Wikipedia:Reference desk/Archives/Mathematics/2018 March 3

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March 3

where do the PI, e, sqrt(2) go on the real number line

The following discussion is closed. Please do not modify it. Subsequent comments should be made on the appropriate discussion page. No further edits should be made to this discussion.

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Hi, https://en.wikipedia.org/wiki/Real_number On this Wiki page a claim is made that Pi , gamma, sqrt(2) or e are on the real number line. I took exception to it and asked the question below :

Assume that you are given a job of placing the algorithmic output(s) of PI on the number line. Where would you put it ? Somehow I ended up in the talk pages where I was put in my place politely but firmly and got sidetracked a bit.

I will ask this question from you, from YouTube users, from the MIT math department from the math geniuses at U of Tokyo until I get a satisfactory answer. Questioning my credentials does not qualify for an answer.

Thanks, Tamas Varhegyi If you could email to me to letting me know that you have responded that would be great.

Counting floats (talk) 02:30, 3 March 2018 (UTC)

Counting floats, the approximations of pi to a certain number of decimal places (namely 3, 3.1, 3.14, 3.141, and so on) are all separate real numbers and they are all different from pi. So, they are located at different places on the real number line. Sure, each approximation will be closer to pi than the previous one, but they are still separate points. Anon126 (notify me of responses! / talk / contribs) 03:00, 3 March 2018 (UTC)

I would like to answer this question but I'm afraid I don't really understand it. You would put it at position π. If that's not a satisfactory answer, what is missing from it? Where would you put, say, 3.1, and how is that different? --Trovatore (talk) 03:47, 3 March 2018 (UTC)

Pi, gamma, sqrt(2) and e are all real numbers. All real numbers are on the real number line, by definition. Bubba73 ^{You talkin' to me?} 03:56, 3 March 2018 (UTC)

Hi, Yeah, You are right on the money!!! No real number line can contain any algorithms only the float outputs of the algorithms, which in the case of PI is about 5 billion digits strong, in the case of 1/7 is endless. My point was this the number line depicted on the page shows PI, e, sqrt(2), gamma as part of the number line. That is not possible. Algorithms do supply valid floats to fill the number line ( with duplicates !!!! ) but they are not part of the real number line. I did not pull this out of thin air as it is the iron rule. One practical aspect of the no algorithms rule is that some of the procedures which define and execute algorithms consists of dozens of pages of computer code. Nobody would ever dream of putting that on any number line, and if they did they would have to rename it the GIANT BOOK OF ALGORITHMS LINE

The no-algorithms rule should be made unequivocally by the Wiki editor who created a page, or somebody who knows how to do it. (Not I !!! ) 1. Should remove pictures of PI, e, sqrt(2) and any algorithm even 1/3 and so on (yes 1/3 is an algorithm where it yields an endless stream of 0.3, 0.33, 0.333, ... 2.a Should state explicitly that the real number line contains nothing but integers (0-9, no leading zeros ) and 2.b Valid floating point numbers or what I call them floats ( digits 0-9 + decimal point in any valid sequence and mix)

Once again I really appreciate your response and understanding the issue. Good night,

Tamas Varhegyi

— User:Counting floats 04:41, 3 March 2018

(Copied above message from my talk page)

Of course the illustration cannot show the exact location of those numbers, because that would require an infinite amount of detail in the image, which is impossible. But the illustration is just that, an illustration of the real number line. The real number line does not exist as a physical object, but we can create a drawing that approximates it. This approximation is useful to show, for example, that pi and e are somewhere on the number line, that pi is between 3 and 4, that e is between 2 and 3, and so on. Anon126 (notify me of responses! / talk / contribs) 04:57, 3 March 2018 (UTC)

Additional response: Actually, one way to create the real number line is in fact a bit like a "giant book of algorithms" as you call it; refer to Construction of the real numbers#Construction from Cauchy sequences. These sequences of "floats" are called Cauchy sequences of rational numbers (all floats are rational numbers), and we can say that the "infinite result" of any such sequence (including 0.3, 0.33, 0.333, ... and 3.1, 3.14, 3.141, ...) is a real number (even if there is no algorithm that can generate it). Anon126 (notify me of responses! / talk / contribs) 05:08, 3 March 2018 (UTC)

Well, if you insist on "algorithms", then even that gets only the computable real numbers. Most real numbers (almost all of them, actually) are not computable. --Trovatore (talk) 05:10, 3 March 2018 (UTC)

Yes, true. I edited my response (strikes and italics). I'm still using the "giant book of algorithms" as a starting point, though. Anon126 (notify me of responses! / talk / contribs) 05:25, 3 March 2018 (UTC)

@Counting floats: Please do not conflate a real number with the set of sequences that converge to that real number, and also don't conflate floating point representations with real numbers in general. I would like to point you to the construction of the real numbers. The "real number line" is a geometric representation for the resulting topological field we call the real numbers. As such, even uncomputable real numbers (like Chaitin's constant) will by definition reside on the real number line. There's nothing in the definition of the real number line that requires a finite or repeating decimal representation, or even a computable one.--Jasper Deng (talk) 07:48, 3 March 2018 (UTC)

In other words, everything which is not an imaginary number is a real number. 92.19.174.150 (talk) 10:28, 3 March 2018 (UTC)

Well, no. Strawberry yogurt, for example, is not an imaginary number, but it is also not a real number. That might sound silly, but I'm making a serious point.

Another attempt that doesn't work is "a rational or irrational number", which is circular, because an irrational number is defined to be a real number that isn't rational.

Actually defining the real numbers is not trivial; it took concentrated effort on the part of some of the smartest mathematicians of the 19th century to get it right. --Trovatore (talk) 10:38, 3 March 2018 (UTC)

If a person knows the radius of a circle and is able to calculate the circumference or area of that circle to an acceptable accuracy, that person can locate pi on the number line. Conversely, if that person finds the irrationality of pi to be an insurmountable problem when trying to locate pi on the number line, then that person is not able to calculate the circumference or area of the circle to an accuracy that the person considers acceptable. Dolphin (t) 10:52, 3 March 2018 (UTC)

Counting floats, you may be interested in our article Constructible number. Sqrt(2) is constructible, though pi, gamma, and e are not. So it is true that the location on the real number line of those last three two cannot be determined using a compass and straightedge in a finite number of steps, but they are still real numbers and thus have a well defined location on the real number line. (You may also be interested in reading about transcendental numbers & algebraic numbers. Note that the constructible numbers are a proper subset of the algebraic numbers.)

Note that by excluding everything except those real numbers which may be represented as a terminating decimal, you are also excluding infinitely many constructible numbers whose decimal expansion does not terminate. A few examples of constructing numbers on the line are given in the § Comparing numbers of our simpler article Number line. While the sixth illustration demonstrates the construction of 3/2, which you already accept since it may be represented as a terminating decimal, a similar construction would yield 1/3. So your restriction is too strict for even the "constructible number line", not just the real number line. You are free to define the mathematical construct "terminating decimal line", and use it to you heart's content, but you are mistaken if you conflate it the the preexisting and well defined concept of the real number line. -- ToE 14:52, 3 March 2018 (UTC)

Yes, but gamma is not known to be irrational, let alone non-constructible.John Z (talk) 16:55, 4 March 2018 (UTC)

Cool! Thanks. -- ToE 17:48, 4 March 2018 (UTC)

@Counting floats:, you said "No real number line can contain any algorithms only the float outputs of the algorithms, ...". Real numbers are not algorithms. In some cases there are algorithms for computing them, but they are not algorithms (and almost all real numbers do not have an algorithm for computing them.) Bubba73 ^{You talkin' to me?} 00:16, 4 March 2018 (UTC)

@Counting floats:, since nobody else mentioned it, you seem to be talking about computable numbers, rather than real numbers. Computable numbers are ones which can be constructed by algorithms, so the "computable number line" is analogous to your "Giant Book of Algorithms". What we call the real numbers is something else, defined differently, and the definition means that there are many more real numbers than there are computable numbers. (There's also definable real number which is a slightly different "Giant Book" approach, though it allows some descriptions of numbers which aren't algorithms. Again this does not give you all possible real numbers.) Staecker (talk) 14:10, 4 March 2018 (UTC)

Hi, This subject, although simple enough, appears to be getting more and more complicated. I am afraid I must insist and keep repeating the question I asked before : "If Pi is on the number line, where would you put it?" Nobody has answered it yet. It is not rocket science : There are two answers : One is a specific numerical value, the second is the admission that Pi is an algorithm which cannot be resolved to a specific number and for that reason can never be placed on the number line. In the second case if that is true, than any picture which shows Pi loitering on the number line is incorrect and must be fixed. So what is your take on it? Please humor me, I am quite capable of differentiating between a redirection to something unrelated (e.g. computable numbers ) and an answer which actually relates to a question. For the record I have nothing against algorithms, I only object to the sloppiness which confuses them with fixed numerical values. I also don't want to banish algorithms from the number line. The two are intertwined in a fundamental manner. The algorithms are an undefinable giant mix of human instructions, computer generated code, rules and derivations. Most of their output is too complex to organize in a consistent easily accessible fashion. There is one exception : Algorithms which produce numerical values conforming to strict syntax rules have found a home : The now ubiquitous number line. The number line is the giant billboard of these algorithms a sort of a meeting place where everybody is welcome to submit their latest and greatest creations provided that they follow the basic rules. But the algorithms themselves cannot stay. (same as creators of actual billboards : they get their instructions, have a purpose, climb up and paint away feverishly until they are done. But then they don't glue themselves to the picture...instead climb down and go away. Like I do now. Counting floats (talk) 17:12, 4 March 2018 (UTC)

Pi goes at π, but that answer does not seem to satisfy you, so lets try to distill your question down to one which makes your objection more apparent. Do you still have the same problem placing 1/3 on the real number line that you do for π? -- ToE 17:30, 4 March 2018 (UTC)

Or, à la Dedekind cuts, you put π to the right of all the rational numbers that are less than π, and to the left of all the rational numbers which are greater than π. But on some level, this is a meaningless question. A number line is just a schematic representation in order to help get a sense of the way that the real numbers are ordered. Writing down the value of π (for example) in base 10 is just one way of representing it. There are others, like $\pi =4(1-1/3+1/5-1/7+\cdots ).$ There are also spigot algorithms which allow you to compute a specific digit (in other bases) without computing the ones before it. These all describe π. –Deacon Vorbis (carbon • videos) 17:53, 4 March 2018 (UTC)

Hi, Thank you for responding. Yes, 1/3 has the same issue, but I let you experience it : take a very fine ball-point pen and draw a vertical line which will intersect the number line. Where will it cross ? 0.3, 0.33 or 0.333... obviously every choice will be wrong. However the whole issue goes away if one follows my advice : if the algorithm has multiple outputs just prefix it with an approximation sign and the whole issue goes away ≈1/3 can be then placed in the vicinity of 0.33 and everybody will know that it is just an approximation. There is another way : have the caption (1/3) (correct to the first 15 digits) that is fine too. Moreover if one loves 1/3 than he can flood the number-line display with hundreds of vertical lines with the annotations of 0.3 0.33 and so on. All logical, informative and correct. But 1/3 just like Pi or sqrt(2) cannot be placed on the number line. That is all I wanted to say. Counting floats (talk) 18:17, 4 March 2018 (UTC)

OK, you don't think 1/3 belongs on the real line. But what about 1/2 or 1/10? You're fine with them, right? -- ToE 18:22, 4 March 2018 (UTC)

Real number line

Pi goes where pi goes; 1/3 goes where 1/3 goes. You seem to be thinking of the real number line as something you can physically draw. The real number line is continuous and all real numbers are on it. There are no gaps (see Completeness of the real numbers) and each real number is represented on the line by a point with no size. There is a one-to-one correspondence between the points on the real number line and the real numbers. Take pi, for instance - all of the real numbers to one side of it on the line are less than pi and all real numbers on the other side of it are larger than pi. Bubba73 ^{You talkin' to me?} 18:51, 4 March 2018 (UTC)

@Counting floats: But you can. You can trisect the line segment between 0 and 1 using a compass and straightedge construction. It even allows drawing the vertical line in the exact correct location.

You seem to be getting confused and/or misled by the decimal representation of real numbers. For your claim to be well defined it cannot depend on the representation of the number. 1/3 is written as 0.1 in base-3 notation, which is obviously a finite expansion. There is no fundamental mathematical reason to choose base 10 over base 3, or any other base divisible by 3, all of which will have a finite expansion for 1/3. This immediately demonstrates the absurdity of what you are trying to suggest. Your notion of "algorithm" is not well-defined. Note that floating point is not the same as a (non-integral) real number.

You may have noticed that we have been pointing you to a definition of the real number line that is contrary to what you suggest. This is because this is the (near-)universal standard of the mathematical community and mathematicians do not like attempted redefinitions of such established terms.

In practical terms, actual points drawn by pen will always have a finite width. So one cannot speak of an exact point at which a given dot lies on a hand-drawn line. Thus we use the mathematical idealization.--Jasper Deng (talk) 19:44, 4 March 2018 (UTC)

That's right. Forget about the digits of a real number - that is just how we write it. And that has nothing to do with the number itself. Bubba73 ^{You talkin' to me?} 00:10, 5 March 2018 (UTC)

Yes, you can do that then mark the spot offset from the origin exactly 1/3 away. But we don't have the same straightedge, so you will have to give us some ballpark. Let's start : 0.3 then 0.33, then 0.333 and so on until we scream : we got it it is going to be a bunch of 3's after the decimal point, but when will it stop ? Why, you say : NEVER ! So did you point us to the exact spot on the number line where 1/3 resides ? Of course not, that spot can be approached from the left from the right or any random fashion but we well never ever get to it. The line you drew over the spot where the parallel lines of the construction intersect the 1/3-rd spot have finite length; compass and straight edge are hampered by nonlinearity, imperfections, expansion and contractions. When you put your picture of it on the computer screen the dot resolution will immediately destroy any fantasies about whether we are looking at the exact 1/3 spot. We cannot. There is no resolution small enough to accommodate infinitesimal distances. So we better stick with the algorithms which does not get corrupted by moisture, dirt, wind, sunshine or magnetic fields or the family dog. Once again, I offer an olive branch it is either ≈1/3 or a stream of 0.3 0.33 0.333 forever. But it cannot be 1/3 Counting floats (talk) 01:26, 5 March 2018 (UTC)

You are not getting what we are saying. For one thing, a straightedge and compass construction is an idealized procedure - not physical as you think. Secondly, forget about digits - you are confusing the number with the way we write a number (in base 10 positional notation, in this case). Numbers are not the digits. And the algorithms you speak of are methods to calculate numbers - they are not the number itself. Bubba73 ^{You talkin' to me?} 01:56, 5 March 2018 (UTC)

@Counting floats: Did you read the rest of my comment? I'd also like to point you to this discussion I had recently. Like in that discussion, you are being hampered by your steadfast adherence to a decimal expansion, when, as I have already pointed out, there is no fundamental reason why base-10 is somehow more natural than any other base, and also your conflation of a real number with the limit of a sequence converging to that real number. In fact, computer screens' pixels almost certainly are not lined up in a way that puts exactly a power-of-10 number of pixels within what is presented on the screen as the interval from 0 to 1. So for example, if 9 pixels are used, the number 1/10 is harder to depict accurately than 1/3.

Please explain exactly how your argument naturally results from a construction of the real numbers. I'll give you a big hint: it doesn't, and any attempt at arguing that is futile.--Jasper Deng (talk) 02:06, 5 March 2018 (UTC)

@Counting floats: Perhaps a refresher on number line will help. Bubba73 ^{You talkin' to me?} 04:48, 5 March 2018 (UTC)

Digression on productiveness of thread

Thanks, Trovatore, for pointing this guy here -- imagine the vast loss to human knowledge that would have occurred if the Ref Desk regulars did not have the opportunity to argue with a crank about whether the point 1/3 exists on a number line. (P.S. to other reference desk regulars: when someone tells you they have a "question" and then their "question" is a manifesto about how lines don't have points one-third of the way between two given points, the underlying issue is not a simple misunderstanding, and it is most certainly not going to be solved by you.) --JBL (talk) 11:54, 5 March 2018 (UTC) This is Counting_floats : Whoa,Joel B. Lewis, you crossed the line, I thought you were supposed to refer to us on Wikipedia either by name, or by "contributor" but never by gender. How do you know I am a guy ? Because you believe that women can't do mathematics ? Please do explain. And the "crank" put-down ? Just pray tell us what your qualifications are ? What did you "contribute" to the wonderful world of mathematics ? Let's have it. I will make an educated guess though : It is my experience that those who have no sound arguments usually substitute name calling just to get noticed. At any rate since you have inject yourself into this scintillating debate, I challenge you : Get a piece of paper, draw a number line on it ( User Bubba73 pointed me to a primer on number-lines if you need a refresher too. ) Make a mark at 0, then a mark at 1/3, then draw a segment between the two and put the distance on it. What is it going to be? Is it 0.3, 0.33, 0.333, 1/3 or ≈1/3, take your pick or select something else. Take a picture of it and publish it on Wiki so that we can comment on it. Please stay focused on the task, it is not rocket science. If you cannot do such a simple task then please disqualify yourself from participating in "where the floats go" debate. P.S. Try to stay away from put-downs like guessing my age, country of origin, political or religious leanings. Counting floats (talk) 14:44, 6 March 2018 (UTC) Gender neutral use of the plural "guys" has been around a long time. Gender neutral use of the singular "guy" is more recent, but is on the rise. Don't take offense where none was intended. As it turns out, JBL is a professional mathematician. While earlier responses to this question were a productive use of this desk, it is now well past the point of diminishing returns. If you insist on believing that God has ten fingers, so be it. Your belief does not change mathematics. Your assertion is incorrect. -- ToE 17:04, 6 March 2018 (UTC)

The discussion above is closed. Please do not modify it. Subsequent comments should be made on the appropriate discussion page. No further edits should be made to this discussion.

Statistical test for determining correlation over multiple time series

Let's say I am interested to see if weather had an effect on population in different countries. I have annual weather data, annual population, a many countries. Is there a statistical test I could run on this data to generate any insights on whether weather has an effect on population for multiple countries over multiple years?--2601:642:C301:119A:3D10:539:40A1:5092 (talk) 06:17, 3 March 2018 (UTC)

There are, of course, several tests for correlation between sets of data, but to understand the concepts involved for their correct application would probably require a course in statistical analysis. You might start with the article on Correlation and dependence. One of the most basic tools is the Scatter plot which is available in most spreadsheet programs. Its trade off is that while it isn't very definitive, you can tell at a glance if an idea is at least worth pursing. You should know however that no statistical test can determine causality; see Correlation does not imply causation. For example even if you did determine a correlation between climate and population, there would still be no way to determine just from the data if climate was affecting population, population was affecting climate, some other factor was affecting both, or some combination of the three. (Btw, I'm assuming that by weather you mean climate; weather being the day-to-day conditions and climate being what happens over the long term. Not that a single storm can't affect population -- look at what's happening in Puerto Rico -- but it doesn't seem likely that it would a global trend.) --RDBury (talk) 22:24, 3 March 2018 (UTC)

For causation you would have to move people to different locations for some reason like jobs or college, keep them there for a while, and see if they decide to stay. That might actually be an interesting thing to study. I know that lots of people came to my school (in California) for academic reasons, then stayed afterwards because of the nice weather. Plus there's the Snowbird (person) phenomenon, etc. There are starting to be useful methods to study probabilistic causation. Judea Pearl's book "Causality (book)" had a major impact in this field and I want to read it someday. 173.228.123.121 (talk) 00:34, 6 March 2018 (UTC)