Aren't irrational numbers... well... irrational?
>>16536633Irrational numbers are numbers that cannot be represented as fractions of two integers. That's it.
>At some point as we add more and more digits to each, the numerator and denominator stop being integers.>No, don't ask me exactly where. Just trust me that it happens.Modern "mathematics" is a religious cult. There are no irrational numbers. None. Zero. Zilch. Nichts.You can have integers that can't be written out at quark scale in the available volume of the entire universe, but they are still just integers.One day you realize that a "real" number N<3 has to actually be N≤3 every time.
>>16536639>"Proof by Contradiction" is not in any of my Bibles.
>>16536633some numbers can be written or symbolically represented or expressed or communicated or codified, in different ways. Some only in one way. Some numbers we know they exist but can't even represent them. In fact, that's most numbers, I believethe number in OP, if you ignore the "..." which I would take to imply a repeated number or some sort of expansion, yes it can be expressed as a ratio and not just as a decimal number (doesnt have to be base 10, dunno what the more general label is here)IF the ... means a[n infinitely?] repeating number, then how do you write that as a ratio?(not recursive rations allowed, like other anon said the top and bottom have to be positive/negative [whole?terminated?] numbers >naive amateur with a mathexplorer spirit
>>16536663oops I see I missed the last digit, 2, in the pic I posted. maybe that changes everything. is this le golden ratio?also, are Normal numbers a disputed thing? I cant find venn diagrams showing it with a quick search but that Math channel made it sound like "we" (not me) confidently know most numbers are Normal but we can't know them?nani?>?(pls no math bully :3 )
>>16536633They are arheton, if you don't know what this means you are uneducated
Rational numbers can be written in this form:2^a * 3^b * 5^c * 7^d * 11^e * ...Where the variables a, b, c, d, e, ... are integers.With the square root of two, a = 1/2 and the other variables equal zero.But a = 1/2 isn't an integer.Thus the square root of two isn't rational.
algebraic numbers aren't even uncountable, OP
>>16536633Sqrt(2) equa..Stopped you there, there is no equality
>>16536633>irrational>numberspick 1
>>16536639>there are no irrational numbersI take a one-inch line and draw another one-inch line perpendicular to it. I draw a line connecting the two ends. How many inches long is this third line?YOU SHOULD BE ABLE TO SOLVE THIS
>>16536815It's 2 inches you dumb nig
>>16536639>Modern "mathematics" is a religious cult.>There are no irrational numbers. None. Zero. Zilch. Nichts.
>>16536633That's right, they aren't ratios.
>>16536700Algebraic numbers aren't well defined.
>>16536707>a = aa is irrational >b/c /= a, therefore a /= aWow, great argument.
>>16536656>>16536815>>16537184>STOP CONTRADICTING OUR RELIGION!!!!!!Makes you feels smart to repeat the mantras in which you were indoctrinated, doesn't it.And of course I have seen the "proof" that √2 is "irrational." Some clueless midwit was surprised to learn that an infinitely long integer can be divided by 2 an infinite number of times, and therefore concluded it wasn't an integer.MAYBE THE SECOND HIS PROOF REQUIRED HIM TO IGNORE ONE OF HIS OWN PREMISES, HE SHOULD HAVE REALIZED HIS ARGUMENT WAS INVALID.(You might want to write that down. I could be important. He didn't prove the numerator or denominator were not integers. All he proved was that his approach doesn't work.)I remember one of my grad school professors going on a rant that we have no way to locate an unending string of 3's on the number line. Yet strangely we can locate an unending string of 0's just fine. I would say with perfect confidence that the unending string of 3's is left of the unending string of 4's and right of the unending string of 2's.
>>16537686>I have seen so much that I will never understand.>Let me extrapolate on that idea.Kek.
>>16537677
>>16536633Yeas, in the sense that they are not ratios (ir-RATIO-nal).
>Assume there are two integers A and B in a ratio such that A/B=√2.*Sigh!* Sure, fine, two integers in a ratio.>Then A^2=2B^2.Yes, yes, I've seen this before. So A^2 is an even number, which means A is an even number, which means we can replace A with 2C where C is an integer.>Right! Therefore 4C^2=2B^2, but that means 2C^2=B^2, so B^2 is even, and we can replace B with 2D!And so on ad-infinitum. *Yawn!*>So that means my premise was wrong! There is no such ratio of integers!Then what are A and B? If they aren't integers, what are they?>W-what? No, I choose the pretend the ratio part is the part I've disproven.Can √2 be multiplied by 10?>Of course it can!So then whatever that result is—let's call it √200—dividing it by 10 would equal √2, right?>Y-yeah.So that's a ratio, isn't it?>Um, yeah, I guess.And we agree 10 is an integer, right?>Y-yeah.So then √200 must not be an integer. Your conclusion was that there is no ratio of two integers A and B that equals √2, but I've just shown that the problem isn't with the ratio part, and the denominator is definitely an integer, so the numerator must not be one. So what is it? >Um, uh, um, there's a decimal point.I'm assume you agree it can written using a string of numerals. We don't have to resort to seance or something, right?>Of course it can be written using numerals, AND a decimal point!And if we keep multiplying by 10, we can keep shifting that decimal point to the right as far as we need to in order to get an integer. Or are you saying that at some length, it ceases to be an integer over a power of 10?>Exactly!But you can't seem to tell me exactly when it stops being an integer.>Only when it goes on forever!You mean like 0.333... as the decimal expression of one-third? That one goes on forever, yet you seem to consider this a valid radio of two integers. Why doesn't that numerator do the magical not-an-integer thing?>SHUT UP! YOU'RE UGLY!
>>16537699I don't think you know how equality works>my software is inaccurateok
>Um, uh, I've got it! It's because 0.333 has a pattern!So do the digits of √2. How the fuck would we know what its digits were if it didn't have a pattern? We even know the pattern for the digits of pi—as long as you're in base-2, but still. We have easy algorithms for getting the digits of √2. Let me guess, you meant a REPEATING pattern?>Um, yeah, that one!Tell you what, bone-head: I'm going to start writing out a number using only numerals and with no pattern, and you tell me when it stops being an integer. Ready? 20348. Is that an integer?>Well yeah, but—20348230983209824039275. Still an integer?>Sure, but—2034823098320982403927531567223662384498405983039485948958494854954. Just stop me when this patternless string of numerals switches to not being an integer anymore.>WHEN IT REACHES INFINITE LENGTH, YOU ASSHOLE!So for all eternity, you will be wrong, but as time ceases, FINALLY you will be right? Is that what you're telling me?>FUCK YOU!That's what I thought. You are doing religion, not math.
At least they're real unlike imaginary numbers (some of them)
>too stupid to understand something, therefore it doesn't exist>every other post is retard talking to himself>>>/x/
>I can't refute you, so I'll just insult you and seethe.>>>/lgbt/
>>16536639Hi Norman
>>16536633yes that's why they murdered the guy who discovered them and literally erased his name from history
>>16536633Irrational numbers are defects of the decimal system. No system can practically describe them better than the conceptual way we currently have.
>>16537797
>>16536633They are, which proves that world is non-euclidean.Take >>16536815 for example. You can't actually drawn perfect lines that connect perfectly irl. These are just convenient approximations.
>>16537772>>16537806>So then whatever that result is—let's call it √200—dividing it by 10 would equal √2, right?>>Y-yeah.>So that's a ratio, isn't it?>>Um, yeah, I guess.>And we agree 10 is an integer, right?>>Y-yeah.>So then √200 must not be an integer. Your conclusion was that there is no ratio of two integers A and B that equals √2, but I've just shown that the problem isn't with the ratio part, and the denominator is definitely an integer, so the numerator must not be one. So what is it?You answered your own question man. How do you go from "So then √200 must not be an integer." to "the problem isn't with the ratio part"? √200/10 = √2 but as you yourself said, √200 is not an integer hence you didn't find a representation of √2 as a/b where both a and b would be integers or whole numbers.>patterns in decimal expansionsThat's not the point, yes there is order in the expansion of √2 but not a repeating pattern, which would be a point after which you get blocks of decimals of some length n that just repeat forever.
>>16537686>muh proof by lobotomization
>>16536667[math] \displaystyle\boxed{ \mathbb{T} \;\boxed{ \mathbb{S} \;\boxed{ \mathbb{O} \;\boxed{ \mathbb{H} \;\boxed{ \mathbb{C} \; \boxed{ \mathbb{R} \;\boxed{ \mathbb{Q} \;\boxed{ \mathbb{Z} \; \boxed{ \mathbb{N}}}}}}}}}}[/math]
>>16538307>The part about any decimal just being an integer divided by some power of ten is beyond my intellectual grasp.Have someone smarter than you explain it to you—a third-grader, for example.
>>16539478Yeah any decimal of finite length or after a finite length consisting of infinite repetitions of 1 single finite-length block. √2 is neither of which.
>>16537828"Real" is meaningless word in mathematics (some folks prefer to say a technical word) especially when considered without the context of complex numbers. It is just a binary opposition, just like "true" is a meaningless word, and what matters (and only this matters) is the binary opposition true/false
>>16536639The dedekind cut {A, B} that satisfies sgn(a)*a^2 < 2 < sgn(b)*b^2 is a real number that is clearly not rational.
>>16536635In other words, ir(ratio)nal numbers are inexpressible as ratios.
>>16539595NTA but is there any way to construct or write down the Dedekind cut of an uncomputable real number?
>>16539611Hmmm not a math guy but I'm pretty sure you can't. Since if you could write down some "rule" for the dedekind cut then the resulting number would be computable. You can't even isolate an uncomputable within a set
>>16539627So how would we construct an uncomputable real number since as I understand it that problem would apply to the other methods I've heard of
>>16539644>constructOr even give an example of one for that matter (and I don't mean some wordcel theoretical one like that chitin computability constant I mean the number as a meaningful object one can do maths with)
>>16539611Euclids elements book 5 proposition 5 might give some insight
>>16539644>>16539646Take this with a grain of salt because I might be completely retarded, but you can't really isolate or describe or give an example of an uncomputable number. Uncomputable numbers are a consequence of the countability of real computable numbers. Since the real set is uncountable that means that most real numbers are actually uncomputable. If you were able to describe or build an uncomputable number, it would not be uncomputable thus a contradiction.
>>16539611>>16539646also nta and idk if this in any way answers some parts of your question but after some searching around also inspired by your post and a remark I saw in a book, what I have gathered is that the view I had of this topic was somewhat "backwards" in relation to the historical progression of events and reasoning for the use of the reals - the reals came first as a tool to do analysis, they were defined by Dedekind and others in order to get a "continuum" to work with, they complete the rationals to get the reals and just got the whole thing as a result of that, and only later was it found out that something like "uncomputable numbers" even exist; then it was also found out that they may apparently appear as limits of sequences of computable numbers and in other places too; the book remark mentioned a computable function attaining its maximum at an uncomputable number.https://en.wikipedia.org/wiki/Specker_sequenceSo it's not like anyone got the idea "let's do math with uncomputable numbers", they just are in the reals the way the reals are defined and there's been efforts like "reverse mathematics" which look into questions regarding what axioms can we do without and how much of the "general context mathematics" will be "left" then. So it seems you could do a lot of analysis absolutely without uncomputables but that's getting way out of my knowledge, but I'm sure more information can be found about it online.
If anyone can talk about euclids elements book 5 proposition 5 and how it relates to dedekind cuts and accompany it with some OC compass and straight edge figures that would be much appreciated
>>16539661>>16539666Thanks. There might be some other anon in the thread who can help us out. But what I was getting at is that it strikes me as strange that we're calling these things we can't actually talk about in a meaningful way "numbers" when they only seem to "exist" insofar as they must for some continuity axioms to be satisfied. Are they really numbers then? I've seen that Specker sequence one before and, being no computer scientist and possibly unqualfied to make the judgement, it just seems like another example like the chitin constant of something that only achieves "uncomputability" by including criteria about halting conditions in the function, which seems kind of trivial (in a conceptual way I mean, I'm sure Specker and Chitin were very smart), like imagine if the only "irrational numbers" I could produce were functions with outputs specified by whether or not certain other numbers are rational.So yeah I know a lot of anons are very rude when they post about this but my concern is basically, if these axioms apparently lead us to the existence of "uncomputables" that are numbers we, in most meaningful senses, cannot even talk about, then it seems to me those axioms seem suspect.I've read briefly about computable analysis (online) and from what I can tell there's a sort of hierarchy of continuum with respect to computability i.e. reals are uncountable; computable reals are countable but not computably countable; finitely computable reals are computably countable but not finitely computably countable (and perhaps so on). Wherever I found this, and I would appreciate if anyone knows where to read more on this, I think the implication was that those uncountabilities correspond to continuums or that those sets are the real number equivalents in the respective domains of computable maths, finitely computable maths.
>>16539685>we're calling these things we can't actually talk about in a meaningful way "numbers"Well that is up to you on what you call a number. If you consider the elements of the set "R" as "numbers" then uncountables are certainly numbers since they provably belong to this set, if not then no.
>>16539691Kind of tautological isn't it? I think numbers are a real thing so it should matter what is and isn't a number. I am not a pure mathematician so maybe it's physics brain getting to me but this doesn't fly for other mathematical objects like vectors or groups, or for physical objects like electrons or phonons, which would all be very confusing to talk about if the consensus was to use the names for whatever you like
>>16536639There's nothing real about integers, either. There are plenty of things that are important to everyday life and ordinary thinking that are nonetheless still no more real than real numbers.Even personal identity is not a physical property or defined in physical terms. You yourself aren't any more real than real numbers. Maybe personal identity is phony but it's real enough for me.
>>16539695>I think numbers are a real thing so it should matter what is and isn't a numberHmmm well the word "numbers" doesn't really reflect any concrete mathematical construct as far as I know. Humanities first understanding of numbers was as of just 1,2,3... the numbers used to count real things in the real world, but as more complicated concepts began being understood (negatives, rationals, reals) the definition of "number" got broadened to include them (probably because all of these included the counting numbers within them). I mean if you went back to the ancient egyptians and showed them a complex number and called it a number they would probably disagree with you that it is. Is a vector with size n a number? Personally I don't know, and don't think it really matters.
Nobody here can elaborate on how elements 5.5 relates to dedekind cuts?
>>16539734https://hsm.stackexchange.com/questions/7882/did-eudoxus-really-set-out-to-present-irrationals-as-dedekind-cuts
>>16539735No
>>16539735Did eudoxus travel to the future and back in order to be aware of dedekind so he could present his work as dedekind cuts to his greek bros who didn't know who dedekind is?
Thoughts on apatomes?
>>16539661What about the number i is it computed or asserted?
>>16540240Surely more of an algebraic object than a number. Like how you can have the matrix representation of i they're just objects that satisfy some properties we chose because they can describe the plane. Writing i with the other numbers is just shorthand for "object that satisfies i*i = -1" in the same way we write e^x which strictly seems like it should mean repeated multiplication as a shorthand for "object that satisfies d/dx e^x = e^x" (or there are alternative definitions of exp(x) you could use)
>>16540240that nigga is algebraic, since it's the solution to (x^2)+1=0, so it's as "asserted" as say root 2 ((x^2)-2=0) or 3 ((x^2)-9=0)
>>16539661>but you can't really isolate or describe or give an example of an uncomputable number.Wrong. You can give many such examples. Here's one: a real number whos i'th binary digit is 1 if and only if i codes a valid program that halts. Its uncomputability is a consequence of the halting problem.
>>16539666>then it was also found out that they may apparently appear as limits of sequences of computable numbersAll real numbers are limits of sequences of computable numbers. In fact, all real numbers are limits of sequences of rational numbers. This follows straightforwardly from the archimedian property>they complete the rationalsYeah kinda not really. You could say rationals have "holes" but you could also then say reals have holes as well which infinitesimal numbers could "fill up".
>>16539685>But what I was getting at is that it strikes me as strange that we're calling these things we can't actually talk about in a meaningful way "numbers" when they only seem to "exist" insofar as they must for some continuity axioms to be satisfied. Are they really numbers then?First of all uncomputable numbers does not mean undefinable numbers. Plenty of uncomputable numbers are definable can can be reasoned about in modern mathematics. You can prove many properties and in modern mathematics there's really little reason to treat them much differently from other numbers.As for whether they should be considered numbers at all, if you accept modern mathematics, then yes, because that's how modern mathematics works. Most objects and notions in modern mathematics rely on the concept of an arbitrary set which does not have a definition, so in this sense the objects are ultimately undefined. They cannot actually be described or exhibited, their existence is simply asserted from an axiomatic system. And defining or constructing something is equivalent to proving its existence (or unique existence in the case of a definition) from the chosen axiomatic system. There is no actual construction going on.That's simply how modern mathematics works, and if that bothers you, or you want the math to be meaningful, you probably are a finitist.
>>16540254>>16540277These guys are misleading. The way complex numbers are defined is not by finding solutions to equations or whatever the fuck.>>16540240"i" is a symbol that denotes the ordered pair (0,1) that is part of the set "C" of complex numbers that is defined as {(a,b)| a,b in R}. These weird explanations about root of -1 or other bullshit is why people are confused and say that math is fake and gay. i^2=-1 follows from the definition of complex numbers and complex multiplication not the other way around.
>>16539685>computable reals are countableComputable reals are actually uncountable also if you require everything to be computable. There is no computable function which enumerates all the computable reals, and a simple application of a diagonal argument proves this.
>>16539701>There's nothing real about integers, eitherThere is. You can actually give a definition of an integer that allows you to distinguish between what is or isn't an integer (for example, as a string of digits 0-9) unlike with real numbers. You also have actual definitions of what it means for two integers to be equal, definition of their sum, product, division etc. None of this can be done with the so-called real numbers.
>>16540278>You can give many such examples. Here's one: a real number whos i'th binary digit is 1 if and only if i codes a valid program that haltSchizo babble. Build me a uncomputable number using ZFC, and no, making vague references to programs that halt is not part of ZFC.
>>16540289>These guys are misleading>(x^2)+1=0>misleadingnigga, [math]{\displaystyle \mathbb {R} [X]/(X^{2}+1){\stackrel {\cong }{\to }}\mathbb {C} }[/math] , i bet you don't even math
>>16540292Sure. Provide your preferred definition of a computable program using ZFC.
>>16540294>definition of a computable program using ZFC.Typo, meant a computable number.
>>16540290>Computable reals are actually uncountableSimply not true.
>>16540293Give me a definition of complex numbers
>>16540298Exhibit a computable enumeration of them.
>>16540294>computable programYou mean computable reals? Can be defined using computable Dedekind cuts.
>>16540300see?, i knew you didn't math>>16540301what you are saying is true, but so is what >>16540298 says, the computable reals ain't computably enumerable but they can be put one to one in corresponace with the naturals & so have said cardinality
>>16540301>computable enumerationNice try schizo, an uncomputable enumeration would also imply that the set of computables are countable. The fact that a computable enumeration does not exist does not imply in any way that they are uncountable.
>>16540306>see?, i knew you didn't mathNo arguments? I accept your concession
>>16540289But complex numbers are actually those ordered pairs + rules about how to multiply and add them. But if you just give someone the definition (which historically we had to work backwards from i^2 = -1 to arrive at anyway) of complex multiplication that's just even more confusing because there's no sense of where this ad-bc stuff came from. Better to start with rotation in the plane first becuase it's a lot less misleading in the explanatory sense IMO>>16540290>Computable reals are actually uncountable also if you require everything to be computableYeah which is why that is immediately followed with>computable reals are countable but not computably countablealthough maybe I should have written computably enumerable
>>16536633Yes, because the universe doesn't contain any infinity so infinities are stupid quirks of math
>>16538337-general TSOH!-Cristiano Ronaldo-QuetZalcōātliN froNt of a bbq grill just shootiN' the shit.TSOH brings the chicken, CRonaldo brought that flawless peri-peri spice rub. QueZa uhh conjures divine tortillas out of thin air.A feast for the ages—or at least, until the prime numbers crash the party.TSOH ~ CR ~ QZ ~ Ninteractive poetry inspired by anon <3
>>16540451(smaller image)
>>16539599sqrt(2)/1
>>16539644You just make a Specker sequence.
>>16539483>AT SOME POINT THE NUMERATOR STOPS BEING AN INTEGER!When?>I DON'T HAVE TO ANSWER THAT!No seriously, when?>INFINITY!You never reach infinity. It's an integer forever. And the denominator is just a power of ten, so ALL numbers are rational. There is no number that cannot be composed in this manner. Only ones that in your imagination switch suddenly to non-integers when you reach the infiniteth digit.You are in a religious cult.
>>16540728You know it's gonna be some retarded shit when they start talking about decimal representations.
>>16539735Why can't you talk about it
The finest, and most concise, proof in this thread:>>16536697
who fucking cares just round it to a few decimal places until its within tolerance
