Currently getting started on my math undergrad, guess i'll have to watch this video to know the ways of abstract tomfoolery.
@SheafificationOfG5 ай бұрын
Don't get me started on abstract nonsense!
@zyansheep5 ай бұрын
@@SheafificationOfGBe you a fellow abstract nonsense enjoyer?
@JakubWaniek5 ай бұрын
A function is just a morphism in the category of topological spaces, so all functions are actually continuous
@JakubWaniek5 ай бұрын
Wanted to talk about discountinuous functions anyway for a laugh? The forgetful functor Top --> Set has a left adjoint, you're welcome
@SheafificationOfG5 ай бұрын
Very discrete, but Top is categorically unpleasant, and I prefer localising at the homotopy equivalences so that the space of real numbers is *really* easy to work with.
@JakubWaniek5 ай бұрын
@@SheafificationOfG I'm very profinite-pilled, so I place my faith in Scholze and Clausen to deliver us from this evil 🙏
@Number6_5 ай бұрын
I see you have mastered select, copy and paste. You might want to work on your math skills now!
@markusklyver62775 ай бұрын
@@JakubWaniek KZbin feels oddly small when my Facebook internet friends show up in KZbin comments. How is your PhD going anyway?
@elygolden5 ай бұрын
It gets worse than this: it's not possible for any real number representation to represent non computable numbers, so any "real number type" necessarily will have gaps you just cannot represent. That's why if I want rigour I will use an integer, rational or other algebraic type and otherwise... "floats are fiiiiine 😅"
@automatescellulaires85435 ай бұрын
you can also try your hand at an epsilon approximation of computable Real numbers. Which in most cases, you can just call "Reals" and be done with it. Alternatively, for day to day use, you can just use a 1024 bits fixed precision (512 bits of integer, and 512 bits of fraction). With that you can write a Kerbal space simulation that will span several time the observable universe at under the plank constant granularity.
@CognitiveOffense5 ай бұрын
As a meandering philosopher, Ultrafinitism is infinitely fun to drop in a math department with whilst armed with delicious controversy popcorn. I like to quip that a good lower bound is the value of the Ackerman function for the number of particles in the observable universe. Great video, thanks for making and sharing it.
@SheafificationOfG5 ай бұрын
Ah, the meandering philosopher, a mathematician's worst nightmare
@leotakkinen48895 ай бұрын
Even though we don't know whether or not pi contains every possible string, I'd expect us to have checked relatively small strings longer than phone numbers and social security numbers, so the clip shown might be technically correct even though misleading.
@SheafificationOfG5 ай бұрын
I was thinking about that, too (but only after I uploaded the video). The clip I yanked from (from an episode of Person of Interest) went on for much longer than talking about security numbers and locker combinations: the guy went on to talk about how, if converted from decimal to "letters" (idk, maybe he meant base 36 or some other mapping to strings), then pi contained Shakespeare, your life story, everything you'd ever say, etc. But you're right, if limited to some small finite sequence length, it's likely from the number of digits of pi that we've computed that we can confirm the existence of each such sequence.
@mido07002 ай бұрын
@@SheafificationOfGI think the point of the clip is to show some intuition about infinity, at least that's what I got from it.
@jkid11345 ай бұрын
My freshman year of college, I presented on p-adic numbers, and at the start I asked if anyone knew how real numbers are defined, which no one did. Literally everyone was operating on the "real numbers are the numbers in between the rational numbers" level instead of thinking of them as sums of smaller and smaller rational numbers. So I totally believe you that other disciplines never learn this 😂
@SheafificationOfG5 ай бұрын
Meanwhile programmers are out there working with 2-adics without even realising.
@hellishbro5 ай бұрын
p adics are basically little endian numbers in base p right
@jkid11345 ай бұрын
@@hellishbro this is my first time hearing the term, so I'm a little out of my depth here, but if you're asking if there's anything more going on than a reordering of the digits, yes, very much so. I'll just explain what I know lol. the distance between two numbers x and y is defined as |x-y|, which is then part of defining convergence for infinite series (each finite step has to get closer and closer to the goal value) and the real numbers are then defined as the numbers you can reach with these convergent sums. But there's another family of absolute value functions that do what an absolute value function needs to (besides the one you know and some degenerate case): |x|=p^-a where x= p^a * r/s (r and s don't divide p). For example the 3-adic norm of 18/5 = 3^-2. Anyway you can use these to flesh out p-adic numbers in the same way the reals are fleshed out, and you get a coherent mathematical system, but there's a bunch of weird stuff going on. I'm talkin fractal number lines, positive expressions for negative values, open balls and closed balls are the same thing, crazy stuff. One of those things is that digits can extend infinitely to the left, but ...111111 does genuinely mean 1+10+100+... It just doesn't diverge. Anyway, I can only hope this was useful at all. Literally everything I know about "endians" is a five minute wiki skim I just did.
@legendgames1285 ай бұрын
@@SheafificationOfG I actually did realize this when I first learned about p-adics and related it to integer overflow.
@legendgames1285 ай бұрын
In ZF, with the Axiom of Infinity and the Power Set Axiom... Natural numbers can be defined as the set of all natural numbers that came before. Integers can be defined as infinite sets of pairs of natural numbers who are related by a-b=c-d or a+d=b+c. Rationals can be defined as infinite sets of pairs of integers who are related by p/q = r/s or ps=rq, notably, q and s cannot be 0 unless you wanna go down the unsigned infinity + the nullity rabbit hole. Real numbers can be defined as a pair of infinite sets of rational numbers either less than (set A) or greater than or equal to (set B) the given number. Taking it a step further, complex numbers can be defined as a pair of real numbers. The details of implementing the operations, particularly for the real numbers, are a little more nuanced and my teenage programmer mind cannot understand those yet. I can understand implementing addition, multiplication, and subtraction for the natural numbers and integers, but where does the decimal place representation come into play, particularly when doing elementary operations on rational numbers?
@treborhuang2335 ай бұрын
The recording noise somehow carries subtle meaning and helps convey the flow of the video, well done!
@treborhuang2335 ай бұрын
And the content is great too! I have wanted to talk about this for a long time, now I can just point to this video
@SheafificationOfG5 ай бұрын
Thank you! Finally, someone recognising the purpose of my audio, and that it's definitely and absolutely not a reflection of how I can't afford a good mic...
@treborhuang2335 ай бұрын
@@SheafificationOfG You can remedy that with some free/pirated noise reduction, but do keep a small amount because I genuinely like that (except it maybe being a bit too loud).
@jichaoyang27675 ай бұрын
can you make a video on the law of the excluded middle? It's really entertaining to see your jokes about it so please throw us more of it
@SheafificationOfG5 ай бұрын
Maybe I will, maybe I won't 🙃
@jichaoyang27675 ай бұрын
@@SheafificationOfG bold of you to assume there’s not a middle 🤔
@quantumgaming91805 ай бұрын
One of the best video I've seen on youtube in my life
@MrAllenmath5 ай бұрын
Wow! This video is amazing! I've been thinking about this sort of issue for a while. Thank you! Have you seen any of Norman J. Wildberger (Mathematician) videos?
@SheafificationOfG5 ай бұрын
Thanks! I feel like I need to be in a very particular frame of mind to watch Wildberger's stuff (and I've never been in that frame of mind).
@markusklyver62775 ай бұрын
Wildberger is on the fine line between mathematics and crankery.
@miguelsouto20912 ай бұрын
The reason why 0.1 + 0.2 != 0.3 is mostly because of the truncation problem when you transform 0.1 and 0.2 into binary. However, usually with squareroots and calculations with transcendental numbers, functions are used to approximate the value.
@derendohoda38912 ай бұрын
This problem is really quite interesting in how it crops up with other models of real numbers, like continued fractions.
@trwn87Ай бұрын
"Close enough is close enough." - every physician ever!
@smellslikeupdog805 ай бұрын
oh no. the level of nuanced shade was an immediate subscribe. Wow, this is brilliantly hilairous.
@ricc35415 ай бұрын
This is the dual of the last video
@SheafificationOfG5 ай бұрын
I wonder what video co/evaluation would mean for this pair of videos 🤔
@legendgames1285 ай бұрын
I wonder if I could implement the unsigned integers in the same way that mathematicians do it: 0 = the empty array [], 1 = [0], 2 = [0,1], 3 = [0,1,2], for some n, n+1 = [all the elements of n, n], and we can keep going to implement integers, rationals, reals, and complex numbers.
@SheafificationOfG5 ай бұрын
You can definitely encode von Neumann naturals as lists, if you like, and work your way up. All arithmetic operations and comparisons can be implemented all the way until real numbers. Once you're at reals, then you run into the issues mentioned in the video: discontinuous operations may spell trouble (e.g. equality of real numbers), no matter your choice of implementation.
@strangeWaters3 ай бұрын
Great vid. If you have a representation of your data stream in some programming language you may *in some cases* be able to tell if it's zero. e.g. 0 : ℝ = loop { yield 0 : ℚ; } = loop { yield 0 : ℚ; yield 0 : ℚ; } by bisimilarity. But you can't guarantee an answer to such questions in turing-complete languages.
@trwn87Ай бұрын
The grade 6 problem is so funny! 😂
@lionelinx75 ай бұрын
As someone taking their second semester real analysis class i never considered this application but its straight out of the textbook in theory lol
@kuzuma45235 ай бұрын
I believe your representation of real numbers by Cauchy successions falls off when considering the distance between x and xn is less than 1/2^n. Mathematically of course it holds since the interval of a_n and b_n halves in distance each iteration; yet the algorithm implementation of such proof (5:31) loses precision when computing *rational mid*, since by dividing by 2 a binary number you are subsequently adding another bit. Now, I don't know how you implemented the *rational* class and if you've overloaded the / operator, but as long as it has finite bit representation, we've got a contradiction. The only silver lining is if *rational* would have size arbitrarily large depending on n, which would indeed be a crazy mad idea: Since you have perfect bit representation for any element in the set A made of 1,2, and the property (x+y)/2€A, with the fact that A is dense in R, we get the assurance that any real number on (1,2) can be the point of convergence of some succession of A (finally proving |x-xn|
@vekyll5 ай бұрын
Rationals are _not_ represented using sequences of decimals, since that would just introduce the same problem. They are represented using a sign bit and a pair of two (arbitrarily sized) natural numbers. Then the implementation of / is straightforward.
@SheafificationOfG5 ай бұрын
Moneybags forgot to mention that the programmer before you already implemented arbitrary precision rational numbers. As @vekyll says, this isn't nearly as challenging as implementing real numbers. Arbitrary-precision integers are okay: you can encode any integer as a (not uniformly bounded) finite sequence of bits (e.g., std::vector, if you like). As long as memory is no issue, you can hold any integer you desire. You can then overload the arithmetic operations and whatnot. (Python already does this sort of thing; memory is allocated according to the number of bits needed to encode the integer.) You can then implement rational numbers as mentioned at the end of the video, as a pair of integers (numerator and denominator). Then, overloading the arithmetic operators for rational numbers will be just like we did it back in grade school. In particular, every rational number is constructible (in finite time and memory). Although rational numbers are (definitionally?) dense in the reals, and every real number can be encoded as a Cauchy sequence of rationals (or whatever other model of real numbers you want to work with), an arbitrary real number cannot be necessarily encoded in finite memory. Some can (even pi, as mentioned in the video), but the set of all "finitely representable / constructible" real numbers will always be countable, and therefore cannot account for all real numbers. I feel like I went on a bit of a tangent, so I hope this addresses your comment.
@dbob1324 ай бұрын
The answer to the interview question is just the naive solution as long as they don't need trigonometry or exponential functions (with reasonable assumptions about the model that has infinite memory). This is why functional containers are used (like streams or channels) in HPC for even the most mundane operations. The main misconception this video leverages is that math and computer science deal with numbers in the same way, they do not. Computer science deals with already extant data held within some model and the way that data is stored or retrieved will determine the type of that data, not some intrinsic property. Thus, to say that there is some real number that is "impossible to represent" is just to say that a given algorithm to store and/or retrieve that data has an issue. Whether that issue is one of computability or complexity depends more on the operations accepted by the algorithm. As a side note, if "runtime isn't an issue" then an infinite amount of time to run isn't an issue. That's what runtime means
@tediustimmy5 ай бұрын
As a computer programmer: most of us can't do math. I would point you to the Linux tool bc, though. It stores numbers in a manner such that .1+.2=.3 correctly, exactly, and with no magic. It is just slow (the version on Android is faster that the GNU version). As for me: if I say arbitrary precision, I mean an a priori precision, so that I can compute the working precision necessary for the final result to have that precision. And then the program would run at a fixed precision.
@SheafificationOfG5 ай бұрын
While fixed precision is essentially unavoidable, one caveat is that it's impossible to maintain a fixed precision: approximation errors will grow as you keep operating on the numbers (this is somewhat reflected by my definitely non-contrived example in 2:38).
@tediustimmy5 ай бұрын
@@SheafificationOfG The theory behind the Table-Maker's Dilemma is frequently not a practical limitation.
@thedude616365 ай бұрын
I have a degree in applied physics, and I am a software developer with 6 years of experience and it's true, I have no idea how to do math
@Number6_5 ай бұрын
I have never met a programer who new anything about math let alone passed a math class. I understand that some degrees required a basic math background to graduate. But not the basement coders and self served programers and hackers. Known as geeks! I say let them wait..
@gSys13372 ай бұрын
First of all: I study computer science in Germany. For the Bachelor we had four lectures purely about maths. One of them being first-order logic. All other lectures that are also maths but in disguise. If someone graduates C's they should have at least a decent understanding of maths. As my second question if I may ask: What is your academic background?
@bes_it5 ай бұрын
Nice video!
@SheafificationOfG5 ай бұрын
Thanks!
@SparkySywer5 ай бұрын
Can you elaborate on how it's impossible to construct a discontinuous function ℝ→ℝ? Maybe I'm misunderstanding, but wouldn't something like f(x) = { 0 if x < 0, 1 if x ≥ 0 } be ℝ→ℝ and have a discontinuity?
@SheafificationOfG5 ай бұрын
So, by "construct f(x)", I mean "determine a finite-time procedure for computing f(x) if given x." While your function f(x) certainly has a discontinuity (in fact, your function is exactly `is_pos`), the hard part is to actually implement it constructively.
@markusklyver62775 ай бұрын
How do you check if f(x) = 0 given your f? You can't know if x < 0 using a Turning machine.
@reddy193614 сағат бұрын
@@SheafificationOfGWouldn't the Dirichlet function (the indicator function of Q) satisfy your criteria?
@SheafificationOfG10 сағат бұрын
@@reddy1936 Dirichlet's function is certainly not computable
@mltyblnd5 ай бұрын
congrats on 315 subscribers
@SheafificationOfG5 ай бұрын
Thanks!
@MrAdamo5 ай бұрын
Wow you guys are smart I just like this because it makes me feel like a genius
Does someone have a reference that explains the part on discontinuous functions. I don't understand at all what he's saying. Isn't it trivial to construct a discontinuous function? I tried googling but I guess idk how to Google it 😅
@pieter-jan26Күн бұрын
I thought about it a bit and I think I understand. A good reference would still be appreciated though.
@NwnwnwNw-z7mАй бұрын
arbitrary precision integer,mantissa and exponent
@vladthemagnificent90522 ай бұрын
After watching this video I'm taking a piece of paper and a pencil and going to solve this problem about watermelons because I just cannot leave it be. Thanks for nothing!
@James-rj8jp5 ай бұрын
I'm currently studying abroad in math. I can confirm that mathematicians are in fact this insane.
@anonymoususer79865 ай бұрын
JapanEat's alt channel
@chri-k5 ай бұрын
You aren't saying that programmers can't do math. You are saying that computers can't do math.
@chri-k5 ай бұрын
And that is clearly false given WolframAlpha exists
@SheafificationOfG5 ай бұрын
Shh, don't give programmers another reason to blame the computer instead of looking inward and realise the deficiency is in their code all along! As for your second comment: although WolframAlpha is a remarkable tool for symbolic mathematics, it's still constrained to constructible real numbers (or constructible approximations of real numbers) when the going gets tough. By definition, it's really difficult for us to hit it with non-constructive real numbers, so this kind of distinction is not important for what happens "irl", for the most part.
@tediustimmy5 ай бұрын
Computers can't do math! They perform algorithms on bits that are consistent with math.
@no_name47965 ай бұрын
Ironically doing precise math is harder on high level languages then on low level ones. Probably doesn't fit with the video, but it's ok, i have still to watch it ;-)
@SheafificationOfG5 ай бұрын
Curious what you mean when you say higher-level languages have a harder time doing precise math? I was originally going to "implement" reals with Python instead of C++, but I quickly realised there wasn't enough gatekeeping this way.
@angeldude1015 ай бұрын
At least Python's negative integer division isn't broken by design like C and C++'s are.
@reallyWyrd5 ай бұрын
"why do programmers settle for the wacky artefacts of floating-point arithmetic?" I mean you could try asking them. And, hey, just because you're unable to explain a Monad to anyone that isn't a mathematician. It's ok. I still choose to believe that it's *not* just something math people made up to sound impressive.
@tediustimmy5 ай бұрын
To answer this question: computing floating point numbers requires multiplying and dividing by powers of your working base. When your working base is your representation base, then this is just shuffling digits around. It's a performance hack that people smarter than most programmers agreed on.
@UFOHESKEY5 ай бұрын
I'm a computer scientist, doing math and physics and math pedantry is just not as fun as CS pedantry but we are at least lazy where it matters. When it feels good physics is not LAZY, it's GAMBLING
@charlesleninja2 ай бұрын
Fun fact: almost every real number is not uniquely describible with a finite sequence of sentences.
@markusklyver62775 ай бұрын
Short answer: no.
@Koroistro2 ай бұрын
I don't get the whole "pi could hold all the information". Lets assume for a second that the normality of pi is a thing. That wouldn't be helpful, you'd need to know WHERE in pi the information can be found. So that's the information you'd need, so you still need information. It'd be like saying "0 and 1 hold all the knowledge of the universe if you combine them in the right way!" Duh, finding the right combination is the problem. (and how you interpret the combination)
@SheafificationOfG2 ай бұрын
You and me both, fam. Even if pi is normal, I won't be able to use it to financially recover from going to grad school.
@Gordy-io8sb2 ай бұрын
@@SheafificationOfGI mean, you could've avoided higher mathematics schooling by just self-teaching.
@mihaleben60515 ай бұрын
A real number is any value that does not contain i In the forms,xi, x+yi, and its negatives
@SheafificationOfG5 ай бұрын
Thanks for your comment! You're defining real numbers as the subset of complex numbers that have no imaginary component (no "i", as you say), but this begs the question: how do you define *complex* numbers (without referencing real numbers)? Usually, people define real numbers first (by "filling the gaps" between rational numbers), and _then_ define complex numbers as numbers of the form x + yi, where x and y are real numbers.
@mihaleben60515 ай бұрын
@@SheafificationOfG oh. Ok A complex number is i^x +i^(x+1) Where x is equal to a non imaginary number that can be represented by a fraction, and the numbers after its decimal points are zero, the first exponent must not result in a imaginary number, and the second results in a non imaginary number.
@SheafificationOfG5 ай бұрын
@mihaleben6051 If you look closely, your definition still relies on real numbers ("non-imaginary"). How do you define non-imaginary numbers without referencing complex numbers and real numbers? The deeper you go with these things, the more you find subtleties if you're trying to avoid being circular.
@mihaleben60515 ай бұрын
@@SheafificationOfG ... You Cant... Wait A number is a constant. A variable is not a constant, and can be a number, but not considered a number itself So a+bi But i know its impossible.
@angeldude1015 ай бұрын
Does this mean that 0x...AAAB is a ℝeal number? I thought that numbers with infinitely many digits to the left of the radix point weren't considered ℝeal even if they don't contain an "i". P.S. The number I gave is specifically 1/3 in hexadecimal.
@WhiteGandalfs5 ай бұрын
Programming has nothing to do with implementing mathematical abstraction. Never. The most it comes in the vicinity of that goal is to implement real world applications of those abstract concepts. And in this, programmer can become really good and useful. Most abstract concepts of mathematicians are completely decoupled from real world applications. Without those programmers, they would seldomly become useful for real life, and mathematicians would barely be as respected and useful as they are nowadays. As a mathematician, avoid going into disrespect against the most useful group of professionals that are out there to make your work into actually useful tools!
@SheafificationOfG5 ай бұрын
😔 what's the point of being both a mathematician and a programmer if I can't dunk on both groups
@user-wv2jc4xm7r5 ай бұрын
Давай!!! ДАВАААЙ УРАА ДАВААЙ ДАВААААЙ
@Dent425 ай бұрын
The information you're sharing is too important to have audio this bad. Where do I sign up to help fund better audio?
@SheafificationOfG5 ай бұрын
Hahaha yes my poor mic is begging to retire I'm on the fence about panhandling the viewers until I figure out my life schedule
@Dent425 ай бұрын
@@SheafificationOfG I mean, if you wanna co-shill, I could start making tuts for folks just starting to dig into maths and send ‘em your way. Anything to hear what you have to say w/o my ears bleeding lol. Alternatively, there are post-process processes that could clean up the audio quite well
@siddanthvenkatesh27445 ай бұрын
Does this mean that you can't check if a value is equal to 0? Or am I misunderstanding.
@tsawy65 ай бұрын
Yeah you can't check equality for any pair, just precision within a level of tolerance, at least for this data structure. Notably, the approximations are not unique!!
@siddanthvenkatesh27445 ай бұрын
@@tsawy6 So is it possible to construct a program where you enter an expression (Eg: Trig functions, Integrals, normal stuff, Factorials etc) and an error margin which returns the answer that is guaranteed to be within that error?
@tsawy65 ай бұрын
@siddanthvenkatesh2744 yeah! For any constructive continuous function! Integrals im not 100% on, and factorials will always be a bit weird cuz yer using reals so youdd have to use the gamma function which id always freaky, but trig would be totally fine
@SheafificationOfG5 ай бұрын
Just to follow up on @tsawy6, once you unwind the definitions of integrals / factorials (another integral) / etc as limits of finite constructions, you'll be able to tease out programs for approximating them to whatever precision you want (though the implementation will depend on the integrand if you're particular about the error bounds).
@janisir45292 ай бұрын
Floating point arithmetic is the devil.
@decare6965 ай бұрын
Sorry, but I gotta say it for the meme: Have you considered rewriting this video in Rust?
@SheafificationOfG5 ай бұрын
I'm surprised I managed to go as long as I have without encountering the rustaceans.
@angeldude1015 ай бұрын
🦀
@tylerfusco74955 ай бұрын
3:10 you forgot to assume a flat earth! 😱
@mihaleben60515 ай бұрын
This has no connectin to real numbers
@colonelmoustache5 ай бұрын
Juste pour être bien sûr, t'es québécois ???????
@SheafificationOfG5 ай бұрын
Nahh, le dernier message c'est en hommage à la loi sur les langues officielles 🙃
@holyshit9225 ай бұрын
Forget the logic, we have breaks and continue Who cares about speed of algorithms
@SheafificationOfG5 ай бұрын
Breaks and continues will ensure the algorithm terminates, at the cost of a bit of "undefined behaviour" at very low orders of magnitude. Of course, that's perfectly fine in real life (otherwise we probably would've already ditched IEEE754).
@thereal48155 ай бұрын
i don’t know how to do math :)
@avg.player5 ай бұрын
nerd
@denizgoksu98685 ай бұрын
LEM bad
@SheafificationOfG5 ай бұрын
LEM not good*
@denizgoksu98685 ай бұрын
LEM some affirmative definition of bad that implies not good
@gustavstreicher48675 ай бұрын
You've just proven that real numbers aren't real. They are an idea that is only formed by looking at rational numbers in decimal notation and saying "Wow, this notation is great, but let's make the amount of digits we can put after the dot unbounded". The moment you make any idea unbounded it ceases to exist in reality. Also, certain kinds of mathematical questions don't have actual answers. They only have rational approximations for answers that don't really answer the question. Like the question: "How many radi of a circle fit into it's circumference?". Your point on the ability to represent pi in a finite way using an integral over the root of 1 - x^2 is wrong. An integral isn't a finite process. It deals with adding infinitesimals together, which is yet another unbounded process. You can only calculate a Riemann approximatiom for it. Or, if you're lucky, you can get an antiderivative, which, when evaluated, can only be evaluated to a rational approximation. The second formula you mention requires you to select a finite number of terms to calculate an answer, which results in another rational approximation of pi. The number pi doesn't exist. All that exists are a bounded (not unbounded) number of approximations that we could feasibly reproduce for different use cases. Now, this doesn't mean that real numbers don't have any utility. They absolutely do in the conceptual/unevaluated domain (in question form). We can work in this conceptual domain until we've come to some representation which represents our question in the simplest way possible and the pull that into the real world by getting a rational approximation to it's answer at the very end. For the astute amongst you, you might realise that even rational numbers don't truly exist given that a ratio is just a question asking "What is the answer of 'a' divided by 'b'?", where we only have answers if the result was a product of a or b with an actual integer. How about negative integers? Show me a negative number. You can't, yet it has utility for when someone finally pays off their debt to know they're square. In fact, show me the number one. You can't, because the difinition of the number one, like beauty, is in the eye of the beholder. A spoon is one as a spoon, but multiple as molecules, even more so as atoms, etc. The only thing you can show me is the number zero because it implies a lack of something to show.
@Errenium5 ай бұрын
you lack imagination if your notion of reality only includes aspects of the physical world, especially if it's only your arbitrary perception of what is possible. The only honest answer to "is the universe finite or infinite?", for example, is "i don't know".
@gustavstreicher48675 ай бұрын
@@Errenium My imagination is definitely not limited to the physical world. That should be obvious of any individual. Anyone can think of things that don't exist. The point of mathematics is to be of utility in the real world. Or do you regard it as a form of imaginary entertainment?
@СергейМакеев-ж2н5 ай бұрын
Were you even listening? As long as you aren't bringing in discontinuous functions, the space of real numbers works just fine! Well, *constructive* real numbers, but that's a technicality. The point is, this space includes Pi and other weird transcendental constants defined by arbitrarily complex formulas.
@gustavstreicher48675 ай бұрын
@@СергейМакеев-ж2н I didn't say that real numbers don't work, in fact I said they have utiliy. What I did say is that they cannot be represented in reality. In the video he has to rely on programming that only has rational approximations with the numbers he's solving for. Maybe you weren't watching.
@СергейМакеев-ж2н5 ай бұрын
@@gustavstreicher4867 A computational device (like a C++ function) that generates digits on demand *is* a representation in reality. The function itself *is* the number. If that's not a representation, I don't know what is.