/sci/ - Science & Math

>>15891105
I do miss the old /sci/

>>15891105
irrational numbers dont exist because you cant add an infinite amount of numbers

number line is infinite
pi is circular and it loops
number line doesn't loop
so translating pi into a infinite number line results in infinite translation

>>15891105
>If the digits of pi are completely random
Proof?
>at some point the digits of pi will coincide with those of e
They might coincide with the digits of e for an arbitrarily long series of digits, but eventually there will be a misaligned digit.

I suppose it's possible for the digits of pi to permanently shift to being e, or vice-versa. However both are not similtaneously true since they've each been proven to be irrational.

>>15891105
>If the digits of pi are completely random
*sigh*
>>15891151
me to

>>15891105
>that means, that at some point the digits of pi will coincide with those of e
No

The real problem is that these numbers are everywhere in the real world and we struggle to define them mathematically. Irrational numbers are still zero rational or else measurement would be impassable.

>>15891290
>The real problem is that these numbers are everywhere in the real world
Are they in the room with you right now?

>>15891105
>If the digits of pi are completely random ad infinitum,
they're not
> that means, that at some point the digits of pi will coincide with those of e
they wouldn't, with probability 1
>but because e also stretches endlessly, that means that pi is in e, which means that the digits repeat, which means that they are rational.
this part actually works but since the premise is false the conclusion is worthless

>>15891290
>measurement would be impassable
it is literally impassable

>>15891290
we have absolutely no problem defining them mathematically, wtf are you on

>measurement would be impossible
As long as your measurement method has a non-zero margin of error then there is no measurable difference between irrational quantities and rational ones. Even if you think irrational numbers are schizo, even the difference between them and other numbers is fundamentally schizo.

>>15891347
>we have absolutely no problem defining them mathematically
What is an irrational number?

>>15891349
an irrational number is any number that can't be expressed as a ratio of two integers.

>>15891357
Is i an irrational number?

>>15891331
You've examined the randomness of every digit of pi? Damn anon, you're so cool.

>>15891362
yeah, its irrational algebraic

>>15891362
he was assuming the real numbers were being discussed.
i is the base unit that extends real numbers into the complex plane. so by a new definition we can also say that any rational real number arithmetic operations on i are also rational complex numbers.

>>15891392
What is a real number?

>>15891357
define one please

>>15891371
the first one's always 3. the second one's always 1. the third one's always 4. etc. I don't know for a fact that pi's digits don't just happen to line up with e's at a certain point, but pi's digits aren't random, at least in a frequentist sense.

>>15891401
the square root of 2, which is geometrically constructible as the length of the diagonal of a unit square.

>>15891409
you cant construct the sqrt 2 using a unit square, try again

>>15891400
equivalence class of Cauchy sequences of rational numbers. Which are themselves equivalence classes of pairs of integers, which are equivalence classes of pairs of naturals, which are funny nested sets.

>>15891412
Take your unit square and draw a line from one corner to the opposite corner. Congratulations, you have constructed sqrt 2.

>>15891412
what's the length of the segment between two opposite corners?

>>15891422
What is an equivalence class? What is a sequence? What is a set?

>>15891429
not possible, lengths must be rational.

>>15891432
no length exists

>>15891443
how can a length be rational if it doesn't exist?

>>15891444
read what i said again before replying to me

>>15891438
>equivalence class
Subset of a set generated by an equivalence relation, which is a type of subset of a Cartesian product of a set with itself satisfying certain axioms. This procedure partitions the original set into disjoint nonempty equivalence classes
>sequence
Function from the naturals into some given set
>set
Object satisfying the ZFC axioms

>>15891449
>Object satisfying the ZFC axioms
So are there only countably many sets? And hence countably many real numbers?

hey hey marcus house with ya here

>>15891454
Given that the axioms include infinity and power set, there is definitely more than countably many and in fact no set of all sets can exist in the theory, certainly not a countable one. And if you're about to try to gotcha me with lowenheim-skolem then please just do us all a favor and kill yourself now.

>>15891447
oh, you're saying that there is no length that is the length of that diagonal. That was so stupid that I didn't consider that as a way to interpret that.

But ok, smart guy, let's say I want a quantity to describe how big a line segment is, that can describe ANY classically constructible line segment. What's the quantity I should be using, if not length?

Artemis 3 SLS will be be ready before Starship. It's time to deliver.

>>15891463
>there is definitely more than countably many
Why? There are only countably many objects describable by ZFC

>>15891471
2^N where N is the set of naturals describes uncountably many such objects. It seems like you are trying to get at unique identifiability by a first-order formula, which is not a standard I have any reason to apply. If you really want to talk about definability you should realize it's a lot more subtle than you think it is, in particular it's only possible to talk about metamathematically using models. For instance there can be no subset S of definable real numbers: fix a well-ordering on R and define y to be the least element of R / S by this order. This means y should have been in S which is a contradiction. Even if you don't want to use well-ordering you can do the same thing with ordinals.

>>15891482
2^N describes just one set: itself. It's a statement of faith to say that it contains or alludes to uncountably many things.

https://www.youtube.com/live/jPTD2gnZFUw?si=5LEasGI5_kB_VF7o
what is that spinning balloon??????

>>15891487
The only statements of faith happened earlier when I agreed to use ZFC (and first-order logic) in the first place. After that, the fact that 2^N contains uncountably many objects is a certainty.

>>15891493
Then you agree that believing in the existence of real numbers depends on your faith that ZFC's statements are statements about reality and the math you're doing is theology.

>>15891464
rational lengths only

>>15891496
Axioms are required for any formal system to work. The ontological status of the reals is the same as that of the naturals, regardless of if you take a platonist or formalist viewpoint.

>>15891499
For rhetorical purposes I will agree with you. But if lengths are good enough, what should I be using?

>>15891501
>>15891499
aren't* good enough, fugg

>>15891500
>The ontological status of the reals is the same as that of the naturals, regardless of if you take a platonist or formalist viewpoint.
No, if you're a platonist about ZFC, the real numbers are as real as the natural numbers and rocks. If you're a formalist, natural numbers have a higher status than real numbers, which are just language games.

>>15891105
>which means that the digits repeat, which means that they are rational.
That is not what rational number means.

>>15891513
you don't think there are formalists who think natural numbers are language games?

>>15891525
OP's idea is that pi's decimal expansion eventually contain's e. In that case, there is some finite prefix [PI] that starts 314159... and pi's decimal expansion is [PI]271828... with all the decimals of e.

But by the same logic that that prefix existed, there would be some finite prefix [E] that starts 271828... such that e's decimal expansion is [E]314159... with all of pi. But that would mean that pi's decimal expansion is [PI][E][PI][E], repeating with a period equal to the total length of [PI] and [E], and since pi has a repeating decimal expansion, it must be rational


obviously there are no such [PI] and [E] prefixes with those properties so the post is useless but that part at least is mostly valid.

>>15891547
>But that would mean that pi's decimal expansion is [PI][E][PI][E]
Not necessarily, e doesn't have to be be finite, pi could just be composed of [PI][E] where it just converges to e at some point or they could both converge to some other irrational number like sqrt(2) by the end.

>since pi has a repeating decimal expansion, it must be rational
That still isn't what rational means.

>>15891561
>Not necessarily, e doesn't have to be be finite, pi could just be composed of [PI][E] where it just converges to e at some point or they could both converge to some other irrational number like sqrt(2) by the end.

yeah but that's not what OP was thinking

>That still isn't what rational means.

it's a theorem that any number with a repeating decimal expansion is rational. I don't remember the proof off the top of my head but iirc it's not that complicated.

>>15891513
Why should a formalist believe there is any difference?

>>15891911
Because he can see 1 dog, 10 fingers, etc.

>>15891105
No it just means with arithmetic we can collapse irrationals into rationals.
It makes perfect sense actually 1 = pi + -pi + 1, this is might seem obvious but if we're looking at the decimal expansion of 1-pi then it's irrational and not immediately obvious.-2.141592... who could possible guess?
Anyways fuck off now, go read a book before you talk.

just because something is infinite, it doesn't mean that everything happens in it.

the probability of getting an infinite series of 1s in pi is zero, same for eternally duplicating the digits of e in pi

>>15892006
>same for eternally duplicating the digits of e in pi
Proof?

>>15891938
To go from 1 dog and 10 fingers existing to 1 and 10 existing is exactly the type of realist/platonist leap a formalist does not take. Instead they regard 1 and 10 as a useful symbolic fictions, like the reals. Given how easy it is to construct R from N, it would be nonsensical to assign a higher ontological status to N unless you disagree with the axioms involved (eg, power set). But that won't stop retards on this board, of course.

>>15891105
Holy shit, the vax really fucked your brain.

>>15891938
I see dogs, how do I prove that I see an integral number of them?

>>15893011
i've got a better one for you, i also see dogs, how do i prove they are real?

Formula for the nth digit of pi discovered last year

>>15891482
>fix a well-ordering on R and define y to be the least element of R / S by this order
That "trick" is using a symbol not available in the original language used to determine what is describable. You can't give a well-ordering of the reals in such a system.

>>15895104
mh, i knew of the spigot algorithm, what is this one about?, also what is that B?

>>15895113
>That "trick" is *fanfiction*
*yanw*

>>15895123
Bernoulli number.

https://arxiv.org/abs/2201.12601

>>15895113
I don't know what symbol you mean, unless you mean the slash which was a typographic error on my part, it should have been \ ie set difference ie {x in R | x not in S} which should be unobjectionable. If you're mad about well-ordering I recommend you deal with it because it's a theorem of zfc, but as I explained you don't even need to because you can generate essentially the same contradiction trying to find a definable subset of an uncountable ordinal, which is naturally well-ordered.

hi everyone

>>15895135
ah, good to know, thank you

>>15895219
Hey Elon