>>315
Let's assume for a moment that a number like ε satisfying
¥ 0 < ε < 1
¥ 0 < ε < 0.1
¥ 0 < ε < 0.01
¥ 0 < ε < 0.001,
and so on indefinitely, exists, and let's look at some properties that number would have.

I won't be proving these in Coq, at least for the moment, because we're talking about number systems that are different from the one I'm describing in the project. The basic assumptions I'm making about a number system containing ε are that it satisfies the properties of an ordered field, which means all the properties listed in >>267 except the least-upper-bound property.

First of all, ε is smaller than any positive rational number.
(A rational number is a number that can be written as a fraction with integers for the numerator and denominator. Since we are not French speakers, positive means greater than zero, and negative means less than zero; zero itself is neither positive nor negative.)
The proof: Given a positive rational number p/q, let n be the number of digits of q. Then 10^n, that is, 1 followed by n zeros, will be greater than q, and 1/10^n will be less than p/q. Since ε < 1/10^n, it is also less than p/q.

Since ε is positive, it's also obvious that ε is larger than any negative rational number.
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Correction: There are some < signs in >>316 that should be ≤ signs.
Instead of
> 2 < (r/s)^2 < (√2 + ε)^2
> 2 < r^2/s^2 < 2 + 2ε√2 + ε^2
> 0 < (r^2-2s^2)/s^2 < 2ε√2 + ε^2 = ε(2√2 + ε) < 3ε < every positive rational.
it should say
2 < (r/s)^2 ≤ (√2 + ε)^2
2 < r^2/s^2 ≤ 2 + 2ε√2 + ε^2
0 < (r^2-2s^2)/s^2 ≤ 2ε√2 + ε^2 = ε(2√2 + ε) < 3ε < every positive rational.

>>315
>>316
So to answer the question of whether there's a number system with a number ε such that
¥ 0 < ε < 1
¥ 0 < ε < 0.1
¥ 0 < ε < 0.01
¥ 0 < ε < 0.001,
and so on, the answer is yes, there are many. There are the hyperreal numbers which are useful as an alternate way to do calculus. There are the surreal numbers which arose out of game theory. And you can even just postulate such a number ε and add it to the reals sort of like how we add i to make the complex numbers. In the case of the complex numbers, you get the complex numbers from all the linear functions of i. If you add ε, your new set of numbers consists of all the rational functions (polynomial divided by polynomial) of ε.

But we can see that a nonzero infinitesimal number like ε will cause trouble if we want our numbers to be expressed by infinite decimals. The whole idea behind infinite decimals is that we can identify a number by comparing it with finite decimals. If we allow ε into our system, then we can no longer distinguish ε from 2ε, or 0.4 ÷ 0.03 from 0.4 ÷ 0.03 + ε, or √2 from √2 + ε by comparing them with finite decimals. We would need something else. For example, in the surreal numbers, ε is the simplest number between {0} and {1, 1/2, 1/4, 1/8, ...}, and 2ε is the simplest number between {ε} and {1, 1/2, 1/4, 1/8, ...}.

Another definition:
¥ Two numbers are infinitesimally close if the difference between them is infinitesimal.

There are many equivalent ways to construct the real numbers, but all of them are designed so that we never construct a real number infinitesimally close to another real number. I don't think anyone knows whether it's possible for two real-life quantities (such as distances, times, weights, and volumes) to be infinitesimally close to each other yet not equal. If it is possible, then when we model these infinitesimally close quantities with real numbers, we are choosing to ignore any infinitesimal difference and assign the same real number to both.
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>>312
>They made use of a preexisting Java library to do the calculations:
>https://www.hboehm.info/crcalc/

I said that like they just used someone else's code, but on second look I see the guy who wrote that library is the author of the CACM paper. He developed it while he was working at SGI and then got hired by Google to work on Android stuff.

>>308
>>311
I've been thinking again about how to do hand arithmetic with infinite decimals, in this case division. I've tried this before, but my calculation of the error on the result at each step was an ugly and unmemorable formula. I think I've found a better way to think about the error on the result, which is to first work out the error on the remainder at each step. There are two contributions, one from the truncation error on the dividend, and one from the error on what's been subtracted from the dividend (due to truncation error on the divisor). Then the error bound on the result is
(|remainder| + |error on remainder|) / (lower bound on |divisor|).
You could also forgo taking the absolute value of the remainder and work out separately the error bounds in each direction, which could occasionally help squeeze out a digit of the quotient a bit earlier.

>>107
Huh? Import to where? You mean from the native 4chan extension to 4chanx? Can't you just copy paste? (I'm not ccd0)

>>108
yeah
thing is I don't think 4chan lets you select your entire filter list and copy everything

Probably something pretty urgent that needs to be fixed. I clicked on one of my 4chan links today and it brought me to a cloudflare check, but unlike usual I was forced to disable 4chanx to proceed since it was somehow blocking challenges.cloudflare

Wish I'd taken a screenshot of it but I was fiddling with things to try and find what the issue was and just happened to find it through disabling 4chanx before the thought to post about it in here came to me.

>>110
Random anon here.
I just passed the cloudflare check. So either the problem is something other than the presence of 4chanX, or it's been fixed.
Can you try again?

opened up on a proxy and i still get the same issue, on regular
yeah the beta version is fixed once i updated it

Is that a music creation program?

>>38
Nightcafe? Barely. No, i asked for an image

Funny, when u aks to paint yourself something anything about sound machine here acts a bit weird. Maybe it curious about sounds because it cant hear. I doubt someone connected a microphone to an image creation "tool" just out of being kindhearted

>>28
>good thing I'm stupid
D-do you like bananas too?!

Banana.

>>32
Bana nã

I don't actually mind JavaScript.

>>7
what programming and what languages have you done before
c++ is my favorite language of all time, but maybe you shouldn't learn it unless you are comfortable with c yet

if you want to learn c++ the best resource ever if you already know how programming works is by bjarne stroustrup and its called 'a tour of c++' and you should get the very latest edition and you should pirate it for free

with c++ you can do everything that every other language can do and you can go right in the guts of the lgnauge. with c++ you know they aren't hiding anything from you. you can look inside/ with c++ you will be the ultimate programmer

that's the kind of username I'd use, so I approve

5GET

10GET for 1.0-tan

I claim this 11 for God

12 for the mafia