#Irrationality of π

1 messages · Page 4 of 1

ember marlin
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...no.

crisp basalt
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the cutting, perhaps, flattens the position where it is cut

ember marlin
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Right, if you delete the cheese at every countable position, 100% of the cheese remains.

gusty burrow
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Says the person, trying to reify an infinite process into a fixed resultant outcome quantity (comprised of countable quanta), which is impossible to do.

ember marlin
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But cutting refers to partitioning the volume of a shape with a plane, which necessarily preserves it.

ember marlin
crisp basalt
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they are not trying to force an irrational into what you call a quantity of countable quanta, it is quite obvious that this is not possible

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the discussion cannot go anywhere

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some people have a different conception of what counts as a number that's it

ember marlin
crisp basalt
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what it means is a rational number, lol

gusty burrow
ember marlin
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...yes, obviously an irrational number is not a rational number, that's what the prefix "ir-" means.

crisp basalt
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yup

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that's why this discussion is not interesting

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and why i was waffling about cheese and finite texts instead

gusty burrow
ember marlin
crisp basalt
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the comparison # to ### doesn't look like a quantity to me, tbh

gusty burrow
ember marlin
gusty burrow
crisp basalt
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. ### is 3 #s, that makes sense

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bruh

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but what is "# to ###" a quantity of

gusty burrow
ember marlin
gusty burrow
gusty burrow
crisp basalt
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yeah we can represent the number 3 as ### where it makes sense to call the number 3 a "quantity" (it's the quantity of #s) but i don't see the same relationship when trying to quantify "# to ###"

gusty burrow
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You cannot do that with the irrationals.

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They are of no quantity.

crisp basalt
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many things can be represented that aren't quantities, though

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we can represent happiness with :)

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etc.

gusty burrow
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Is happiness a number?

crisp basalt
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i don't think so... i don't think it's a quantity, certainly

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seems like rationals are something that shouldn't be called quantities

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but you can use words however you feel like ig you're already way off the beaten path

gusty burrow
crisp basalt
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incredible

gusty burrow
gusty burrow
crisp basalt
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run doom in Q

gusty burrow
ember marlin
hazy iron
merry lava
crisp basalt
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the goal is to make operations in Q (addition etc) correspond to execution steps

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i think

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john conways fractran is a good start

merry lava
hazy iron
supple spire
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Can we lock this down when we reach 3142

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I think that would be hilarious

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It's not active either way anymore and the discussion has ended

lapis marten
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I know it's rounding but

supple spire
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eh it doesnt matter to me both are fine

crisp basalt
supple spire
crisp basalt
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ah

vast spearBOT
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scared of the later person 😭

1262418762444046389.png
gusty burrow
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Can I share my Pythagorean Discord at 3142? || #advertisement message ||

pythag.png
crisp basalt
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in #advertisement

hazy iron
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We’re quite close to 3142

crisp basalt
# hazy iron

this is your 7th time sending this image maybe stop

hazy iron
merry lava
supple spire
gusty burrow
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----√2 φ π e √3-----

Closing statement, the great Pi debate of 2025

The Pythagorean framework that I have been arguing for since the beginning challenges the conventional view of irrationals as complete numbers by insisting that true numbers must be finitely constructed, measurable, and composed of countable elemental units (a quality entirely absent in irrationals, whose existence is predicated solely on infinite procedures like convergent sequences, Dedekind cuts, and limit processes).

I holds that the methods used to "fill the gaps" on the number set of the reals rely on axiomatic abstraction and reification fallacies - such as completeness - that reify the abstract concept of convergent infinity into a static entity, based on a priori rules established from their incommensurable counterparts. This is despite irrationals never having a discrete, measurable landing points like the rationals.

During this debate by arguing that quantities, by definition require countable quanta and must have explicitly defined, finite measurements (which irrationals fail to possess due to their inherent incommensurability and lack of common measure with rational units), I deconstructed traditional proofs like the diagonal of a square and the Pythagorean theorem as misapplied in this context, concluding that irrationals are, at best, useful fictions rather than genuine quantities grounded in the finite, elemental foundation of mathematics.

Using infinite processes to define irrational numbers obscures their lack of tangible discreteness. In this regard, procedures like the bisection method or Cauchy sequences only generate a never-ending sequence of approximations rather than a finalized entity that can be pinned down by a finite collection of rational units. This falls into the reification fallacy.

Irrationals, I argue, are not numbers, nor measurable distances, nor quantities comprised of countable quanta.

They are Failures of Measurement.

pythag.png
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||If anyone wants to hear or debate more of my constructivist perspective, I am somewhere in #advertisement message and there is a lot of cool other ones there as well. 🙂This was a very joyful debate where people can see the best arguments from both sides. Ὑγίαινετε ἕως ὅτου ἀνταμώμεθα, φίλοι μαθηματικοὶ φιλόσοφοι. ||

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❤️ 🙏 📐 🔢 🚫 ♾️

supple spire
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Coolio

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Atp we're just waiting for 3142 or 3141

cerulean junco
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3142 or 3141?

supple spire
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I'm neutral against either

tawdry swift
sinful adder
hazy iron
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3076 🔥🔥🔥

cerulean junco
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3077 🔥🔥🔥

hidden jay
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Nah i think we need to add a digit

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31415

supple spire
crisp basalt
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(since this was on sunday this is the hardest problem of the week)

sinful adder
supple spire
sinful adder
supple spire
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So we just throw in any approximation?

sinful adder
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you need to say what the error is tho if you want to use an approximation argument

supple spire
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How would I do that?

sinful adder
# supple spire How would I do that?

dunno, let's see.
let's first define pi
we can say pi is:
pi/2=min{t: cos(t)=0 and t>0}
equivalently
pi=min{t: e^z = 1 iff z ≡ 0 (mod 2ti) and t>0}

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hmm idk

crisp basalt
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one thing to do is put polygons on the inside of the circle

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i tried this once, and it takes a very long time to get as close as 3.14

sinful adder
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because cos has an alternating series on the reals, the error is easier to bound

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oh but we're inverting it

crisp basalt
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so cos(1.57) > 0, cos(1.575) < 0?

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probably would work

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though, idk if this would require you to prove that taylor expansion is a thing that works

sinful adder
crisp basalt
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but why

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i only say this because pi = 3.14 is commoner knowledge than that

sinful adder
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It's easy to prove with a computer because even if a series converges slowly to it, you can just calculate enough terms to make the error small enough, if you have a bound for the error

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proving it by hand is not easy

sinful adder
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For an alternating series
Sum{k=0 to infinity} (-1)^k a_k, for all a_k > 0, a[k+1] \le a[k] for all k

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with lim{k to infity} a[k] = 0

crisp basalt
sinful adder
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then after calculating the first n terms of the series, the error, which is the sum of the tail of the series, call it R_n, satifies |R_n| le a[n+1]

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mm-hmm

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And cosine is analytic, meaning it's exactly it's power series, well known result

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if I call C_n(x) the partial sums of the cosine series,

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|R_n|<x^n/n!
using a computer to see how many terms we need

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up to C8, 4 non-constant terms (because cosine only has even powers)

crisp basalt
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1.575^4

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maybe 11/7 is better

sinful adder
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C8(1.570) > 0.008 (error < 0.00003)
C8(1.575) < -0.004 (error < 0.00003)

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So you just need to calculate
C8 of those numbers
Which is very annoying without at least a simple calculator, forget a scientific calculator.
But if you allow yourself a simple calculator, your idea works!

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For C6, the error is large enough that it doesn't guarantee the approximation is the same sign as the true value

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there should be a root finding method for cosine that's faster

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ah but then you have to be able to do trig on the guesses, so, not doable without a calculator really even in theory

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Convergent fractions might be faster

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no, but hen you need to assume you know it begins with 3.14, that's circular

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there's a method for evaluating polynomials by nesting that's generally easier than calculating powers x^2, x^4, etc, called Horner's rule

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1 - (x^2)/2 + (x^4)/24 - (x^6)/720 + (x^8)/40320
=
1 + x^2 · (-1/2 + x^2 · (1/24 + x^2 · (-1/720 + x^2 · 1/40320)))

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well this method is more for what's faster for a computer to do
It's not necessarily what's easier for a human with a pen and paper

hazy iron
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Just use a power series for arctan smh

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Omg we’re so close

sinful adder
crisp basalt
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arctan doesn’t converge outside |x|<=1 am i wrong?

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oh yeah it does work at 1

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that’s the 1-1/3+blah that converges really slow

hazy iron
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Use Newton’s approximation instead

tawdry swift
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Should I spam the thread with LaTeX

cerulean junco
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5 more messages to go

rain heath
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?

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wdym 5

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I guess now 2 after this lol

supple spire
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