Rational, Irrational, and Integer. The correct definition.  - Page 1
Forum section: electronics / general

author: Peter Taylor
author_rank: Regular Contributor
timestamp: December 21, 2022, 03:43:02 am
content: This is the new and correct definition of a number.1/ A number must be an integer.2/ The term rational and irrational don't apply to a number by definition 1/.An integer is any number that can be defined wholly. For instance, 7, and 1 / 7 can be defined wholly and are therefor integers.An equation that cannot be defined wholly, such as root 2, where we don't know what number multiplied by itself equals 2, remains a question, and is not a number by definition 1/.A number can't be rational or irrational. These terms have no meaning.Thankyou.

author: TimFox
author_rank: Super Contributor
timestamp: December 21, 2022, 03:48:08 am
content: No.

author: gnuarm
author_rank: Super Contributor
timestamp: December 21, 2022, 03:54:39 am
content: Why are you making up your own math? I'm not following what you are trying to do.

author: SiliconWizard
author_rank: Super Contributor
timestamp: December 21, 2022, 04:01:29 am
content: Is this a philosophical issue you have with non-integer numbers (and well, why not, although that's a bit odd, but a deep philosophical approach about this could be interesting, maybe), or is it because you have a problem fully understanding them?If that's the former, you may want to elaborate (although it's not guaranteed to get much traction.) Otherwise that's just your own definition with nothing to back it up except itself, which, you'll have to admit, is a bit circular.

author: gnuarm
author_rank: Super Contributor
timestamp: December 21, 2022, 04:17:44 am
content: I know there are different levels of infinity. There are an infinite number of integer numbers. You can pick two integers so there are not integers between them. Then, between these two integers, there are an infinite number of rational numbers.Here's where it gets tricky. There are an infinite number of irrational numbers between any two rational numbers. But can you select a pair of rational numbers, that have no other irrational numbers between them?Clearly, integers and rational numbers are different sizes of infinity. But I'm not sure that is true for rational and irrational numbers.Don't even get me started about transcendental numbers.

author: Tomorokoshi
author_rank: Super Contributor
timestamp: December 21, 2022, 04:33:00 am
content: All numbers are imaginary anyway.

author: ledtester
author_rank: Super Contributor
timestamp: December 21, 2022, 04:37:02 am
content: Quote from: gnuarm on December 21, 2022, 04:17:44 amThe cardinality of the integers and rationals are the same. They are both "infinite" but the size of the infinity is the same. The cardinality of the irrational numbers is strictly larger according to how mathematicians define it.Between two rational numbers there are an infinite number of irrational numbers.Between two rational numbers there is always another rational number, e.g.: (a+b)/2.

author: IanB
author_rank: Super Contributor
timestamp: December 21, 2022, 04:42:38 am
content: This is a rehash of an old topic. Let's not go there again.https://www.eevblog.com/forum/chat/integers-pi-and-number-lines/

author: magic
author_rank: Super Contributor
timestamp: December 21, 2022, 10:12:18 am
content: Quote from: SiliconWizard on December 21, 2022, 04:01:29 amHardly anyone who hasn't done at least a BS in maths understands what the irrationals are so it's the latter obviouslyAnd yeah, Deja vu

author: jpanhalt
author_rank: Super Contributor
timestamp: December 21, 2022, 11:29:15 am
content: More of this same rubbish:https://www.eevblog.com/forum/chat/integers-pi-and-number-lines/50/

author: pcprogrammer
author_rank: Super Contributor
timestamp: December 21, 2022, 11:46:23 am
content: Well in the other thread Peter Taylor does give a hint of where he livesIn Lala LandAnd that was already clear with his superb website that he made

author: TimFox
author_rank: Super Contributor
timestamp: December 21, 2022, 02:43:54 pm
content: Mathematics is something that happened to other people.

author: rsjsouza
author_rank: Super Contributor
timestamp: December 21, 2022, 02:52:14 pm
content: Quote from: gnuarm on December 21, 2022, 03:54:39 amIt is to "decolonize math", as these vicious numbers are transphobes since they don't believe in trangenderism and categorize themselves in binary terms (rational/irrational). Burn the books!

author: Nominal Animal
author_rank: Super Contributor
timestamp: December 21, 2022, 04:18:47 pm
content: I've always found itcomfortablehow the classification of numbers could be used as a tool. They are not arbitrary, and they areuseful.You start with integers, then rationals, and then irrationals. You extend these into complex numbers, and you'll find integers and rationals and irrationals have interesting features. For example, consider \$e^{-2 i \pi x}\$, where \$x \in \mathbb{R}\$, and what it means for \$x\$ to be integer, rational, or irrational.You start with integers, and extend these into vectors, and you get regular rectangular lattices. Integers correspond to lattice points, rationals to lines that cross an infinite number of lattice points, and irrationals to lines that intersect with a single lattice point. Examine the properties of such lattices, extending to non-rectangular lattices (say, face-centric cubic, body-centric cubic, hexagonal, and so on), and you've arrived at crystallography.From rationals, you can represent both the numerator and the denominator with a polynomial, and you arrive at rational polynomials, which have a lot of practical applications.You start with integers, think about rationals and divisibility, and arrive at prime numbers, prime number decomposition, and all sorts of discrete math.You combine the idea of irrational numbers, prime numbers, and rational polynomials, and you arrive at ideas like elliptic polynomials, which can lead you to elliptic cryptography.Everything else like discrete logarithms, factorization, and so on, used everyday in cryptography and computers and networks, is extensible from the first principles like this, and easily understood at the conceptual level, even though the actual mathematical machinery may be hard to fully understand and utilize. The space of such concepts has an infinite number (hah!) of dimensions, with each conceptual approach their own dimension, and the elements along that dimension are the concepts in which it is useful to divide the number or value into. For scalars, it is integers, rationals, and reals. For complex numbers, you have unity (\$\lvert z \rvert = 1\$, i.e. \$z = e^{i x}\$, \$x \in \mathbb{R}\$), roots of unity, pure reals, pure imaginary numbers, et cetera.Instead of a vast continuous infinite-dimensional space of concepts we'd be hopelessly lost in, these give us reasonable and trustworthy stepping stones we can walk along. They may not lead to everywhere yet, but you can always build new ones to reach places not yet reached.The only reason you'd want to redefine these concepts is if you insist on sitting at the singular starting point, with your eyes closed, hoping for an universe that was less complicated so you could fool yourself into believing that you understood it all, fully. So foolish.

author: T3sl4co1l
author_rank: Super Contributor
timestamp: December 21, 2022, 04:36:23 pm
content: Quote from: rsjsouza on December 21, 2022, 02:52:14 pmGot just a little chip on your shoulder there?Tim

author: TimFox
author_rank: Super Contributor
timestamp: December 21, 2022, 04:39:44 pm
content: Quote from: magic on December 21, 2022, 10:12:18 amI learned about irrational numbers in high-school mathematics at a US public school.

author: eugene
author_rank: Frequent Contributor
timestamp: December 21, 2022, 05:57:51 pm
content: Quote from: Peter Taylor on December 21, 2022, 03:43:02 amI doubt anyone would object to you inventing your own, private version of math. I also doubt that very few people would be interested, but I'm pretty sure nobody would object.What lots of people object to is redefining terms that are already well defined in the old (and still correct) math. So, my advice, if you hope to get any reaction other than negative, is to stay away from existing terms like "integer", "rational", and "irrational" and coin your own NEW terms to go along with your NEW math. You don't always need to tear down something old to build something new.

author: magic
author_rank: Super Contributor
timestamp: December 21, 2022, 06:06:40 pm
content: Quote from: TimFox on December 21, 2022, 04:39:44 pmCool.So I gather that the naturals are an extension of counting fingers beyond the actual number of fingers you might have, the integers are the naturals mirrored around zero, the rationals are fractions of the integers, and complex numbers are what happens when you imagine that sqrt(-1) is a thing.So what the hell are the irrationals, then?Something about continuous divisions and stuff?But you have heard about those "atoms" and "quantum" stuff, right?There is nothing real about real numbers.The theory that they are an inside joke of heterosexual European men invented to confuse Americans doesn't even seem so far off

author: TimFox
author_rank: Super Contributor
timestamp: December 21, 2022, 06:08:34 pm
content: "rationals" are ratios of integers"irrationals" are not ratios of integers

author: magic
author_rank: Super Contributor
timestamp: December 21, 2022, 06:10:20 pm
content: Quote from: TimFox on December 21, 2022, 06:08:34 pmThis definition is not a ratio of integers either

author: TimFox
author_rank: Super Contributor
timestamp: December 21, 2022, 06:11:49 pm
content: Take it up with the Encyclopedia Britannicahttps://www.britannica.com/science/irrational-number

author: magic
author_rank: Super Contributor
timestamp: December 21, 2022, 07:01:48 pm
content: Quote from: TimFox on December 21, 2022, 06:11:49 pmNeat, let's see if we need to further restrict the Privileged Boys Club to continental Europe onlyQuoteOK, this definition is not an irrational number itself, but let's see the reals...QuoteSo say I have a sequence of infinitely many zeros, followed by seven. Like, 0.(0)7.What sort of real number is that?

author: Nominal Animal
author_rank: Super Contributor
timestamp: December 21, 2022, 07:06:19 pm
content: In my opinion, complex numbers are between scalars and vectors. (There are others there too, like quaternions.)The \$i^2 = -1\$ thing is just a "mathematical" way of expressing the idea of rotation by 90°. If you think of the one-dimensional number line concept, then such a rotation by itself doesn't make any sense, but if applied twice, you mirror the number line. And that's exactly what happens.The math boffins examined the concept for a bit, and realized that it also naturally aligned with the exponential function, so that \$e^{a + i b}\$ corresponds to a scaling (multiplication) by \$e^a\$, and rotation by \$e^{i b}\$. Everything else stems from there.So, while the "imaginary number" concept doesn't make much sense for real numbers (integers, rationals, irrationals), it does work as a stepping stone for extensions to more complex concepts.In certain ways, complex numbers remind me of homogenous coordinates. Cartesian coordinates \$(x_1, \dots, x_N)\$ are represented by \$(X x_1, \dots, X x_N, X)\$ in homogenous coordinates, and somewhat similarly to complex numbers, homogenous coordinates can be used to express both translation and rotation (full transform) in a single operation. (In fact, the vast majority of computer 3D graphics libraries use homogenous coordinates at least internally for transformations.) A particularly interesting facet of homogenous coordinates is that they let one trivially distinguish betweenposition vectors(\$(x_1, \dots, x_N, 1)\$) anddirection vectors(\$(x_1, \dots, x_N, 0)\$). Mathematically, there is no reason to distinguish between the two, but in 3D visualization, the ability to easily do so while applying the exact same transformations (with translation only affecting position vectors and not direction vectors) simplifies things quite a bit.Quote from: magic on December 21, 2022, 06:06:40 pmAny number that you cannot express as the ratio of two integers (or an extension of the same concept for e.g. complex numbers).A step further are transcendental numbers. They are numbers you cannot obtain as a root of a finite-degree nonzero polynomial with rational coefficients, like \$\pi\$ and \$e\$.Quote from: magic on December 21, 2022, 06:06:40 pmThe two 'real' mean completely different things: one is 'physically real', and the other is 'the set of numbers \$\mathbb{R}\$ that we call "reals"'.Quote from: magic on December 21, 2022, 06:06:40 pmExcept they weren't initially European and never even "mostly" European, men, or even heterosexual.That's what makes "decolonizing math" so ridiculous, and extremely racist and bigoted concept.In particular, Indians and Arabs are quite,quiteoffended by your suggestion, and for good reason. Sthananga Sutra is about 2400 years old, and covers things like fractions and elementary number theory. In the 800s Arabs made huge leaps forward in maths, especially people like Muhammad ibn Musa al-Khwarizmi.

author: magic
author_rank: Super Contributor
timestamp: December 21, 2022, 07:59:00 pm
content: Quote from: Nominal Animal on December 21, 2022, 07:06:19 pmCan the number of all real numbers be expressed as a ratio of two integers, or is it an irrational number?Quote from: Nominal Animal on December 21, 2022, 07:06:19 pmYes, and hardly anyone knows what the latter actually are, that's my point which started this whole exchange.All I hear is "irrationals are not rational this", "reals are not imaginary that".So what theyare, if they arenotall those other things?Quote from: Nominal Animal on December 21, 2022, 07:06:19 pmNo, it's precisely truePeople of Color invented Mathematics.Heterosexual European Males came up with the formal concept of "Real Numbers".Heterosexual European Males teach that their "Real Numbers" are no more, no less, but all numbers that are real.Reality is a social construct now.This is Colonization.

author: SiliconWizard
author_rank: Super Contributor
timestamp: December 21, 2022, 08:29:47 pm
content: Good lord.

author: TimFox
author_rank: Super Contributor
timestamp: December 21, 2022, 09:02:11 pm
content: Although mathematics may have been dominated around AD 1920 by white males, there were also people of color at the highest end of mathematics.Seehttps://www.britannica.com/biography/Srinivasa-RamanujanAlso note that the Encyclopedia Britannica is now published in the US (headquartered in Chicago for several owners, including the University of Chicago at one time), no longer in UK.

author: Nominal Animal
author_rank: Super Contributor
timestamp: December 21, 2022, 09:04:24 pm
content: Quote from: magic on December 21, 2022, 07:59:00 pmNo, there areinnumerablereal numbers, anuncountableamount, which means there is no number – and there can be no number – that expresses how many unique real numbers there are. Your question is therefore based on an erroneous assertion and thus invalid.Even the concept of innumerability/uncountability and the cardinality of innumerabilities and infinities is over 2000 years old; the old Jain texts have been preserved to this day. Those semi-darkie boffins, eh?Quote from: magic on December 21, 2022, 07:59:00 pmOf course they do.Youdon't, but we both know you're atypical anyway.Any philosophical examination of what the realness of real numbers might mean is as useless as wondering what the "Nominal" in my pseudonym means. The former case is just a label and the definition is written in the language we call math; the latter is a pun that I feel close to. Neither warrants any philosophical or sociological analysis, because they just do not have any presence in any such domain.If we start by agreeing on the concept of anumber, then I could walk you through it. But everything stems from that. No philosophy or other sophistry needed.(But please, don't get into nitpicking on how the English language allows one to construct sentences where the word 'number' is used in different senses, for example in the sense of 'something that is countable'. Not all languages have that fault.)Quote from: magic on December 21, 2022, 07:59:00 pmI know you're yanking my chain, so I'll just say that I will never, ever accept that, and nobody can make me.

author: SiliconWizard
author_rank: Super Contributor
timestamp: December 21, 2022, 09:10:09 pm
content: Well given the type of opinions that I've gathered from magic's posts so far, I'm pretty sure he is just trolling here. Although you never know.

author: CatalinaWOW
author_rank: Super Contributor
timestamp: December 21, 2022, 10:02:25 pm
content: There is some minor merit in the OPs position. In terms of computing there is no way to correctly represent either irrational numbers or infinities. So these mathematical concepts have no existence in that subject matter area. The expressions and concepts that require irrational numbers can be adequately computed by rational approximations to both real and imaginary numbers. These expressions range from simple things like the area of a circle on up through vey complicated concepts, things like contour integrals and beyond.In the broader world of mathematics and engineering there is real use for all of these extra concepts that cannot be exactly computed. The identities and relationships between these computable quantities are heavily dependent on theorems and proofs using these concepts. If you choose to ignore all of this rich body of knowledge that is fine. Most of the human race does. But aren't you at least a little bit curious about the limitations or your "real world" view? They can occasionally jump up and bite you. Look up Gibbs phenomena for a relatively simple and benign example.

author: TimFox
author_rank: Super Contributor
timestamp: December 21, 2022, 10:11:17 pm
content: The Gibbs Phenomenon falls directly out from the mathematics of a Fourier series, but is always surprising the first time that one encounters it as a student.Most digital computation is glorified (integer) arithmetic, and the precision is limited by the available word size (etc.), but can be expanded to achieve a required precision for an engineering application.In his other post, the OP claimed that the circumference of a circle could be a rational value, if one merely cut a string to the length required to encircle a cylinder that had been machined.A parable: A flying saucer lands at the UN, and the little green men negotiate peacefully.In exchange for their information on curing cancer and other diseases, humans allow them to visit the important libraries and scan the vast collection of human knowledge contained there.The aliens convert their scans into Intergalactic ASCII binary code, then place a radix point in front of the huge binary number to make a rational fraction.To ship the information back to their planet, they engrave an extra line between the original two lines on a Pt-Ir alloy rod corresponding to that fraction.

author: magic
author_rank: Super Contributor
timestamp: December 21, 2022, 10:54:14 pm
content: Quote from: Nominal Animal on December 21, 2022, 09:04:24 pmAnd you believe it because some men in position of authority told you so.Meanwhile, I recall a first semester proof that there is precisely 2ℵ0real numbers, although things get murkier when you ask whatthatthing actually is.Quote from: Nominal Animal on December 21, 2022, 09:04:24 pmAnd yet still no proof has been posted that real numbers are a sound concept and anything more than an in-joke between a bunch of privileged European menEverybody knows how Q is built on Z and how C is built on R and somebody even namedropped quaternions in this thread, but how precisely R extends Q is apparently a mystery."It's just the set of all the numbers there are, trust me dude"."But wait, except for those others, they aren't real numbers because I said so".So what is 0.0000.... 0007 - a simple, infinite decimal number, apparently perfectly fine according to Encyplopedia Laymanica?Quote from: Nominal Animal on December 21, 2022, 09:04:24 pm17th century called and wants its calculus backeditReally, it's quite an ironic response to the question how todefinereal numbers.We will never agree, I could keep producing progressively more bizarre examples.Like, take a unit square. Remove all points whose either coordinate is rational. Is the total area of the reminder a real number?

author: Nominal Animal
author_rank: Super Contributor
timestamp: December 21, 2022, 11:16:15 pm
content: This will sound like nit-picking, but isn't: I'm focusing on the underlying concept, and there is an important concept-level distinction here.I'm trying to say that there are fundamental concepts related to "number" at play here: the number itself, itsrepresentation, and itsapproximation. For many irrational numbers, we canrepresentthem exactly in many types of computations on a computer, but can onlyapproximatethem in output useful for us humans.Quote from: CatalinaWOW on December 21, 2022, 10:02:25 pmNot exactly correct.We can correctly represent many irrational numbers. For example, \$\sqrt{2}\$ as \$2^0.5\$. To convert to a decimal or binary form, we can only show an approximation, but internal calculations can be done exactly, using this and various other representations.(Background: I have used this exact trick for regular cubic (SC, FCC, and BCC) lattice exact distance histograms. The distances along each coordinate axis are always integers (or halves), so the distance is an expression of form \$d = \sqrt{\sum_{k=1}^N x_k^2}\$ (possibly multiplied by \$2^{-N}\$). Because the distance is nonnegative, we can do all computation in the squared domain, i.e. using \$s = d^2 = \sum_{k=1}^N x_k^2\$, and only use the approximations in the labels when displaying the data. Essentially, my histogram had variable-width bins, but each such bin was exact. Thus, the histogram itself was exact, and only needed to be approximated when output in a form useful to us humans.)Similarly, arithmetic infinities can easily be represented; even standard IEEE 754 floating-point types do so correctly (in the arithmetic domain).Quote from: CatalinaWOW on December 21, 2022, 10:02:25 pmThe expressions and concepts that involve irrational numbers can be adequatelyapproximated, yes. But, we can also compute many of them in a closed form (including in other arithmetic expressions) that allows us to calculate the approximation to whatever precision we like, only limited by available memory and processing time. \$\pi\$ is a perfect example of this, especially if you want to calculate its representation in binary or hexadecimal instead of decimal, because theBailey–Borwein–Plouffe formulaexists, making it trivial to calculate thek'th hexadecimal digit of \$\pi\$ without having to know any of the preceding or succeeding digits.This does not mean we cannot use entities thatrepresentthe irrational number exactly in computations. Like I explained for square root of two above, there are ways of representing many irrational numbers –– perhaps not all, like some of the physical constants that might turn out to be irrational; consider thefine structure constantfor example –– using alternate, non-binary/non-scalar/non-rational representations in computations; and only need to be approximated for display or final result, for human use.Quote from: TimFox on December 21, 2022, 10:11:17 pm(It is extra funny when one happens to know that you only need 206 bits to express the size of the observable universe in units of Planck length. If the lines were half a meter apart, there would be fewer than 2³² atoms in the lattice between the marks, so you might be able to store the first four ASCII characters that way. Apologies to TimFox for me trying to explain the joke..)Quote from: magic on December 21, 2022, 10:54:14 pmNope, Iproved it for myselfdeduced the logical necessity of it being so for myself, from simple principles.Quote from: magic on December 21, 2022, 10:54:14 pmNot really. I've seen variants of "Have you stopped beating your wife yet?" that were better posed than any of your questions are. If you include an error in your question, you cannot expect correct answers to conform to your erroneous logic/understanding that lead to posing the question.Quote from: magic on December 21, 2022, 10:54:14 pmAgain, a question with an erroneous concept at its core. By definition, the area of a point is zero. So, no matter how countably many points you remove, the area does not change.

author: T3sl4co1l
author_rank: Super Contributor
timestamp: December 21, 2022, 11:16:47 pm
content: Quote from: CatalinaWOW on December 21, 2022, 10:02:25 pmWell, numerically perhaps. They can be represented symbolically, which is as good as anything when you get into higher level forms.To wit: irrational numbers are constructed by taking the rationals and adding an operation (a root, for example). This makes a set of numbers adjoined with the new operation. Then adjoin again so you can have roots of roots, plus offsets inside and outside each root. And so on and so forth, building whatever kind of number you like. And finally add the irrationals that don't have any radical representation at all (just straight up roots of algebraic equations; I forget how these are supposed to be otherwise constructed or ordered?). Rather than a numeric approach, you might represent these numbers as a tree (the sequence of operations starting from rationals on the leaf nodes), or as the equations they are the roots of. In any case, they don't take any more storage space than any other number, given that it takes countably infinite storage to represent almost any random element from that infinite set."Real" numbers are the standout here, with absolutely no representation other than the (transcendental, if not algebraic) equations that represent not even a minuscule fraction of them -- indeed that's only the computable reals. There's an immesurable infinity of them which are so thoroughly and utterly unknowable as the proverbial elder gods. We can only infer their existence from the nature of the continuum, minus the few holes we can poke in it with these mere countable sets.And of the computables, there's no need for any of them to have any relationship to anything else; numbers like pi and e are just about unique in their promiscuity, that they show up all over the place. But you can also define numbers that have seemingly no possible meaning in relation to anything else (say, take an arbitrary random sequence of digits, as generated by some certain process so we aren't subject to the limitations of rationals; say, the concatenated hashes of all primes?), and, there's nothing insightful about that, it doesn't connect to anything, it's not like you'll have a billion ways to compute it like you do the digits of pi.So, imagine that, but not even any way to even begin to identify the number, to compute the digits; that's the true unknowableness of the reals.So, "imaginary" set is only the second worst named, among the most common sets of numbers.Tim

author: TimFox
author_rank: Super Contributor
timestamp: December 21, 2022, 11:17:49 pm
content: A simplified discussion of real numbers that starts by ignoring their formal definition:https://abstractmath.org/MM/MMRealNumbers.htmThe important bits are found under "Properties of the Real Numbers", including mathematical concepts such as "closure" and "closed under limits".An important mathematical concept that is not taught at the elementary level is "limit theory", which is the basis for much of the relevant discussion here.The formal definitions can be found in the Wikipedia articlehttps://en.wikipedia.org/wiki/Real_numberin the section "Formal definitions".Note that the concept of "icky" is not found in standard references.The mathematical term "real" is historical yet universal among educated people.It is not the same as the English word "real" used in politics, history, or other non-mathematical contexts.

author: SiliconWizard
author_rank: Super Contributor
timestamp: December 21, 2022, 11:22:57 pm
content: So it looks like we're going over it all over again - as some have already pointed out - with some (that I remember at least) new concepts introduced about decolonization and some other woke stuff. So it's getting even better than the first time. It's like fine wine. Hats off to Peter for that.

author: tggzzz
author_rank: Super Contributor
timestamp: December 21, 2022, 11:40:29 pm
content: I think your question is easier to understand and answer appropriately if you annotated it as shown below.Quote from: Peter Taylor on December 21, 2022, 03:43:02 am

author: TimFox
author_rank: Super Contributor
timestamp: December 21, 2022, 11:45:03 pm
content: To which, I reply "says who?".

author: bdunham7
author_rank: Super Contributor
timestamp: December 21, 2022, 11:49:04 pm
content: Quote from: TimFox on December 21, 2022, 11:45:03 pmOur future dystopian leaders who will insist that all knowledge and arts should be dumbed down to the lowest common denominator so that they don't have to feel inadequate because some things are beyond their grasp.

author: Nominal Animal
author_rank: Super Contributor
timestamp: December 21, 2022, 11:54:07 pm
content: Quote from: SiliconWizard on December 21, 2022, 11:22:57 pmYou're right. And I'm sorry for adding to the flames. Apologies; I should know better.I just get so darned excited when people talk about stuff that I've found can be effectively used as a tool for both real world work as well as for investigation and learning and science, since I want them to find that utility too, and build stuff I myself never could...When it comes to learning, I'm an incorrigible romantic optimist.

author: free_electron
author_rank: Super Contributor
timestamp: December 22, 2022, 12:24:38 am
content: non-binary binary numbers ? next we'll have transgender gender changers...

author: Alex Eisenhut
author_rank: Super Contributor
timestamp: December 22, 2022, 12:27:27 am
content: in da house

author: free_electron
author_rank: Super Contributor
timestamp: December 22, 2022, 12:43:07 am
content: -wrong video-... look for Russel peters, invention of zero

author: SiliconWizard
author_rank: Super Contributor
timestamp: December 22, 2022, 12:47:32 am
content: Quote from: Nominal Animal on December 21, 2022, 11:54:07 pmWell, can't blame. It is interesting per se and I happen to be re-diving into advanced algebra and calculus lately, great stuff. Saddening to see all this turning into a political matter, but I guess it's the sign of the times, as they say.

author: CatalinaWOW
author_rank: Super Contributor
timestamp: December 22, 2022, 01:03:57 am
content: Quote from: Nominal Animal on December 21, 2022, 11:16:15 pmSorry about not taking the time to dis-assemble this quote to just the points I disagree with you, but here goes. I completely agree that these more exotic types can be represented symbolically. And many operations can be performed correctly and exactly on these representations. Maple and other symbolic operation systems do this. But both inputs and outputs are symbols and not an exact representation of the number. And writing an expression root(2) exactly and correctly defines one of these numbers. But if you generated a histogram with this expression what happened in the machine was that an approximation was created and compared with the data. Unless the data was all symbolically generated.The fact remains that the only numbers that computers can deal with are the rationals, and only the subset of the rationals which is within the numerical representation limits of the machine and language used. Of course those limits are fairly broad, as the specialized software for computing very extended approximations of pi, and those searching for prime numbers demonstrate.

author: IanB
author_rank: Super Contributor
timestamp: December 22, 2022, 02:24:06 am
content: The reason I am skipping over this thread and not reading it in detail, is that the whole premise is a fool's errand.Mathematics is agame, or a collection of games. For each game there arerulesyou must follow, and for the game to be interesting the rules must be satisfying, and lead to internally consistent logic and conclusions without contradiction. Games with rules that don't meet this criterion are not interesting and are discarded.An example of interesting rules is for Euclidean geometry, where one of the rules is that the surface we draw on is flat and not curved. There are other games where the surface may be spherical or hyperbolic, and these games are also interesting, but they are different games.If you want to invent new mathematics that other people can engage in, you need to come up with a set of rules that leads to an interesting game. Otherwise, your new mathematics will not find a following, and you will be doomed to be the only practitioner. Which could be fine for you, but don't expect others to be interested.

author: magic
author_rank: Super Contributor
timestamp: December 22, 2022, 08:46:09 am
content: Quote from: Nominal Animal on December 21, 2022, 11:16:15 pm"Your question is trivial and the answer is one. Therefore your question is ill-posed and I refuse to answer your other questions".And again, the reason I ask about a lot of stupid things is not because I expect a sensible answer.It's to show that you don't know why they are stupid and can't prove them stupid.You could equally well say they are stupid because some heterosexual dude told you so, which is apparently how good chunk of population perceives mathematics.I'm not even surprised

author: ledtester
author_rank: Super Contributor
timestamp: December 22, 2022, 09:22:04 am
content: Quote from: magic on December 21, 2022, 10:54:14 pmWhat you are describing is I x I where I is the set irrational numbers between 0 and 1. The unit square is [0,1] x [0,1].According to standard measure theory, the set I is measurable and has measure 1. The cross product, therefore, also is measurable and has measure 1*1 = 1.

author: TimFox
author_rank: Super Contributor
timestamp: December 22, 2022, 04:52:21 pm
content: As stated above, the only numbers that can be used in digital computers are rationals, which are treated as binary integers.All digital computation is binary arithmetic, a small subset of the broad universe of discourse in mathematics.Progress in computation theory and practice is mainly development of algorithms to improve the efficiency of the arithmetic.When I started grad school in physics, living in a dorm, one guy on my floor was a mathematics student who disliked the University's placement of the Department of Mathematics in the Division of Physical Sciences, instead of in the humanities where he thought it belonged.That was a defendable statement; I asked him what he was studying, and he stated topology and differential geometry, and he was proud that neither subject was of practical use.I had to point out that physics used both: topology in Feynman diagrams and differential geometry in General Relativity.The word "stupid" has been flung about in this discussion.Ignorance and stupidity are different concepts: we are all ignorant, since there is far too much for one person to know; stupidity is pride in ones ignorance.

author: Tomorokoshi
author_rank: Super Contributor
timestamp: December 22, 2022, 04:55:06 pm
content: Quote from: TimFox on December 22, 2022, 04:52:21 pmUnless one is using an IBM 650!

author: TimFox
author_rank: Super Contributor
timestamp: December 22, 2022, 05:02:22 pm
content: Quote from: Tomorokoshi on December 22, 2022, 04:55:06 pmThe first computer I used (ca. 1968) was an IBM 1620, which, along with the 650, used BCD (binary-coded decimal) internally.That is another type of binary arithmetic, with slightly different rules.

author: Nominal Animal
author_rank: Super Contributor
timestamp: December 22, 2022, 05:54:56 pm
content: Quote from: CatalinaWOW on December 22, 2022, 01:03:57 amIt was all symbolically generated, yes. That's what "representation" as opposed to "approximation" means, at its core.Quote from: CatalinaWOW on December 22, 2022, 01:03:57 amNo, that's my point. Any numerical object you can represent in arithmetic operations, you can use in a computer.For example, consider the case of representing numbers using format \$S\cdot M^{X/F}\$, where \$S = \pm 1\$, \$M\$ and \$F\$ are unsigned integers, and \$X\$ is a signed integer. You can represent many irrational numbers exactly with this form, although the algebraic operations you can do are a bit limited (only multiplication and division being trivial and exact). An approximation is only needed when e.g. a human wants to see what the number is in decimal.Almost all the computer hardware we use now does use integers, fixed-point integers (\$M/B\$, with \$M\$ a signed integer and \$B\$ an unsigned integer constant), and floating-point numbers (\$M\cdot B^X\$, with \$M\$ a signed integer, \$B\$ an unsigned integer constant most typically 2, and \$X\$ a signed integer). These are all very efficient and effectivee representations for our typical needs –– that's why they're popular now, instead of the other possibilities! ––, but they are not the only ones, and it is a grave error to think of them as somehow a "requirement" or "definition".

author: TimFox
author_rank: Super Contributor
timestamp: December 22, 2022, 08:13:50 pm
content: A late reply to Nominal Animal's post above about my parable:"(It is extra funny when one happens to know that you only need 206 bits to express the size of the observable universe in units of Planck length. If the lines were half a meter apart, there would be fewer than 2³² atoms in the lattice between the marks, so you might be able to store the first four ASCII characters that way. Apologies to TimFox for me trying to explain the joke..)"Hamlet: "There are more things in heaven and earth, Horatio, / Than are dreamt of in your philosophy."There is a whole lot more data extant than the extent of the universe. There are unreliable estimates of the data content of the holdings (digitized or not) of the Library of Congress, but they range up to the petabytes, or roughly 253bits.

author: SiliconWizard
author_rank: Super Contributor
timestamp: December 22, 2022, 11:13:04 pm
content: The Hamlet quote is great.

author: Nominal Animal
author_rank: Super Contributor
timestamp: December 23, 2022, 12:51:46 am
content: Quote from: TimFox on December 22, 2022, 08:13:50 pmYup, I only meant to illustrate how little data one can encode in a single length.If I try to consider how much information the universe contains, I get genuinely scared/overwhelmed.For illustration, consider a cubic meter of iron, about 7874 kg, containing about 8.5×1028≃ 296iron atoms. Iron has four stable isotopes, so in theory we could encode two bits per atom just by selecting the isotope, so that cubic meter of iron would then encode 297bits of information.How much is that? Well, if you unraveled that cubic meter into a monomolecular thread, just one atom wide, and you could read it at light speed, your bandwidth would be about 262bits per second, some 500 Libraries of Congress per second. It would take about 1088 years to read the entire iron cube.About 30% of Earth is iron, or about 269cubic meters of it. That's just on Earth, mind you. Since iron is the end product of both fusion and fission chains, there is a lot of it in the universe. And this is just a single information storage method with zero compression, just picking between four well-known stable iron isotopes. There are much more dense methods; I just picked iron because it is utterly stable, easy to grasp, and Stephen Baxter wrote a story about it.

author: SiliconWizard
author_rank: Super Contributor
timestamp: December 23, 2022, 01:04:04 am
content: I know I shouldn't add anything more here and it's not unlikely that the OP is just yanking our chain. With that said...Universe is thought to be infinite, although we don't know that for sure, nor what this would really mean. But until we can prove it finite, it looks useless to me to try and estimate how many "elementary" particles it contains. If we have even an idea of the smallest ones, which we probably don't yet.Secondly, to me the only real crucial point about "numbers" is whether, precisely, you admit infinity or not. (Which admittedly can itself be a philosophical matter since we have no means of grasping the reality of it.)Once you admit infinity, pretty much any number can be derived from that. If you have integers, basic arithmetic operations and infinity, then you have rational numbers, irrational numbers (which can be expressed as infinite sums of rational numbers), complex numbers, which are just an extension of that all, and higher-dimension numbers, and so on. Just a thought. So do you admit infinity?

author: EPAIII
author_rank: Super Contributor
timestamp: December 23, 2022, 08:03:36 am
content: Yea, yea, yea! And the earth is flat. And the moon landings never happened. And ...Quote from: Peter Taylor on December 21, 2022, 03:43:02 am

author: SiliconWizard
author_rank: Super Contributor
timestamp: December 25, 2022, 08:27:36 pm
content: Some food for thought for anyone interested. (Some people call that education, but maybe saying this is oppressive now, so I dunno.)

author: Circlotron
author_rank: Super Contributor
timestamp: December 25, 2022, 08:54:04 pm
content: Quote from: Nominal Animal on December 23, 2022, 12:51:46 amLooks like Big Iron, the euphemism for large scale computer hardware, has been redefined.