Wikipedia:Reference desk/Mathematics

From Wikipedia, the free encyclopedia
Jump to: navigation, search

The Wikipedia Reference Desk covering the topic of mathematics.

Welcome to the mathematics reference desk.
Want a faster answer?

Main page: Help searching Wikipedia

How can I get my question answered?

  • Provide a short header that gives the general topic of the question.
  • Type ~~~~ (four tildes) at the end – this signs and dates your contribution so we know who wrote what and when.
  • Post your question to only one desk.
  • Don't post personal contact information – it will be removed. We'll answer here within a few days.
  • Note:
    • We don't answer (and may remove) questions that require medical diagnosis or legal advice.
    • We don't answer requests for opinions, predictions or debate.
    • We don't do your homework for you, though we’ll help you past the stuck point.

How do I answer a question?

Main page: Wikipedia:Reference desk/Guidelines

  • The best answers address the question directly, and back up facts with wikilinks and links to sources. Do not edit others' comments and do not give any medical or legal advice.
See also:
Help desk
Village pump
Help manual

August 24[edit]

August 28[edit]

Why is Regression analysis called regression analysis?[edit]

Regression means to go back but I don't see what that has to do with regression analysis. (talk) 16:42, 28 August 2014 (UTC)

This is explained in the history section of the article. This came from using the word "regression" for the phenomenon (also known as "regression towards the mean") that children go back to the average relative to their parents. The statistical procedures that came out of this observation have therefore been called "regression". -- Meni Rosenfeld (talk) 16:50, 28 August 2014 (UTC)
That seems to be basically correct, in light of Regression_analysis#History. SemanticMantis (talk) 15:09, 29 August 2014 (UTC)
There goes my theory that people who do regression analysis are all thumb-suckers who wet the bed at night. :-) StuRat (talk) 00:05, 30 August 2014 (UTC)

August 29[edit]

notation of differentiation versus notation of integration[edit]

We have an article notation for differentiation but no article of notation for integration which is telling. Why? — Preceding unsigned comment added by (talk) 17:14, 29 August 2014 (UTC)

Because one is a big enough subject for its own article and the other is a small topic best dealt as a subsection in Integral. Dmcq (talk) 17:58, 29 August 2014 (UTC)
I mostly agree with that, but it doesn't really answer the underlying motivation: why are there several conventions for notation of differentiation still in modern use, but only one for integration (at least restricting to functions of a single real variable)? Put another way, why do we still use sometimes use Newton's notation for derivatives, but not for integrals? I suspect the answer is that the various options for differentiation have different strengths and weaknesses, while in contrast, the integral notation doesn't have any real downsides. Of course, there are a few different notations for different types of integrals, e.g. path integral, double integral, surface integral, Ito integral etc. In that light, it wouldn't be so strange to have an article that mentions each of these briefly. Checking the articles, the notation is fairly consistent, but sometimes in text books the integral symbol gets adorned in different ways, depending on context. SemanticMantis (talk) 21:40, 29 August 2014 (UTC)
Let's look at the question in another way: 1/1 = 1^1 = 1^{-1}. 1^{-1} is the inverse of 1/1 but there is only one way to write this. Differentiation on the other hand has many different ways, but the inverse, integration, has one way. Why? (talk) 22:17, 29 August 2014 (UTC)
Actually there's an article Integral symbol. I just remembered about that as it describes how the Germans and Russians use much more upright versions. Dmcq (talk) 22:18, 29 August 2014 (UTC)
Don't forget the physicists habit of writing the d-whatever right after the integral sign as opposed to after the integrand. YohanN7 (talk) 22:19, 29 August 2014 (UTC)
I'm quite liable to leave it out altogether sometimes ;-) Dmcq (talk) 14:16, 30 August 2014 (UTC)
This mention makes me think of differential geometry, where the integral does not form a notational pairing with a formal variable of integration (as in Exterior derivative#Stokes' theorem on manifolds); it is only over a region of a manifold. This might be relevant in that while it looks similar, it is a distinct notation. —Quondum 20:54, 30 August 2014 (UTC)

Ok, let's ask another question. We know that "n" is any number, "b" is any number. Newtonian notation uses "dt". Is "dt" = "dx"? Why? (talk) 16:07, 30 August 2014 (UTC)

I can't quite make out what you are saying but Newtonian notation does not use dx or dt. It assumes a single independent variable, t normally but something else can be assumed instead. For instance \ddot x = -cx describes simple harmonic motion with time as the independent variable but \dot y=y might describe the exponential function with x as the independent variable - but in mechanics it would just be time again. Dmcq (talk) 17:50, 30 August 2014 (UTC)

Ok, my situtationsituation is at a Math 31 level, which is a grade 12 calculus course in Alberta. I am stuck on the quotient rule. I need a proof. I believe where I was stuck uses Leibniz notation. I think the quotient rule is one multiplied by another, but I don't understand why. (talk) 20:13, 31 August 2014 (UTC)

This is quite different from your original question. Try reading quotient rule and product rule. —Quondum 01:20, 1 September 2014 (UTC)
That is a poor explanation of my question. Here's another question, why is d over dx? (talk) 01:28, 1 September 2014 (UTC)
The purpose of the reference desk is not to act as a tutoring service, but is primarily to provide references such as I gave you; in particular, you need to be prepared to take the information and links given and extract the information that is relevant to your question. If you cannot frame your questions so that it is clear what information you seek, and especially if you are so dismissive, you can't expect much of a response. You are not demonstrating that you are trying to synthesize the information that you have been given. —Quondum 01:51, 1 September 2014 (UTC)

August 30[edit]

Integers/whole numbers vs decimals[edit]

The advantage of using integers instead of decimals would seem obvious to most (9 mm instead of 0.09 cm, 1500 metres instead of 1.5 kilometres). But is preference for integers/whole numbers over decimals when using SI units an established principal?--Gibson Flying V (talk) 03:22, 30 August 2014 (UTC)

It is more that people like to use a system where their measurements have a whole number part but not be too big and to use the largest unit like that they can. 1500 meters is an example where one tries as far as possible to use the same scale for all ones measurements. In athletics one would say 1500 meters but in a car one might say 1.5 kilometers. Dmcq (talk) 07:31, 30 August 2014 (UTC)
Right, but for whatever reason 9mm and 1500m were chosen. Similarly, drinks are in 700ml bottles, not 0.7l bottles, snacks are in 200g packs, not 0.2kg packs, films are 90 minutes, not 1.5 hours. It seems that where integers can be used, they are, and I was curious to know from those knowledgeable in mathematics if this apparent preference has ever been acknowledged anywhere (or does it just go without saying).--Gibson Flying V (talk) 07:40, 30 August 2014 (UTC)

Note that 9 mm = 0.9 cm, (not = 0.09 cm). Integers are more elementary and were historically used before fractions, and so an integer number of subunits were preferred to fractions of larger units. The prefix c = 0.01 is usually considered part of the unit, cm = 0.01 m, rather than part of the number, 0.9c = 0.009 . Of course 0.9 cm = 0.9c m. Bo Jacoby (talk) 20:25, 30 August 2014 (UTC).

  • Medical professionals are taught to avoid working with decimals, particularly when measuring dosages.[1][2][3][4][5]
  • The UK Metric Association's Measurement units style guide says, "Use whole numbers and avoid decimal points if possible - e.g. write 25 mm rather than 2.5 cm."
  • In his book entitled The Fear of Maths: How to Overcome It Steve Chinn opens the chapter entitled "Measuring" with I am sure that most people would rather avoid decimals and fractions. This is the reason we have "pence" rather than "one hundredths of a pound". The metric system allows us to avoid decimals by using a prefix instead of a decimal point. If £1 is the basic unit of money, then 1 metre is now the basic unit of length. The metre is too long for some measurements, so we use prefixes, as in "millemetre" as a way of dealing with fractions of a metre.
  • This article cites the Australian construction industry's standardisation on millemetres for all measurements in 1970 as having saved it 10-15% in construction costs due to the eilimination of errors associated with decimals.
That's all I could find so far.--Gibson Flying V (talk) 01:12, 31 August 2014 (UTC)

Absurd or meaningless rate[edit]

I couldn't decide what desk to post this question to. It's kind of a logical/mathematical question but it's also a semantic/linguistic question, so if this is the wrong place to ask this question, please forgive.

Consider the following statements: 1) "I can run fast, up to 10 miles an hour" 2) "I can run at least one mile in at least an hour"

The first statement refers to a maximum possible speed or rate or ratio. But the second statement appears to be absurd or meaningless (I think). Can someone explain to me in a quasi-systematic way *why* the second statement is meaningless.--Jerk of Thrones (talk) 06:51, 30 August 2014 (UTC)

The Humanities reference desk would probably have been the right place for a question like this.
The first asserts that you can run at that speed for a short distance at least. The second is not meaningless, it says you can run one whole mile but sets no limit on the speed. The meaningless bit is because of the very reasonable expectation that the speaker actually meant something more otherwise they wouldn't have said so many unnecessary words, that implies they made a mistake in what they said. In English that sort of sentence can easily be the result of a common habit of duplicating a superlative and one would suppose they just made a mistake and meant "I can run at least one mile in an hour", but there may be some other explanation depending on the circumstances. Dmcq (talk) 07:21, 30 August 2014 (UTC)
It is absurd because it seems as if it should be a statement about how fast someone can run, but isn't. It could be paraphrased as 'I can run for some unspecified distance of a mile or more - but it will take me an hour or more to do it.' It isn't actually meaningless, just less informative than it first appears. AndyTheGrump (talk) 07:24, 30 August 2014 (UTC)
I think the odd part is claiming one can move a mile in a period of time without any upper limit. Unless the person is infirm, that should be true of everyone. Of course, just what constitutes "running" is open for debate, but most wouldn't call a mile in an hour to be a run at all, only a slow walk. If you said it as "I can travel at least a mile in at least an hour", then that might be a reasonable statement from somebody with some type of injury, or carrying a heavy load. StuRat (talk) 02:35, 31 August 2014 (UTC)
Running vs. walking isn't defined by the speed, but by the gait. When walking you have 1-2 feet on the ground at any time, when running you have 0-1. -- Meni Rosenfeld (talk) 07:50, 2 September 2014 (UTC)
The meaninglessness comes with both the over-generalization of the sentence (mentioned above, effectively weakening the statement to "I can run 1 mile before I die") and the contrast with the listener's expectation (" at least one hour? That doesn't help one bit").
Advertisers do this a lot, throwing a heap of positive-sounding phrases which don't actually synergize at the audience. ("Save up to 50%, and more" is the textbook example. It could be 50%, 99%, or only 1%, and due to the illogical structure of their promises, they didn't really lie even if most customers save much less than 50%.
Some politicians use similar patterns, usually for similar reasons (to suggest, rather than actually make, promises).
Sometimes employed for comedy ("A messy death is the last thing that could happen to you" – literally) or by a "lawful" character who would never lie. TV Tropes calls it a "false reassurance" . - ¡Ouch! (hurt me / more pain) 10:43, 1 September 2014 (UTC)
That's a good one. Absolutely true but conveying no information. I like those in my speech, like 'If I don't go to sleep I'll never wake up in the morning'. I think there's a term for those but I've forgotten it. Dmcq (talk) 11:07, 1 September 2014 (UTC)


I read the article Coequalizer, and feel a little bit stupid, because even after repeatedly thinking about it, it evades my grasp.

The article tells me:

In the category of sets, the coequalizer of two functions f, g : XY is the quotient of Y by the smallest equivalence relation ~\sim such that for every x\in X, we have f(x)\sim g(x).[1] In particular, if R is an equivalence relation on a set Y, and r1, r2 are the natural projections (RY × Y) → Y then the coequalizer of r1 and r2 is the quotient set Y/R.

Firstly I have trouble understanding what the smallest equivalence relation is. I assume, it's the finest?

To make a simple example, assume X=Y is the set of real numbers and f(x)=|x| and g(x)=x. What would be the coequalizer? (talk) 13:08, 30 August 2014 (UTC)

Yes, smallest means finest. The term smallest is justified by thinking of an equivalence relation as a set of pairs. Then the smallest one with property X is the intersection of all equivalence relations with property X.
Another way to view it is to start with f(x) \sim g(x) for all x, then make it reflexive and symmetric and close under transitivity.
Using your example, for every nonnegative x, x = f(-x) \sim g(-x) = -x, so we start with x \sim -x for all x. Of course, we also add symmetry and reflexivity. Normally we'd need to close under transitivity, but this is already transitive. So now we take the quotient of the reals by this, which gets us a set which can be naturally identified with the nonnegative reals.-- (talk) 14:38, 30 August 2014 (UTC)
Excuse me, it really looks like I have some extraordinarily mental block on that subject. Please tell me what the morphism of this coequalizer would be. (talk) 14:57, 30 August 2014 (UTC)
I'm not sure what morphism you're asking for. The equivalence relation from your example is given by x \sim y if x = y or x = -y. We get the coequalizer by taking the quotient of the reals by this, so the coequalizer is the set \{ \{x,-x\} \ : \ x \in \mathbb{R}_{\ge 0}\}. The natural identification I mentioned earlier is given by \{x,-x\} \mapsto x.-- (talk) 17:04, 30 August 2014 (UTC)
Thank you so far. I guess my problem is some misunderstanding deep inside my head, probably mixing limits and colimits. At least I now have an example that is not tainted by this fault inside my brain. Thanks. (talk) 20:18, 30 August 2014 (UTC)
Got it! I finally got my brain bug fixed. Having been trained on resolving equations, my mind was tied on thinking about the domain, but, as the name co-equalizer strongly suggests, we are rather forcing equality on the codomain. Nice koan. (talk) 08:53, 31 August 2014 (UTC)
{{reflist-talk}} added here for clarity (talk) 11:58, 31 August 2014 (UTC)
  1. ^ Barr, Michael; Wells, Charles (1998). Category theory for computing science (PDF). p. 278. Retrieved 2013-07-25. 

August 31[edit]

Trilateral symmetry[edit]

My question relates to a hypothetical sentient lifeform based on trilateral symmetry. Assume their mathematics to be base-9 (since they have 3 digits on each of their 3 appendages; the only reason humans created the decimal system is that we happened to be created with ten "digits").  —The question is: Are irrational numbers such as π and φ irrational for all base systems –in the sense that they cannot be expressed with a finite set of ordinal digits, (or whatever the proper terminology is)? Does this relate to Commensurability, and would this be applicable to all number-base systems (specifically, base-3 and base-9)?  —I might not be expressing myself clearly, but hopefully you get the idea. A second (tangentially related) question might best be asked on the computing or science desk, but I'll give it a try here: is there such a thing as a trinary computer based on (null, +/-); translated as (0,1,2) or base-3 (?)     ~: (talk) 11:08, 31 August 2014 (UTC)

Actually, humans developed place-value arithmetic three times, with three different bases. The first place-value system was that of the ancient Babylonians, with base 60. The Mayans used base 20. We use so-called Arabic numerals, which were actually invented in India before being adopted by the Arabs, with base 10. The connection of the arithmetic base with evolutionary anatomy would appear to be sort of random. There are still a few vestiges of Babylonian mathematics, such as 60 minutes to a degree and 60 seconds to a minute, reflecting the use of Babylonian mathematics in astronomy and astrology. Except for that specialized use, Babylonian mathematics did not displace the use of non-place-value systems such as Egyptian, Greek, and Roman numerals. It had the advantage (as do Arabic numerals) of permitting calculations with an arbitrary amount of precision. (That is, you can always carry out a long division to as many decimal places or sexagesimal places as you need, which is important for calculating astronomical events.) It had the disadvantage that it was difficult to memorize the addition and multiplication tables.
However, the question about rational, irrational, and transcendental (incommensurable) numbers has already been answered, which is that rationality does not depend on the base. The axiomatic formulation of mathematics, with Peano postulates, Dedekind cuts, etc., does not depend on the base. Robert McClenon (talk) 19:21, 31 August 2014 (UTC)
The definition of irrational is that such a number cannot be expressed as the ratio of two integers. Since being an integer doesn't depend on base, being irrational does not depend on base. The fact that the decimal expansion of irrationals is infinite without repetition is a theorem. If you go through the proof, you'll see that it can be repeated in whatever integer base you like. So yes, π's expansion is infinite without repetition in base 9.
Since being irrational (and similarly, being rational) does not depend on your base, commensurability does not depend on your base. Otherwise, I don't see much of a way in which it's related.-- (talk) 12:58, 31 August 2014 (UTC)
(E.C.) Yes, they are still irrational. An irrational number is one that can't be expressed as a fraction -- or ratio -- of two integers, and this definition is irrespective of base. One consequence of this definition, discussed in Irrational number#Decimal expansions, is that an irrational number cannot be expressed as a terminating or repeating expansion in any natural base (decimal, binary, ternary, whatever), while a rational number can be expressed as a terminating or repeating expansion in every base, although any given rational number may have an infinite but repeating expansion in one base and a terminating one in another. For instance, 1/3 = 0.333333... in base 10 and 0.010101... in base 2, but 0.3 in base 9 and 0.1 in base 3.
For base 3, see our articles Ternary numeral system, Balanced ternary, and Ternary computer. -- ToE 13:09, 31 August 2014 (UTC)
Thank you, everyone, for your informative replies and links!   ~: (talk) 00:16, 1 September 2014 (UTC)

Defining a perfect number[edit]

Go to Perfect number. It says:

In number theory, a perfect number is a positive integer that is equal to the sum of its proper positive divisors, that is, the sum of its positive divisors excluding the number itself (also known as its aliquot sum). Equivalently, a perfect number is a number that is half the sum of all of its positive divisors (including itself) i.e. σ1(n) = 2n.

It's a provable theorem that the 2 definitions equate. But what I want to know is why the latter definition is preferred by some modern mathematicians. Georgia guy (talk) 13:42, 31 August 2014 (UTC)

I can't speak for all of those modern mathematicians but moving out any one exception from a definition looks well worth trading in an additional factor somewhere. (talk) 14:22, 31 August 2014 (UTC)
While I would not think of a uniform exclusion as an exception, there is a pleasing symmetry between:
  • A perfect number is a number for which its positive divisors sum to twice the number, and
  • A perfect number is a number for which the reciprocals of its positive divisors sum to 2.
The second statement becomes rather awkward when the reciprocal of the number itself is omitted. —Quondum 19:15, 1 September 2014 (UTC)

Total degree of elementary symmetric polynomials[edit]

One can think of the Fibonacci numbers as the number of integer solutions to x1, x2, ..., xn ≥ 0, x1+x2, x2+x3, ... xn-1+xn ≤ 1, the number solutions being Fn+2. Define S(n,k) as the number of integer solutions to x1, x2, ..., xn ≥ 0, x1+x2, x2+x3, ... xn-1+xn ≤ k. So S(n,0)=1, S(n,1)=Fn+2. (S(n,k) is the value at k of the Ehrhart polynomial of polytope defined by the first set of inequalities.) I computed S for n and k ≤ 7 and found a matching set of values in OEISA050446, but I don't understand the description of the entry "total degree of n-th-order elementary symmetric polynomials in m variables," Also, some insight on how S(n, k) might be related to the degree of an elementary symmetric polynomial would be appreciated. --RDBury (talk) 19:45, 31 August 2014 (UTC)

September 1[edit]

Addition help[edit]

Hi guys

How can you do 1+1 WITHOUT a calculator?

tks — Preceding unsigned comment added by (talk) 15:01, 1 September 2014 (UTC)

Principia Mathematica 54-43.png
Perhaps this from Principia Mathematica might help? ;-) Dmcq (talk) 15:56, 1 September 2014 (UTC)
If you're asking how to teach children to add, the usual first step is to take one object, say an apple, then add another object, then have them count the total. Repeat this with various objects, and eventually they will understand that if you add 1 + 1 of any objects, you always get 2. StuRat (talk) 16:20, 1 September 2014 (UTC)
Unless of characteristic 2Face-smile.svg YohanN7 (talk) 16:47, 1 September 2014 (UTC)
Not Unless, but rather Including when: that 2 = 0 for characteristic 2 does not change the validity of 1 + 1 = 2 (2 is normally defined as 1 + 1 in the general case). —Quondum 18:06, 1 September 2014 (UTC)

September 2[edit]