Integers, Pi, and Number Lines. - Page 1
Forum section: electronics / general

author: Peter Taylor
author_rank: Regular Contributor
timestamp: June 29, 2022, 11:01:29 pm
content: In classic education, we are taught that a number line contains an infinite number of irrational numbers between each integer. I will show that it should contain only integers.dy / dx is never evaluated because dx is infinitely small and zero.When working with Fourier Transforms for example, the terms are introduced into a formulae as a pair to help solve it, and then disappear at the end without ever being evaluated.The imaginary number "square root -1" is never evaluated, but introduced into a formulae at the start to help solve it, and disappears at the end.Pi also is not evaluated, but remains the ratio of the circumference of a circle to its radius."Square root 2" likewise is not evaluated, but remains a question: "What value multiplied by itself equals 2 ?".These are termed irrational "numbers", having no exact value.But these are not numbers, or values, but questions, or formulae.If we remove all these non-numbers, or questions, or ratios, or any unevaluated formulae from our number line, we are left only with integer numbers.So, a number line should contain only integer numbers.QED.

author: aeberbach
author_rank: Regular Contributor
timestamp: June 29, 2022, 11:46:27 pm
content: Similarly you can take an opaque tube and put a ball bearing in it, and seal the ends. Then because you can no longer know the exact position of the bearing, it is no longer there! (Ignore that rattling noise.)

author: John B
author_rank: Frequent Contributor
timestamp: June 29, 2022, 11:49:24 pm
content: Quote from: Peter Taylor on June 29, 2022, 11:01:29 pmBut they do have exact values, or positions on the number line (complex numbers excluded), just they can't be expressed as the ratio of 2 integers.

author: RJSV
author_rank: Super Contributor
timestamp: June 30, 2022, 01:55:50 am
content: Actually, 'Infinitely small' and 'zero' are not the same.In CALCULUS mode, you need a really good school instructor. You see; Calculus being a methodology, a new method of approach to arriving at usable numeric result. It's NOT instinctual, and so it seems impractical. For example, the mind-excercise that says things like; "dx goes to zero, as you repeat forever..."But that is only one fragment. With cleverness, it can be seen, that as that 'dx' portion moves a certain way, that another 'fragment', 'dy' also, in turn, is moving some characteristic way.It's almost as nebulous a concept as saying, for example, one WIZARD shrinks (you) to half size, and then another WIZARD makes you 4X bigger, with end result; you are twice original size. The little, localized exchanges make no real sense, but combined the two actions DO make a nice, clean result.A lackluster, dopey but competent MATH teacher might not be able to teach this essence.Now you've got me wondering about Issac Newton and his thoughts on inventing The Calculus...

author: Someone
author_rank: Super Contributor
timestamp: June 30, 2022, 02:02:39 am
content: Some nonsense to reply to the nonsense:Quote from: Peter Taylor on June 29, 2022, 11:01:29 pmOnly because people tend to think in units of integers, other problems are thought of in other units:https://www.explainxkcd.com/wiki/index.php/1292:_Pi_vs._Tauhttps://en.wikipedia.org/wiki/Radian_per_secondhttps://en.wikipedia.org/wiki/Normalized_frequency_(signal_processing)Their ranges overlap on the continum of scalar values with integers, but are equally "nice/round" as integers. Engineering/physics is all about the units you choose to define the problem.

author: SiliconWizard
author_rank: Super Contributor
timestamp: June 30, 2022, 03:15:51 am
content: Quote from: Peter Taylor on June 29, 2022, 11:01:29 pmWhy did you leave out the rational numbers though? They must be feeling lonely now.Quote from: Peter Taylor on June 29, 2022, 11:01:29 pmIrrational numbers can only be expressed indirectly indeed. Why would that mean they are not numbers? What's your definition of a number? Do you include infinity in your imaginary "number line"? What is infinity anyway?And even rational numbers can only be expressed indirectly, if all you have are integers. I guess you don't see them as numbers either?What are really integers anyway?Isn't it time for a beer?

author: ledtester
author_rank: Super Contributor
timestamp: June 30, 2022, 03:38:19 am
content: Quote from: Peter Taylor on June 29, 2022, 11:01:29 pmNot every real number is describable. Even after you remove all describable numbers you are still left with an uncountably infinite number of numbers.Here's a relevant blog post:http://www.goodmath.org/blog/2014/05/26/you-cant-even-describe-most-numbers/and here's one by someone who quit studying EE and instead turned to math when he came upon this revelation:https://blog.ram.rachum.com/post/54747783932/indescribable-numbers-the-theorem-that-made-me

author: hamster_nz
author_rank: Super Contributor
timestamp: June 30, 2022, 03:47:50 am
content: But if you view Pi as being a number line wrapped around a unit circle, then you can quite easily have Pi as "half way around"... and 2 Pi will take you back to where you started from.Or are circles just "questions, or formulae" and need to be removed?

author: Someone
author_rank: Super Contributor
timestamp: June 30, 2022, 07:34:31 am
content: Quote from: hamster_nz on June 30, 2022, 03:47:50 amNah nah nah that's old 2d thinking, you need to start with spheres!

author: newbrain
author_rank: Super Contributor
timestamp: June 30, 2022, 11:34:18 am
content: Quote from: ledtester on June 30, 2022, 03:38:19 amIn addition to the links from ledtester, here's a simple video from Numberphile with Matt Parker:

author: snarkysparky
author_rank: Frequent Contributor
timestamp: June 30, 2022, 12:06:41 pm
content: pi has an exact value. No one will ever know what it is though.

author: TimFox
author_rank: Super Contributor
timestamp: June 30, 2022, 01:52:16 pm
content: Remember: "irrational" in this context means that it is not a ratio of integers, it does not imply insane or other English meanings of "irrational".

author: xrunner
author_rank: Super Contributor
timestamp: June 30, 2022, 02:10:58 pm
content: Quote from: TimFox on June 30, 2022, 01:52:16 pmAnd imaginary numbers are not "imaginary".

author: kosine
author_rank: Regular Contributor
timestamp: June 30, 2022, 03:40:39 pm
content: The ancient Greeks referred to "irrational" numbers as "alogos" (although other terms were used). A good translation would be "inexpressible" in terms of whole numbers. Many such quantities can be constructed geometrically, however, and were then called "magnitudes" to distinguish them from "arithmos" (numbers).Gauss thought imaginary numbers should have been termed "lateral", with positive numbers being called "direct" and negative numbers "inverse".

author: Nominal Animal
author_rank: Super Contributor
timestamp: June 30, 2022, 06:55:54 pm
content: Even though $$\varphi = \frac{1 + \sqrt{5}}{2}, \quad \psi = \frac{1 - \sqrt{5}}{2}$$ are irrational numbers, I can calculate the number \$F_n\$,$$F_n = \frac{\varphi^n - \psi^n}{\varphi - \psi} = \frac{\varphi^n - \psi^n}{\sqrt{5}}$$for any positive integer \$n\$ without ever calculating any kind of an approximation for \$\varphi\$ or \$\psi\$.I bet you can, too, because the sequence \$F_0=0, F_1=1, F_2=1, F_3=2, F_4=5, \dots, F_n = F_{n-1} + F_{n-2}\$ is the sequence ofFibonacci numbers,OEIS A000045.Thus, while you cannot specify the value of irrational numbers exactly, you can specify their behaviour and use, which makes them exceedingly useful as numbers. Many other irrational numbers (like \$\pi\$, \$e\$ for example) have similar use cases.Let's say we have a continuous function \$f(x)\$.The slope of this function is \$f^\prime(x)\$:$$f^\prime(x) = \frac{d ~ f(x)}{d ~ x} = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$$The right side does not mean that \$\frac{d f(x)}{d x} = \frac{f(x + h) - f(x)}{h}\$; you can't just ignore thelimit operator\$\lim\$ like that!What that right side actually means (being the definition of aderivative), is that the value of the expression at \$x\$ is determined by thebehaviourof the ratio between the change in \$f\$ and the change in \$x\$, as the change in \$x\$ tends towards zero.That is, we never, ever try to divide anything by zero there. Instead, we can think of the process as looking at how the ratio changes as \$h\$ gets closer to zero, andextrapolatethe value the ratio would be if we could calculate it at \$h = 0\$.The power ofcalculusis that instead of having to do that rigmarole for every value of \$x\$ we're interested in – especially considering how careful one would need to be and how much work it would be to obtain the correct ratio! –, calculus tells us directly how we can obtain the slope as a function, \$f^\prime(x)\$, from the mathematical expression of \$f(x)\$ itself, with very little work.For example, if we have $$f(x) = 3 \sin(x) + 2 \cos(3 x^2 - x + 2)$$calculus tells us that $$f^\prime(x) = \frac{d f(x)}{d x} = 3 \cos(x) + (2 - 12 x)\sin\left(3 x^2 - x + 2\right)$$In short, because the limit operator tells us the behaviour of a function at a value where the function itself is not defined, there is absolutely no "division by zero". And it is a powerful operator, because even in cases where we have something like \$\lim_{x \to c} \frac{f(x)}{g(x)}\$ with \$f(c) = g(c) = 0\$ (or \$\pm\infty\$) and both \$f(x)\$ and \$g(x)\$ continuous (or differentiable) around \$x = c\$ but perhaps not at exactly \$x = c\$, it can still yield the correct answer (vial'Hôpital's rule).Simplifying all this hard work done by mathematicians over the centuries by claimingmy number line is differentis, well, pretty arrogant. Does your number line work as well as the one we use right now does?

author: mathsquid
author_rank: Frequent Contributor
timestamp: June 30, 2022, 08:40:22 pm
content: None of this makes any sense.

author: tggzzz
author_rank: Super Contributor
timestamp: June 30, 2022, 08:58:40 pm
content: Quote from: mathsquid on June 30, 2022, 08:40:22 pmKAll will become clear if you look at the other topic starter emitted by the OP.TL:DR your statement is correct.

author: TimFox
author_rank: Super Contributor
timestamp: June 30, 2022, 09:00:48 pm
content: 19th century mathematics is something that happened to other people.

author: eugene
author_rank: Frequent Contributor
timestamp: June 30, 2022, 09:09:17 pm
content: Quote from: Peter Taylor on June 29, 2022, 11:01:29 pmWhen I was taught about the number line in grade school we also learned about different number systems. Specifically real numbers, integers, and so-called natural numbers. Do recall that part of your classical education?QuoteNeither dy or dx are ever zero, but they do approach zero. In classical education the concept is introduced in the form of a limit.QuoteI don't know what you mean by that. Nothing disappears without being evaluated. I assure you that Mr. Fourier was not a fool.QuoteAgain, I don't understand what you mean. If you're working with mathematical statements and surprised that terms have disappeared, then either you lack a basic understanding of the math or you're doing it wrong.QuoteI evaluate pi regularly.QuoteLikewise, I evaluate sqrt(2) regularly. The fact that I can't write it down with a finite number of decimal digits is nothing more than a minor inconvenience.QuoteYes they do! Who told you they don't have exact values? Don't trust that person to teach you math. They must be incurably stupid!QuoteNo, they're numbers, just like 1 or 10.25. Again, don't trust the person that told you they aren't.QuoteIf we remove those numbers from the set of real numbers then we'll have a different set of numbers. You can ponder that set all you want, but I don't see how it will be useful to me. I need all the reals to get much of my work done.QuoteNow we're back to the elementary school concepts of real numbers, integers, and natural numbers. Nobody is stopping you from working with only the set of integers. If you want to talk about it with someone you might find a third grade math teacher. That's where I learned this stuff.EDIT: the campaign to stop people from driving while drunk is for the safety of others. Posting while high doesn't present a safety problem, but it does make you look like a fool.So don't post while high.

author: SiliconWizard
author_rank: Super Contributor
timestamp: June 30, 2022, 09:10:05 pm
content: When you see that:https://equitablemath.org/everything becomes possible.

author: jpanhalt
author_rank: Super Contributor
timestamp: June 30, 2022, 09:19:39 pm
content: @SiliconWizard"DEI" is sweeping the known universe. Thanks for the link.I don't remember where I saw this first, but it certainly seems consistent with that scenario:

author: Nominal Animal
author_rank: Super Contributor
timestamp: June 30, 2022, 10:01:15 pm
content: Quote from: mathsquid on June 30, 2022, 08:40:22 pmYou! You.. ALT-numeric bigot, you!Oops, sorry, my new-speak translator was accidentally left on. What I was actually trying to say was, I think these threads are further proof of how little humans make sense in general... Even in universities, the weight has shifted to the first syllable,uni. All Shall Think And Speak As One, with as little information content as possible, so as to not offend anyones sensibilities.

author: RJSV
author_rank: Super Contributor
timestamp: June 30, 2022, 10:15:17 pm
content: Math Squid;It's sometimes hard to make sense, but Nominal Animal has best grasp of how 2 or more 'weird' things can still be resolved, if you can somehow postpone doing exact calc on the individuals, and, instead focus on the relationship between the two:Maybe try 3x Pi / 4x Pi. Quick glance, and just CANCEL the weirdos, resulting in concrete answer, without having to solve EVERY individual term.

author: RJSV
author_rank: Super Contributor
timestamp: June 30, 2022, 10:20:57 pm
content: Maybe could install a 'CONFUSED' icon, along with 'thanks' and 'reporte de la Moderatique'!'Je suis Reporte de dis-raespect la familla'(He takin bout your MAMA).

author: tooki
author_rank: Super Contributor
timestamp: July 01, 2022, 07:35:01 pm
content: Quote from: Peter Taylor on June 29, 2022, 11:01:29 pmHaving yourother threadabout this closed by a mod wasn’t enough, eh?

author: tom66
author_rank: Super Contributor
timestamp: July 01, 2022, 08:57:31 pm
content: A number being irrational does not make it non-existent. It simply means no fraction exists to represent it. Do you claim that Pi can be represented with a fraction precisely?If so, you either have the greatest mathematical breakthrough known to man (and can surely supply proof upon request) -- or you're wrong.I'm betting the latter is true, but please, FEEL FREE to show us your Pi fraction. Be aware that Pi has been calculated to over 100 trillion digits now, and therefore any such fraction is liable to be rather large.

author: magic
author_rank: Super Contributor
timestamp: July 01, 2022, 09:13:07 pm
content: Quote from: SiliconWizard on June 30, 2022, 09:10:05 pm50% of that stuff is actually common sense if you replace "racism" with "poverty", but it's written like that because the latter wouldn't sell nearly as well in America's egalitarian fundamentalist environment.50% is promoting the mental gymnastics concept of "PoC" which enables them to excuse dumbasses with the fact that some tribe in some remote forest did something differently and it simply is unreasonable to expect "PoC" to learn things the Eurasian way (which they call "European" and "white" due to their own internalized white supremacy bias).50% is ignoring the reality that a teacher's first job in a pathological school is keeping the kids from putting a trash can on his head and maths always comes second to that.Case in point, right from the 1st PDF:QuoteYep, "don't worry, base 10 is just a special case of base N"

author: Peter Taylor
author_rank: Regular Contributor
timestamp: July 04, 2022, 01:19:13 pm
content: Quote from: Nominal Animal on June 30, 2022, 06:55:54 pm"Square root 5" is an unevaluated question: "What number multiplied by itself equals 5" ?So the result of your formulae is also unevaluated.An evaluated number is an integer. "Square root 5" is a question. It can't be evaluated as an integer.Its a question that has no answer because we can only know integers.

author: tooki
author_rank: Super Contributor
timestamp: July 04, 2022, 01:44:32 pm
content: Honey, you can’t just go around making up your own rules and definitions about mathematics.

author: tom66
author_rank: Super Contributor
timestamp: July 04, 2022, 02:01:22 pm
content: Saying sqrt(5) has no solution is like saying 1/5 has no solution because you need to evaluate the fraction to calculate the decimal value.

author: TimFox
author_rank: Super Contributor
timestamp: July 04, 2022, 02:09:00 pm
content: Also, 1/7 = 0.14285714... can't be expressed as a finite number of decimal digits. Big deal! Live with it!

author: ebastler
author_rank: Super Contributor
timestamp: July 04, 2022, 04:20:26 pm
content: You guys are operating with those absurd "fractions", which according to P. T. don't exist either...And I assume "zero" is an illusion as well, since how can "nothing" be a thing? And let's do away with the stupid "infinity" concept while we are at it. And don't get me started about "negative" numbers!

author: tggzzz
author_rank: Super Contributor
timestamp: July 04, 2022, 04:44:46 pm
content: Quote from: ebastler on July 04, 2022, 04:20:26 pmDon'tstopat infinity -startthere with the aleph numbers introduced by Georg Cantor.ℵ0(aleph-nought, also aleph-zero or aleph-null) is the cardinality of the set of all natural numbers, and is an infinite cardinal. A set has cardinality ℵ0if and only if it is countably infinite. FFI seehttps://en.wikipedia.org/wiki/Aleph_number

author: tggzzz
author_rank: Super Contributor
timestamp: July 04, 2022, 04:47:26 pm
content: Quote from: Peter Taylor on July 04, 2022, 01:19:13 pmClearly someone that can't understand thebasic premiseof algebra. Nuff said

author: tggzzz
author_rank: Super Contributor
timestamp: July 04, 2022, 04:48:20 pm
content: Quote from: tooki on July 04, 2022, 01:44:32 pmPoliticians and others looking to deceive can and do

author: Nominal Animal
author_rank: Super Contributor
timestamp: July 04, 2022, 06:47:21 pm
content: If you believe only integers exist, you're welcome to proverbially blow your mind by the fact that there exist several simple algorithms ofextracting digits of Pi–– in hexadecimal, base 16. For example,David H. Baileyand Helaman Ferguson discovered in 1997 a formula they called PSLQ:$$\pi = \sum_{k=0}^{\infty} \frac{1}{16^k} \left( \frac{4}{8 k + 1} - \frac{2}{8 k + 4} - \frac{1}{8 k + 5} - \frac{1}{8 k + 6} \right)$$where each term \$k\$ defines exactly one hexadecimal digit of \$\pi\$.In general, this subfield is described as"finding integer relations among more than two numbers".In the physical sense, more than 52 or so hex digits is unnecessary precision, because the diameter of the universe is about 5.4×1061Planck length units, and there is nothing physically larger nor more precisely measurable. For computation, especially for intermediate results, more precision is often useful, of course.The key realization is that just because you cannot write the exact value down in any base, does not mean it isn't a specific value: it is onlyunwritable, but not unexpressible. It is perfectly acceptable to define say \$x = \sqrt{5}\$, because there are many formulae where you only ever need \$x^2 = 5\$.A particular example is Euclidean distances. In \$N\$ dimensions, the distance \$d\$ from origin to point \$(x_1, x_2, \dots, x_{N-1}, x_N)\$ is \$d = \sqrt{x_1^2 + x_2^2 + \dots + x_{N-1}^2 + x_{N}^2} = \sqrt{\sum_{i=1}^{N} x_i^2}\$. In many subfields of physics, especially those dealing with lattices, we can determine certain useful properties of materials by just examining the various possible distances between lattice points. Numerically, it is preferable to do this using functions that returnthe distance squared, which in a lattice is an integer, because that way we avoid rounding errors. (Essentially, if you examine a \$(2 L)^N\$ cuboid of points, or rather an \$1/2^N\$ positive cuboid of that with \$L^N\$ lattice points, you get the lattice point distance distribution for distances not exceeding \$L\$, and the points above that distance are discarded. You save potentially a lot of time (when \$N\$ is large) by only doing the square root operation for display purposes, i.e. when showing the distance histogram.)Same applies to many cases in 3D computer graphics, especially raycasting and raytracing. For example, if you're interested in ray-cylinder (right circular cylinder) intersections, check out the formulae at myWikipedia user page(from 2010); in many cases the square root operation can be omitted. Note that these formulae are so simple only because the origin of the ray (ie. the focal point of the camera or the eye of the viewer) is at origin!Because square root is a monotonically increasing function for nonnegative distances, one can use az-buffercontaining squared distances instead of distances themselves, without any effect on the resulting visuals.

author: SiliconWizard
author_rank: Super Contributor
timestamp: July 04, 2022, 06:53:23 pm
content: Well, there is something philosophical (probably a bit too much) about the OP's questioning, and it's not completely irrelevant. It's been a very common consideration for centuries. It's not really anymore, but hey. Why not.The problem with the OP's approach is that it looks more like certainties than questioning.But on a deeper level, we can still fundamentally question whether the "universe" is really "continuous" or whether it's kind of discrete. It's all a complex matter. No pun intended. We can think of the Planck constant, and stuff. And if it's really discrete, all that would be needed would be integers. Somehow.But as I hinted earlier, as long as you consider "infinity" to be part of your "number line", then it opens a whole world of peculiarities that bridge the gap between rational and irrational numbers. So, does the OP consider infinity? But here again, since we still don't know for sure whether the universe is infinite or not, if we can't get infinitely small or infinitely big, then what is infinity really?While this can be all fun as a late-night, drunk discussion, mastering basic mathematics would still be a useful skill here.Beyond the lunacy, there is something in this topic that kind of reminds me of the topic about electrons.

author: magic
author_rank: Super Contributor
timestamp: July 04, 2022, 10:04:15 pm
content: Two pages and still no one has pointed out OP's fundamental error:Almost all real numbers cannot be described as a solution to any equation, and yet mathematicians will insist that they are there

author: nigelwright7557
author_rank: Frequent Contributor
timestamp: July 04, 2022, 10:26:36 pm
content: I have used Fourier transforms in a scope I designed.It simply outputs buckets of numbers representing frequency and amplitude.I didn't need to understand how it worked just what needed to be input and what the outputs meant.Interestingly when I looked online for FFT code the first three examples didn't work right.It was only fourth that gave good results.

author: ledtester
author_rank: Super Contributor
timestamp: July 04, 2022, 11:36:35 pm
content: Perhaps this is the angle the OP is trying to express:QuoteIndeed, you could even create the natural numbers from the basic axioms of set theory. Once you have the natural numbers you can create the integers, then the rationals, the reals, complex numbers, etc.From:https://www.cantorsparadise.com/kronecker-god-and-the-integers-28269735a638

author: Someone
author_rank: Super Contributor
timestamp: July 05, 2022, 12:18:00 am
content: Quote from: tooki on July 04, 2022, 01:44:32 pmSeem to work for people in other fields on this forum... "no no, what I define that word to mean is XXXX"

author: Nominal Animal
author_rank: Super Contributor
timestamp: July 05, 2022, 12:41:54 am
content: Quote from: Someone on July 05, 2022, 12:18:00 amYes, human communication is complex.When you want to discuss mathematics or physics, you need to agree upon the terms used to refer to things. That means there aretwolayers of definitions: there is the English (or human language) definition, and then there is the definition of the physical or mathematical theorem, quantity, object, or rule. It is the latter that you cannot make up as you go; and the former you are allowed to redefine as you wish, as long as you use specific, detailed, unambiguous human language to do so.For example, the metal called "Tungsten" in English, is "volframi" in Finnish. "Addition" in English is "yhteenlasku" in Finnish, and so on. No problem there.Trouble happens, when terms are redefined without expressing the definition in precise language. Graphics artists, for example, can have huge problem grasping a correct intuitive understanding about physical measurables likegravityandvolumeand especiallyweightversusmass, because they traditionally redefine these terms with their own definitions that are close enough to cause serious headaches, and have a hard time switching to a new set of definitions. (And vice versa, I guess!)This is also why you see myself mentioning things like"quaternions are multiplied using theHamilton product", because in the context of quaternions, multiplication is a very specific operation with very specific rules, even though it is just "multiplication". It isalwaysuseful to define the human-language terms one uses for mathematical and/or physical references, unless a single commonly accepted definition exists.The same goes with acronyms: the first layer is the words the acronym expands to –– SPI can refer to Serial Peripheral Interface or to Service Provider Interface in programming, for example –; and the second layer is the thing those words refer to. For example, if we agree that MOSI refers to Master Out, Slave In, we still need to be in agreement that Master is the one who provides the clock (SCL) line, and Slave is the other one. Again, the former can be redefined as needed, but the latter must not, or confusion results.

author: ledtester
author_rank: Super Contributor
timestamp: July 05, 2022, 02:22:30 am
content: Quote from: magic on July 04, 2022, 10:04:15 pmfor the record, I did point this out early on:https://www.eevblog.com/forum/chat/integers-pi-and-number-lines/msg4270255/#msg4270255

author: IanB
author_rank: Super Contributor
timestamp: July 05, 2022, 03:01:12 am
content: Quote from: Peter Taylor on June 29, 2022, 11:01:29 pmSince I don't think anyone has really pointed this out yet, I will do so:You talk about irrational numbers, but you overlook the rational numbers. If you remove all the irrational numbers from the number line, there is still an infinite number of rational numbers, that are not integers, left over.For example, 1/7 is a rational number that is not an integer. You may protest that is is not exact, and cannot be evaluated. But on the contrary, if we evaluate 1/7 in base 7 arithmetic we get 1/10 (base 7) = 0.1 (base 7) exactly. So 1/7 is clearly exact, and can be evaluated, and is not an integer. The same is true of all the other (non-integer) rational numbers.Furthermore, if you take any two non-integer, but rational, points on the number line, however close together, you can always find an exact, rational, number that lies between them. So you can zoom into the number line as much as you like, and still find numbers filling the space. Any time you think there might be a gap, you can find infinitely many exact numbers occurring inside that gap.

author: magic
author_rank: Super Contributor
timestamp: July 05, 2022, 08:27:52 am
content: Quote from: ledtester on July 05, 2022, 02:22:30 amWell,thenThis is the only thing that really matters. Irrational numbers are filler junk which is postulated to exist solely for the purpose of ensuring that every Cauchy sequence actually has a limit, because somebody foundthatto be convenient.All irrational numbers that one actually calculates with are simply solutions to some equations and they could be regarded as such without insisting that they are "numbers". Indeed, when you handle them, you don't process the numbers, you process the equations they are solution to and thus sqrt(3)·sqrt(5) becomes sqrt(15). And if you need a real-world actionable result, you just pick a nearby rational like 3.87298334620741688518.So OP isn't even far off the mark, but haters gonna hate.Quote from: nigelwright7557 on July 04, 2022, 10:26:36 pmYour FFT function only works with rational numbers, too.

author: ebastler
author_rank: Super Contributor
timestamp: July 05, 2022, 09:14:44 am
content: Quote from: magic on July 05, 2022, 08:27:52 amHmm, not sure which original post you read, but the one I see finishes with the following. Which I don't hate, but find rather dumb:Quote from: Peter Taylor on June 29, 2022, 11:01:29 pm

author: eugene
author_rank: Frequent Contributor
timestamp: July 05, 2022, 12:47:24 pm
content: Quote from: magic on July 04, 2022, 10:04:15 pmI propose that we stop using a base-10 number system and switch to one that is base-pi. Now, when we draw a number line '1' will land exactly where pi used to. Pi does have an exact value. We couldn't write it down before, but now we can. It's as simple as this: 1. Now all the mathematicians can stop worrying about whether or not pi is really "there" or if it's just a question that never gets evaluated. It really is "there."Of course the old decimal 1 is gone now. It's no longer "there" or describable as a solution to an equation. Now the old decimal 1 is just a question that never gets evaluated. Sucks if you want to buy some eggs. But still, I think I should win a prize in mathematics.

author: kjpye
author_rank: Contributor
timestamp: July 05, 2022, 12:49:49 pm
content: Actually, in a base-pi number system, pi would be represented as 10. Not quite as simple, but convenient.

author: Peter Taylor
author_rank: Regular Contributor
timestamp: July 05, 2022, 01:26:05 pm
content: Quote from: IanB on July 05, 2022, 03:01:12 amI'm cutting a length of pine to a rational length of 24.6 cm's, but I'm also cutting it to an integer length of 246 mm's.The rational number 0.142857 ... recurring may repeat infinitely but it is still an integer, 1 / 7.A "nose zing" equals 7 "whisker zings" and I want to cut my timber to a rational length of 3 / 7 th's of a "nose zing", but I'm also cutting it to an integer length of 3 "whisker zings".A rational number refers to how an integer is represented in our number system, not to the type of number it is.In LaLa Land, where they use a base 7 number system, 1 / 7 in our base 10 number system is an integer.My conjecture states that irrational numbers are not numbers, but questions, and that only integers are numbers.

author: TimFox
author_rank: Super Contributor
timestamp: July 05, 2022, 01:36:25 pm
content: Your use of "integer" in the above post is absurd.

author: tooki
author_rank: Super Contributor
timestamp: July 05, 2022, 01:38:53 pm
content: Quote from: Peter Taylor on July 05, 2022, 01:26:05 pmNo, that’s not an integer. 1, 2, 3, 4, 5, 6, 10, 11, 12, 13, 14, 15, 16, 20, … are the first 14 positive integers in base-7.7 isn’t even a digit in base-7, 0-6 are. (Like how base-10 is 0-9.)

author: TimFox
author_rank: Super Contributor
timestamp: July 05, 2022, 01:42:34 pm
content: Quote from: tooki on July 05, 2022, 01:38:53 pmIn base-7 land, 1/7 (decimal) would be 0.1 (a septimal fraction),not an integer.If he wants to make up new terms, he shouldn't re-define existing nouns such as "integer".

author: eugene
author_rank: Frequent Contributor
timestamp: July 05, 2022, 02:28:06 pm
content: I just experienced something that might be apropos. I had lunch at my favorite local restaurant. After enjoying the meal, the server came to the table and asked, "would you like desert?""Yes!" I replied. "I would like 1/4 pi radians of your delicious cherry pie!"To which the server replied, "I'm sorry sir. I cannot serve you 1/4 pi radians of pie. Pi is an irrational number, so I cannot measure any quantity of them exactly. I can serve you 1/8 of an entire pie. Would that be okay?""No, thanks anyway" I said.

author: xrunner
author_rank: Super Contributor
timestamp: July 05, 2022, 02:43:31 pm
content: Oh we're getting weird now, OK well ...I give somebody an exact diameter, say it's x. I ask them to draw a circle with a circumference of pi * x. I only want an exact circle with the exact circumference.Draw the circle on a piece of paper with a compass. Or you can use a computer with graphics if you want to. Is the circle you draw going to meet at the ends?If you can't calculate pi exactly then how can the circle even exist before your eyes, it shouldn't have been able to be drawn in this universe - but you will do it ...

author: Peter Taylor
author_rank: Regular Contributor
timestamp: July 05, 2022, 04:41:15 pm
content: An integer number (or just number), should be any value that is wholly defined.An irrational "number" is not a number, so there can be no rational "number".These terms hark back to the days of the Pythagoreans, when they thought the Earth was the centre of the Universe.

author: Peter Taylor
author_rank: Regular Contributor
timestamp: July 05, 2022, 04:53:39 pm
content: Quote from: xrunner on July 05, 2022, 02:43:31 pmA circle can be made to an exact (integer, or number) circumference. Cut a piece of string to that circumference, and make a circle.A circle can be made to an exact (integer, or number) radius. Cut a piece of string to that radius, and make a circle.What can't be done is find how long to cut a piece of string to that radius, given the length of string to make the circumference, because the ratio of the circumference of a circle to its radius is not a number, but a question.

author: eugene
author_rank: Frequent Contributor
timestamp: July 05, 2022, 04:55:59 pm
content: Quote from: Peter Taylor on July 05, 2022, 04:41:15 pmSo now's the time to get out the toga that's been in your closet!

author: TimFox
author_rank: Super Contributor
timestamp: July 05, 2022, 05:01:42 pm
content: Quote from: eugene on July 05, 2022, 04:55:59 pmMathematics has progressed past the time of Pythagoras, and the terms that offend the poster are now very well defined, and not in a circular method that boils down to "These concepts are icky."The current definitions of "natural numbers", "positive integers", "integers", "rational numbers", "irrational numbers", "transcendental numbers", etc. are useful in mathematics and consistently defined.Otherwise, one might as well ban "improper fractions" as being immoral.

author: eugene
author_rank: Frequent Contributor
timestamp: July 05, 2022, 05:07:11 pm
content: Quote from: eugene on July 05, 2022, 04:55:59 pmPeter, if you're frustrated that I'm just not taking you seriously, it's because I just can't take you seriously. So instead, I'm trying to find a good use for something that otherwise seems useless. I'm trying to find comedy in things that are otherwise not obviously comical.

author: Peter Taylor
author_rank: Regular Contributor
timestamp: July 05, 2022, 05:15:05 pm
content: Love u all.

author: xrunner
author_rank: Super Contributor
timestamp: July 05, 2022, 05:32:34 pm
content: Quote from: Peter Taylor on July 05, 2022, 04:53:39 pmBut I can draw a line from the circumference to the center of the circle I just made (I must know where the center is or I could not have made the circle with an exact piece of string. So all I have to do is measure the line I just drew, and that's the diameter; not a question anymore but an answer. How can that happen?

author: IanB
author_rank: Super Contributor
timestamp: July 05, 2022, 07:05:58 pm
content: Quote from: Peter Taylor on July 05, 2022, 04:53:39 pmAbsolutely it can be done. Turn a cylinder on the lathe to an exact diameter of 1 unit. Wrap a piece of cord snugly around the cylinder, and cut it where it overlaps, so that the two ends meet. You now have piece of cord exactly pi units long.

author: Vtile
author_rank: Super Contributor
timestamp: July 05, 2022, 07:36:10 pm
content: Quote from: Nominal Animal on June 30, 2022, 06:55:54 pmIIRC Newton had this concept of using something almost a zero (taste, feels and quaks like zero) as divisor when forming the concept and ideas of modern calculus. Much more intuitive, but unfortunately not mathematically rigorous.That was fixed and reformed much later with idea of limits. ....In sixties the duck did born again with hyper-reals.Quack quack.

author: Vtile
author_rank: Super Contributor
timestamp: July 05, 2022, 07:53:05 pm
content: Quote from: IanB on July 05, 2022, 07:05:58 pmOnly if the string has diameter of zero.

author: RJSV
author_rank: Super Contributor
timestamp: July 05, 2022, 08:18:21 pm
content: Everyone:EITHER Fly, or get out of the Toga Closet.(Just trying to fit in, by lookinstuupid)

author: Nominal Animal
author_rank: Super Contributor
timestamp: July 05, 2022, 08:47:01 pm
content: Quote from: Vtile on July 05, 2022, 07:36:10 pmThe way I intuitively grasped it in comprehensive school was by realizing that if one graphed the function$$g(h) = \frac{f(x+h) - f(x)}{h}$$(with a specific value of \$x\$, of course), then the limit \$h \to 0\$ is what \$g(0)\$ would be if it could be evaluated. Thus, extrapolation.If you start that by drawing a curve, picking a point on that curve, and asking the students how they'd find the tangent of that curve at that point, it's very easily graspable concept.It becomes even more so, if you tabulate or plot the graph \$g(h)\$ with a logarithmic \$h\$ axis (\$h = 0.1, 0.01, 0.001, 0.0001, 0.00001, \dots\$).Quote from: Vtile on July 05, 2022, 07:36:10 pmEww.Well, what else can you expect from mathematicians? I onlyuseapplythe stuff they come up with, to solve problems.

author: Vtile
author_rank: Super Contributor
timestamp: July 05, 2022, 10:03:08 pm
content: The duck:https://en.m.wikipedia.org/wiki/Infinitesimal

author: tggzzz
author_rank: Super Contributor
timestamp: July 05, 2022, 10:05:48 pm
content: Quote from: Peter Taylor on July 05, 2022, 05:15:05 pmI'm failing to distinguish you from a conventional troll.Am I missing something? If so, what?

author: tooki
author_rank: Super Contributor
timestamp: July 06, 2022, 05:34:08 am
content: Quote from: Peter Taylor on July 05, 2022, 04:41:15 pmYou don’t get to decide what you think the definition of a wordshouldmean. They have established, clearly defined meanings. An integer is a number that can be written without a fractional component.An irrational number most certainly is a number, as are non-integer rational numbers.You need to shut the fuck up and go back to middle school (it was it primary?) math, rather than trying to tell anyone anything.