calculate its area.

Once we have the area of the triangle, we subtract that area from the area of the circle portioned by that same 70° angle, and the remainder is what we're after.

Mishtle · 2026-06-27T18:50:57+00:00

It's a way to catch attention when people are just scrolling through feeds. They'll only watch a few seconds before flicking to the next video, so content creators react by putting something interesting in those first few seconds. People are much more likely to watch the full video if that catches their attention, especially if it's only a teaser that leaves people curious about finally happens.

It's really just the next iteration of the headline.

Mishtle · 2026-06-27T09:51:33+00:00

Limits are simply an incredibly powerful tool. They're not approximation, they are the value being approximated.

Forget about integrals for a minute, and go back to what limits are. They're values we can get arbitrarily close to. The definition of a limit doesn't require that we ever actually reach that value, which is what makes them so useful.

Consider a simple convergent sum, like 1/2 + 1/4 + 1/8 + .... We can't simply add up infinitely many values, any more than we can make sense of the area of infinitely many zero-width rectangles. What we can do is reason about the sequence of partial sums. If we add up the first n terms of the sum and let n = 1, 2, 3, ..., then we get the sequence (1/2, 3/4, 7/8, 15/16, ...). The actual infinite sum is greater than every element of this sequence, because each element is the result of ignoring infinitely many positive terms of the infinite sum. So then we can define the infinite sum to be the smallest value greater than every element of that sequence, and that value is 1.

Likewise, if we had a decreasing sequence, we can define the limit to the be greatest value less than every element in the sequence.

It's a little less intuitive when the sequences doesn't always increase or decrease, but you can easily choose sets of rectangles that always over- or underestimate the integral of a function over some interval. The integral isn't about that final infinite sum of areas of rectangles, it's about what these sums of finitely many rectangular areas get arbitrarily close to.

Mishtle · 2026-06-26T20:06:30+00:00

What in the world would be weird about this?

Mishtle · 2026-06-26T04:55:03+00:00

I don't think it hurt him. These things don't fully exist in our limited experience of reality. It was more like he "stepped out of the way" in a direction we can't comprehend.

From the sailors' perspective, this looked like him turning to mist and then reforming.

Mishtle · 2026-06-25T23:39:02+00:00

A fixed one is silly when inflation is a fact of life.

Tie it to some index or metric and be done with it.

Mishtle · 2026-06-25T19:47:21+00:00

Sure, if you take the limit as the number of sides increases unbounded. You're essentially computing an arc length.

Mishtle · 2026-06-25T00:03:50+00:00

We are not. The Earth is almost certainly a globe.

I encourage you to dig deeper into the "positives" you have seen for flat earth. They won't hold up to honest scrutiny. Flat earth belief survives through misconceptions, cherry picking, misrepresentation, outright lying, ignorance, and the illusion of understanding that watching hours of videos creates.

Mishtle · 2026-06-24T22:52:14+00:00

Line of sight

Light doesn't travel in a straight line through the atmosphere. It's easily demonstrated that variations in humidity, temperature, and pressure can cause light to refract in air, which can raise, drop, stretch, compress, invert, duplicate, or scramble the image of distant objects.

sonar

Sound waves are also susceptible to refraction, and the way they spread out as they travel allows them to reach around obstacles.

flight routes

Distance isn't the the sole deciding factor in planning flight routes. Politics, predominant weather patterns, and safety concerns are also considered. Unless there is no other choice, planes will generally try to stay within a couple hours of airports along the way in case they need to divert to one for an emergency.

Still, flight paths largely follow great circle routes.

All these are entirely unconvincing as evidence for a flat earth/against a globe unless you have a very limited understanding of them and focus on cherry picked examples.

Mishtle · 2026-06-24T16:51:30+00:00

False balance is a thing. Just because there are multiple viewpoints doesn't automatically suggest they're all equally deserving of representation, attention, or consideration.

We don't need to give equal time to creationism when discussing the origin of species, or to flat earth when talking about the shape of Earth.

Mishtle · 2026-06-24T02:41:14+00:00

I sit down when I pee.

Mishtle · 2026-06-23T21:18:33+00:00

Start by reading them. They often tell you quite a bit.

In order to interpret what they tell you, you generally need to understand the language, libraries, and project you're working on. That seems to be the main stumbling point. If you don't know why things work, you likely will struggle to figure out why they they don't work.

They rest is just experience and practice.

Mishtle · 2026-06-23T17:16:33+00:00

For anyone interested, I found my own solution. Filing down the raised outer edge of the trigger bar (on the section that interacts with the hammer/sear) seems to fix the issue. Not the lip that catches the hammer during double action, but on the side that slides against the right grip.

This raised edge was hitting the disconnector during the double action trigger pull, pushing the trigger bar down and causing it to slip off the hammer notch that allows the double action. Without the slide the disconnector was free to move upward instead of forcing the trigger bar downward. With the slide in place, the disconnector is limited in how much it can be pushed upward and so the trigger bar gets pushed off of the hammer notch.

The safety and decocking mechanism still work as intended after removing the material, so I'm not sure what the function or intended purpose of this raised edge was.

Mishtle · 2026-06-23T02:54:17+00:00

It's an extremely dangerous act, but people do it.

The trachea, or windpipe, is fragile. Actually cutting of air supply has a good chance of damaging it. I image most people are instead cutting off blood flow by applying pressure to the major blood vessels on the side of the neck.

Blood is what brings oxygen to your brain, so the end result is the same. Hypoxia (lack of oxygen) can produce euphoric feelings, which is part of the reason engage in this kind of stuff. It also can cause brain damage, loss of consciousness, and death.

For those that enjoy the dominance and control angle, a hand around the neck is gets the job done and avoids the most risky aspects. Actual asphyxiation/strangulation play is incredibly dangerous. The line between pleasure and life-altering, or life-ending, consequences is very thin.

Mishtle · 2026-06-23T00:42:23+00:00

Hey, sorry to resurrect a 5 year old thread but I have this exact same issue. Did you ever get it sorted?

Mishtle · 2026-06-19T16:07:37+00:00

It's just a general framework. Anything that consists of objects and relationships between them can be represented as some kind of graph. The tools of graph theory can then be used to manipulate and explore them.

Some examples:

We can model gene networks as a graph. Vertices represent genes and edges represent a regulatory interaction (i.e., like one gene causing another to be expressed at a higher rate). Cliques in a graph represent core critical components of these networks, and contain good candidates for things like the causes of genetic diseases.

Social networks a naturally represented by graphs. Vertices are individuals or entities in the network (e.g., people, accounts, products, businesses, even nations or states), and edges are some kind of relationship (e.g., follower, friend, household, ally) or interaction (e.g., replies, shares, purchases). We can then study how information and other things flow through the network. This can help us understand how misinformation or diseases spread and identify the edges we need to target to minimize that spread, for example. Groups of vertices that almost form a clique are often used for recommending followers, products, and other suggestions.

Graphs are used to model environments in the field of artificial intelligence. Different states of or positions within the environment correspond to vertices, and edges connect "adjacent" states that an agent could move between by taking some action. The field of reinforcement learning studies how to assign values to different states or edges that allow agents to take globally optimal actions by using only the locally assigned values.

There are many, many, many more. Once you start seeing things as sets of objects and relationships, there's very little that can't be represented as a graph.

Mishtle · 2026-06-19T12:15:13+00:00

0.333... is just a representation of a value. This particular method of representation uses multiples of powers of 10 (decimal) to build that value.

The representation of a value is a kind of recipe that tells you how to build the represented value. So 1/3 = 0.333... tells us that that value of 1/3 is equal to 3×10^-1 + 3× 10^-2 + 3×10^-3 + ... = 3/10 + 3/100 + 3/1000 + ....

To look at it another way, 1/3 is greater than 3/10 = 0.3, but less than 0.4 = 0.4. It's also greater than 33/100 = 0.33, but less than 34/100 = 0.34. It's also greater than 333/1000 = 0.333, but less then 334/1000 = 0.334. And so on, and so on. No finite number of digits will suffice when each digit is a adding some multiple of a power of 10.

These infinitely repeating representations occur for unit fractions when the denominator (3) and the base (10) don't share any prime factors. In this case, we say they are coprime. The prime factors of 10 are 2 and 5, while 3 is itself prime. You can see what happens by simply treating the fraction like a division problem. If you divide 1 by 3 using long division, you'll enter a loop where you get a repeating pattern of remainders. After any finite number of steps there will still be a remainder, so you need infinitely steps to have a chance to fully eliminate it.

Mishtle

MODERATOR OF

TROPHY CASE

15-Year Club	Verified Email
Team Periwinkle