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February 27

Material conditional

I must have asked this question before: Is $p\to q\equiv \neg (p\land \neg q)$ "intuitively" defined, or can be proven by logic? (not truth-tables)

I found these 2 proofs here and here – are they valid? I do not quite understand them. יהודה שמחה ולדמן (talk) 00:18, 27 February 2018 (UTC)

Depending on which system of logic you're talking about, p→q might be something you define via a truth table, or as an abbreviation for an expression such as ~p∨q or ~(p∧~q), or → might be taken as a fundamental operation whose behavior is determined by axioms and rules of inference. The problem with something like Proof Wiki is that you can't necessarily pull a single proof out of context; a proof of a given statement depends on other statements which in turn depend on others. The proofs you linked to seem to be using a particular flavor of a subgenre of logic systems called natural deduction, and I'm guessing that the proofs are valid withing the context of one the sources listed as references. Specifically, the linked proofs seem to be taking the → symbol as fundamental with behavior determined by the Rule of Implication and Modus Ponens, but even within natural deduction it might be a defined symbol where the Rule of Implication and Modus Ponens are proved as theorems. So the short answer to the question is that it depends on which system of logic you're talking about. The good news is that different systems of propositional logic are equivalent in the sense that they prove the same statements, at least for classical systems. (In intuitionist logic e.g., the linked proofs are not valid.) But there can be a lot of variation in how you get there; you could build a system of logic built just from the ↓ (nor) operation, or a system using just truth tables with no normal proofs at all. --RDBury (talk) 10:36, 27 February 2018 (UTC)

The thing is, I understand pretty well why the $\land ,\lor$ functions give their answers; it is called "by definition".

But it seems to me that $\to$ is not "Well-defined". And now you tell me that Modus Ponens can be proven in some systems... does this ever end?

So I wonder: Is it $p\to q:=\neg (p\land \neg q)$ , or is it $\equiv$ ? יהודה שמחה ולדמן (talk) 11:25, 27 February 2018 (UTC)

As an example of a similar situation, you could define an even number as one that can be written as x+x, or you could define an even number as one that is divisible by 2. They are two different definitions that turn out to be equivalent. Euclid used the first definition but most modern authors use the second. But it's like you're asking which is 'the' definition of even; the answer is it depends which book you're reading, but in the end it doesn't really matter because the two definitions give the same results. It's the same with logic; five books on logic will have five different sets of axioms, rules of inference, and definitions of the operations. The book here defines → in terms of ~ and ∨, another book might define it a different way, and another might define ∨ in terms of ~ and →. Again, it doesn't matter in the end because the same set of statements can be proved. --RDBury (talk) 12:43, 27 February 2018 (UTC)

OK, so what are the axioms and rules of inference for mathematical logic? יהודה שמחה ולדמן (talk) 13:41, 27 February 2018 (UTC)

As RDBury said above, that depends on which system of logic you have in mind. The axioms of propositional logic are different from the axioms of intuitionist logic, for example. Gandalf61 (talk) 16:06, 27 February 2018 (UTC)

I finally found a proof on youtube, but I keep getting a posting-error. יהודה שמחה ולדמן (talk) 00:29, 2 March 2018 (UTC)

Meter to French foot and vice versa

Our Units of measurement in France before the French Revolution says that in 1799 one meter was set to 443.296 French lines, or 3 feet 11.296 lines. That corresponds to 9,000/27,706 m. A note follows: This can be shown by noting that 27706 x 16 = 443296 and that 9 x 16 = 144, the number of lignes in a pied. I feel that it must be entirely correct, but I simply can't get it. How does it work? Could you elaborate, step by step, how we get from 1 m = 443.296 lines to 1 ft (144 lines) = 9,000/27,706 m? --Lüboslóv Yęzýkin (talk) 17:49, 27 February 2018 (UTC)

It's simple arithmetic. If 1 foot is 144 lines and 1 meter is 443.296 lines, then 1 foot is 144/443.296 meter. Multiply both numerator and denominator by 1,000 to get rid of the decimals: 144,000/443,296. Now divide both by the common factor of 16 and you have 9,000/27,706. That is what the footnote is clumsily trying to say. I have no idea why the remaining common factor of 2 was not eliminated to write the fraction in lowest terms as 4,500/13,853. --69.159.62.113 (talk) 00:44, 28 February 2018 (UTC)

• One more point. One of the motivations for adopting metric in France was that each region had its own definitions of different units. So speaking of the "French foot" is wrong; it's only the "Paris foot" or the "French royal foot" or some such. In the article, look back a few paragraphs from where it mentions 9,000/27,706. --69.159.62.113 (talk) 00:47, 28 February 2018 (UTC)

March 1

Math homework question.

From a tenth grade math book.

A man and his wife walk up a moving escalator. The man walks twice as fast as his wife. When he reaches the top he has taken 28 steps. When she reaches the top she has taken 21 steps. How many steps are visible in the escalator at any one time?

Hint: It was in the "Fractional equations" section. — Preceding unsigned comment added by 2600:1700:2ED1:4020:AD71:72DF:DA0A:A11E (talk) 15:34, 1 March 2018 (UTC)

I'll give you a starting point. After the man gets to the top, he's covered 28 steps, but x1 steps have been skipped over because the escalator is moving. after the wife gets to the top, she's covered 21 steps, but x2 steps have been skipped over because the escalator is moving. As the escalator doesn't change, 28+x1 = 21+x2. Figure out the equation linking x1 and x2 together from the speeds of the 2 people, and the rest is simple algebra. Iffy★Chat -- 16:39, 1 March 2018 (UTC)

Here is a worked answer using lots of algebra.

A      answer aka number of steps visible on the elevator
V      velocity of elevator  (independent of speed of man or speed of wife)
L      Length of elevator
TM     time for man
TW     time for wife
SM     Speed for man
SW     Speed for wife

SM = 2 * SW

28 == A - V * TM
TM == L  / SM

21 == A - V * TW
TW == L / SW    

Putting it together

28 == A - V * L  / SM      next substitute SM with 2 * SW 
28 == A - V * L  /  ( 2 * SW )     call this AAA

21 == A - V *  L / SW      call this BBB

Next we calc AAA - BBB

7 == - V * L  /  ( 2 * SW ) -  ( - V *  L / SW )     Next we pull out the common factors  (V * L  / SW )
7 ==  (V * L  / SW ) * (-1/2 + 1 )
7 ==  (V * L  / SW ) * (1/2)                Next we multiply both sides by TWO
14 ==  (V * L  / SW ) 

recall BBB
21 == A - V *  L / SW        Next we substitute (V * L  / SW )  with 14
21 == A - 14                 Next we add 14 to BOTHSIDES
35 == A

110.22.20.252 (talk) 06:50, 2 March 2018 (UTC)

Unfortunately this answer is wrong. The time taken by the man to get to the top is 28/SM = 14/SW, and the time taken by the woman is 21/SW. putting these in to the equation I derived above gives us 28+14/SW = 21+21/SW. These are equal when SW=1, and gives us our answer of 42 steps. Iffy★Chat -- 09:52, 2 March 2018 (UTC)

That's what I got. (Insert Hitchhiker's joke here.) I think the problem is ambiguously worded (not to mention a bit sexist) when it says that the man walks twice as his wife. To me that means the man walks twice as fast on solid ground. But it could be interpreted to mean that the man moves twice as fast while on the escalator. --RDBury (talk) 10:12, 2 March 2018 (UTC)

PS. The problem seems to be originally from Essentials of Mathematics by Russell V. Person (1961). There is a worked solution here and here. --RDBury (talk) 10:28, 2 March 2018 (UTC)

The mistake in 110.22.20.252's solution is that when walking up the escalator, we must add the escalator's speed V to the man's speed SM or the woman's speed SW. So we have TM = L / (V + SM) and TW = L / (V + SW). Gandalf61 (talk) 10:40, 2 March 2018 (UTC)

The difficult bit here is turning the words into equations (which may then be solved fairly mechanically).

It was initially suggested that if the escalator moves $1x$ steps during the man's journey then it must move $2x$ steps during the women's - or going back one step in the thinking, that the women is on the escalator twice as long as is the man. So is this correct?

One way to test it is to consider extreme (limit) cases:

1. Is it correct when the escalator is moving much (1,000,000 times) faster than either (when they are both effectively standing)?
2. Is it correct when they are both walking much (1,000,000 times) faster than escalator is moving (when the escalator is stopped)?
3. If the women was walking at 1/1,000,000 instead of 1/2 the speed of the man (effectively standing), would she be on the escalator 1,000,000 times longer (for ever)?

The idea fails 2 out of 3 of these tests, so we must think again. It is often possible to use these sort of checks on the final answer too. --catslash (talk) 12:53, 2 March 2018 (UTC)

Assuming v is the speed of the woman, V is the speed of the escalator and k is total number of steps, you can write the share of the steps which she covered as

${\frac {v}{v+V}}={\frac {21}{k}}$

As for the man,

${\frac {2v}{2v+V}}={\frac {28}{k}}$

You then get a trivial quadratic equation which has only one solution as the other is eliminated be the requirement that v ≠ 0 ≠ V, and that solution is indeed k = 42. 93.142.83.134 (talk) 20:27, 2 March 2018 (UTC)

March 3

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

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)

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)

March 4

A design wind is defined as a 50-year wind (p=0.02)

The rest of the question, the answer and my attempt follow.[1] I don't understand why my attempt is wrong. Thank you in advance as always. 151.202.5.26 (talk) 01:15, 4 March 2018 (UTC)

But you're not wrong. There can be more than one way to enumerate the possible outcomes, which is what happened here. Hint: it might be a good idea to double check the book's computation of the numerical approximation. –Deacon Vorbis (carbon • videos) 02:14, 4 March 2018 (UTC)

I've performed the two calculations many times. The book's answer gives 1.24E-4, while mine gives 7.76E-5. The professor didn't want to go into details, perhaps because he hadn't covered it, but he said that my method calculates a different scenario. 151.202.5.26 (talk) 02:43, 4 March 2018 (UTC)

No, if you plug p = 0.02 into the book's answer, you get the same value that you're getting for your answer too. The book's answer is right, but its computation of an approximate value for that answer was wrong. I don't know where that number is coming from, but everything else is otherwise in agreement. The book's solution is only counting up to the 3rd success, whereas yours is counting all possibilities within the first 5 trials (or rather you're actually counting the complementary probability and subtracting from 1, but that doesn't really save you anything for this problem). But both approaches are correct and give the same answer. –Deacon Vorbis (carbon • videos) 02:53, 4 March 2018 (UTC)

The book's final answer is not an estimation, just a rounding.[2] 151.202.5.26 (talk) 03:00, 4 March 2018 (UTC)

Oh, the book forgot to divide 4*3 by 2 as I did? Thank you. 151.202.5.26 (talk) 03:04, 4 March 2018 (UTC)

The two expressions are equal for all p. So if the book gives a different value then either they are using a different value of p to start with, hard to tell without the first part of the problem, or they simply made a mistake in calculation. I get 7.76192E-5 for the expression when p=.02. --RDBury (talk) 04:00, 4 March 2018 (UTC)

What is the solution of this circuit by using thevenin theorem?

Here is a QUESTION — Preceding unsigned comment added by 39.40.141.28 (talk) 11:14, 4 March 2018 (UTC)

I don't know what thevenin theorem is, but basic circuit analysis gave me 63/13 mA and 108/13 V, no guarantee of correctness however.→31.53.2.226 (talk) 12:51, 4 March 2018 (UTC)

As 31.53.2.226 suggests, the problem can also be worked via basic circuit analysis, but they seem to have made a mistake in their calculation and their numerical answer is incorrect. To work this problem via Thévenin's theorem, you are to consider the points where the load attaches above and below R₂ to be terminals A and B. The easiest way to determine the source's equivalent voltage and resistance is to consider the two extremes of R_L=∞Ω and R_L=0Ω. In the first (open circuit) case all the current in the actual circuit passes through R₂ and it is easy to calculate the voltage across the A-B terminals. Since there would be no current in the equivalent circuit under this condition, that voltage must be V_Th. In the second (short circuit) case, no current in the actual circuit passes through R₂, so it is easy to determine the current through the shorted load. Since R_Th is all that is limiting current in the equivalent circuit under this condition, that resistance must be V_Th divided by the current you calculated. Now consider the equivalent V_Th/R_Th/A-B circuit with your given load R_L=4kΩ and determine the load's current and voltage. Once you have done that you may wish to verify your work by basic circuit analysis, showing that the answers (which are not what 31.53.2.226 gave) match. That should get you going, and you will learn more if you can work this problem through on your own, but let us know if you need more help. While it took some time to explain how to apply Thévenin's theorem, it is quicker and easier (for me, at least) to do the calculation that way than via basic circuit analysis, and it is particular useful if you wish to know the current and voltage for a variable load. But remember that it only tells you what is happening with the load, not with the actual source. -- ToE 14:27, 4 March 2018 (UTC)

The current through the system is $36/(6+3*4/(3+4))=14/3$ mA. The voltage at the load is $36-14/3*6=8$ V. Then the load current is $8/4=2$ mA. Ruslik_Zero 20:36, 4 March 2018 (UTC)

March 5

volume in Hilbert space?

Is it customary to speak of the volume of some region of Hilbert space? I have no doubt one can concoct measures that make it possible, but I'm asking if it's done in practice. I'm particularly thinking of the notion that quantum states correspond to vectors on the unit sphere in Hilbert space. If n>7 or so, the volume of the unit n-ball decreases as n increases, with a limit of 0 as n approaches infinity. So would we say the unit sphere in Hilbert space has zero volume? What about the unit cube?

I thought about the unit sphere because if it had positive volume, one could say from the curse of dimensionality that the volume was entirely concentrated at the surface, so in some sense quantum states as unit vectors couldn't be distinguished from random points in the unit ball, and I wondered if that could mean anything physically. I don't actually know any QM so am trying to make some sense of it by understanding little aspects like this. Thanks.

173.228.123.121 (talk) 11:17, 5 March 2018 (UTC)

@173.228.123.121: There is no extension of the Lebesgue measure to an infinite-dimensional Hilbert space other than the uninteresting assignment of either zero or infinity. --Jasper Deng (talk) 11:49, 5 March 2018 (UTC)

Jasper, thanks, that makes sense. (Edited): is that a theorem, about not being able to put a non-trivial measure on Hilbert space? What about on the unit sphere or unit ball in Hilbert space? Those are not themselves Hilbert spaces. The thought is that picking a random point on the unit sphere (i.e. making a physical measurement under the Born rule) implies there's a probability measure on the unit sphere to pick from, and I guess the Born rule itself gives a measure. 173.228.123.121 (talk) 22:49, 5 March 2018 (UTC)

You can probably come up with a measure, but it won't satisfy the properties of a Lebesgue measure. That's all I'm saying. See [3]. Basically, compact sets are hard to come by in an infinite-dimensional topological vector space.--Jasper Deng (talk) 03:32, 6 March 2018 (UTC)

Not that I understand this stuff, but we have an article about Infinite-dimensional Lebesgue measure's; it basically states the theorem above and gives an outline of a proof. It also points to some alternative measures which don't have all the properties of Lebesgue measures. (The Stack Exchange post points to the same article but it seems worth adding it here too.) --RDBury (talk) 06:53, 6 March 2018 (UTC)

Roots of cyclotomic polynomial for z^7 = 1

Our article Root of unity#Algebraic expression says, concerning the sixth-degree cyclotomic polynomial giving the primitive 7th roots of unity,

As 7 is not a Fermat prime, the seventh roots of unity are the first that require cube roots. There are 6 primitive seventh roots of unity; thus their computation involves solving a cubic polynomial, and therefore computing a cube root. The three real parts of these primitive roots are the roots of a cubic polynomial....

• Why does this involve a cubic polynomial, rather than a 6th degree one?
• How is this cubic polynomial found?

Thanks in advance, Loraof (talk) 16:41, 5 March 2018 (UTC)

After factoring out the obvious root of $z=1$, the remaining roots form 3 conjugate pairs, with 3 distinct real parts - the roots of a cubic:

$8(\mathrm {Re} (z))^{3}+4(\mathrm {Re} (z))^{2}-4\mathrm {Re} (z)-1=0$

Use $z=x+iy$ and then $y^{2}=1-x^{2}$

${\begin{aligned}0&=z^{7}-1\\&=(z-1)(z^{6}+z^{5}+z^{4}+z^{3}+z^{2}+z+1)\\&=(x+iy-1)((x+iy)^{6}+(x+iy)^{5}+(x+iy)^{4}+(x+iy)^{3}+(x+iy)^{2}+x+iy+1)\\&=(x+iy-1)(y(6y^{4}x+y^{4}-20y^{2}x^{3}-10y^{2}x^{2}-4y^{2}x-y^{2}+6x^{5}+5x^{4}+4x^{3}+3x^{2}+2x+1)i-y^{6}+x^{6}+15y^{4}x^{2}+5y^{4}x+y^{4}-15y^{2}x^{4}-10y^{2}x^{3}-6y^{2}x^{2}-3y^{2}x-y^{2}+x^{5}+x^{4}+x^{3}+x^{2}+x+1)\\&=(x+i{\sqrt {1-x^{2}}}-1)({\sqrt {1-x^{2}}}(-8x^{3}-4x^{2}+4x+1)(2x+1)(-2x+1)i+x(-4x^{2}+3)(-8x^{3}-4x^{2}+4x+1))\\&=(-8x^{3}-4x^{2}+4x+1)(x+i{\sqrt {1-x^{2}}}-1)({\sqrt {1-x^{2}}}(2x+1)(-2x+1)i+x(-4x^{2}+3))\end{aligned}}$

--catslash (talk) 17:23, 5 March 2018 (UTC)

Wow, catslash, thanks for the detailed answer! Loraof (talk) 18:04, 5 March 2018 (UTC)

March 6

Limits of partial functions

At first glance it looks like

$\lim _{x\to 0}x{\sqrt {\sin {\frac {1}{x}}}}=0$

since whatever is under the square root is bounded and x→0. But both of my undergraduate analysis texts say no because the domain of the function does not include a neighborhood of 0. (One-sided limits are defined separately in these sources.) If

$D:=\{x:x\neq 0\,{\rm {{and}\,\sin {\frac {1}{x}}\geq 0\}\cup \{0\}}}$

and

$f(x):={\begin{cases}x{\sqrt {\sin {\frac {1}{x}}}}&x\neq 0\\0&x=0\end{cases}}$

for x∈D, then we have the odd situation where f is continuous on D, f(0)=0, but $\lim _{x\to 0}f(x)$ is undefined. Are there alternative definitions for limit which avoid this issue? Note that the WP article Limit of a function, and presumably most freshman calc texts gloss over this issue since they use the expression | f(x) − L | < ε without explicitly requiring that it be defined. --RDBury (talk) 07:59, 6 March 2018 (UTC)

In my view, $\lim _{x\to 0}f(x)=0$ here, because for all x in D with $|x|<\delta =\epsilon$, $|f(x)-0|<\epsilon$. D is a metric space (as a subset of the real line), and I would use the definition of convergence in that metric space. —Kusma (t·c) 12:03, 6 March 2018 (UTC)

See the section More general subsets at Limit of a function#More general subsets, IMHO it precisely addresses your problem. --CiaPan (talk) 12:41, 6 March 2018 (UTC)
Forgot to ping (again!): @RDBury:. --CiaPan (talk) 12:42, 6 March 2018 (UTC)

Thanks; I missed that. There are no sources listed for the section though so it's hard to tell if that notation is generally accepted. @Kusma:, the more general topological approach does include the 'More general subsets' definition as a special case. I can see why it's not used as an initial definition because of the limit point requirement, needed to guarantee the limit is well-defined. I'll check my texts again to see if they generalize their definition later on. --RDBury (talk) 15:22, 6 March 2018 (UTC)

Principle

What's the principle called when you choose one individual out of a sample and infer that it will be representative of the entire sample? 93.140.132.12 (talk) 08:24, 6 March 2018 (UTC)

Inductive reasoning. Bo Jacoby (talk) 12:54, 6 March 2018 (UTC).