Talk:Real number
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I've just (mistakenly) did an update to this entry. No changes have been made at all. The update I meant to do was that of the Portuguese version, which I have just forked off the English one. --Doshell
"Real numbers" has just been redirected to "Real number". Fine with me, but presently "Rational number" redirects to "Rational number".
What do we want to do, Wikipedians?
- Thanks for noticing that "rational number" has the same problem--it's fixed now. The singular form is preferred for simple nouns like this to make linking easier. For example, "...probability can be expressed as a real number in the interval [0,1]..." --LDC
A real number is one that can be expressed in the form 'DDD.ddd'. DDD is zero or more decimal digits ddd is zero or more decimal digits Of course, DDD must be finite in length. This restriction does not apply to ddd.
Why must DDD be finite in length? If a sequence of real numbers goes to infinity, then there must be an (countably) infinite number of digits in ...DDD. What am I missing?
- I don't really understand the question. The sequence 101,102,103,104,... goes to infinity, but none of the numbers have infinite digits.
- In effect, a sequence of numbers may go to infinity, but a single number can't. (Consider the problem of comparing two such "infinite integers". How could you decide which was bigger without calculating all the (infinite) digits ..DDDD for both numbers.
If we have a space where Cauchy sequences are meaningful (a metric space, i.e. a space where distance is defined), a standard procedure to force all Cauchy sequences to converge is adding new points to the space (a process called completing). When applied to the rational numbers, it gives the following useful construction of the real numbers:
It should point out that this only works with a Euclidean metric or one equivalent to a Euclidean metric; using other metrics gives you the p-adic numbers instead.
The following comment was moved from the main page:
RB: The dimension is actually difficult to define: the reals have dimension 1 for pretty much any sensible definition, but the best definition I know is that cohomology with compact support is non trivial in dimension 1 and vanishes above it.
- That's the cohomology dimension; I don't see why it's better than any other.
Tarquin, do you plan to add to the symbols for subsets of the real line? Otherwise, I'll add to it, including the uses of the symbols. Right now, it's not very clear. — Toby 07:16 Aug 3, 2002 (PDT)
As I understand it, local field means a field complete with respect to a discrete valuation. Are the reals and complexes really local fields? --alodyne
I moved the statement here for the moment, until we sort this out:
- The reals are one of the two local fields of characteristic 0 (the other one being the complex numbers).
See also Talk:Local field. AxelBoldt 19:38 Nov 9, 2002 (UTC)
Oh and by the way: at least one change certainly must be made in that statement since the p-adic numbers are local fields of characteristic 0 by all the various definitions I've seen. Any dispute? --alodyne
- No. The Encyclopedic Dictionary of Mathematics defines a local field as a field that's complete with respect to a discrete valuation and such that it's residue field is finite. They mention that the the reals and complex numbers are also sometimes considered as local fields, but they explicitly exclude them. AxelBoldt 19:14 Nov 12, 2002 (UTC)
To tie up this article with the entries on Model Theory, can someone tell me whether or not there is a first-order theory model for the real numbers? A maths teacher friend of mine told me that there is, but this would imply, by Lowenhein-Skolem theorems that the reals are denumerable, or have a countable model, which which seem to be inconsistent with Cantor's diagonal proof of uncountability of the Reals. --B. Smith.
No, it would not imply that. It would imply only that there is a countable field satisfying all of the same first-order sentences that the real numbers satisfy. The Loewenheim-Skolem theorem does imply the existence of such a model. If the model and the language are elaborate enough, one could write a first-order sentence saying the reals are not countable. It would be true within the countable model. That means that although there would be a sequence containing every member of the model, such a sequence could not itself belong to the model; there could be no enumeration within the model. Michael Hardy 00:12 Apr 11, 2003 (UTC)
"The term 'real number' is a retronym coined in response to 'imaginary number'." That hardly seems likely, since extension to the reals should always have seemed more urgent than extension to complex, and the reals need a name as soon as you consider them.
IMO, real is intended to contrast with rational, and the motivation for "real" is that the rationals are an unrealistically limited set. That follows from the mind necessarily regarding all integer-legged right triangles as precise models of something from reality, if anything in math beyond counting is such a model; from irrational hypotenuses, it follows that rationals aren't quite the full set of numbers needed to correspond to reality, so the choice of the term "real number" reflects the hypothesis that there are no problems in the real world that the reals don't suffice to express the answer to. (In contrast, wanting imaginaries and quaternions takes some sophisticated mathematical abstraction.)
In fact, the best evidence that "real" led to "imaginary" (not the other way around) is that "real" is a better fit to the reals than "imaginary" is to the imaginaries: IIRC how one of my physics profs put it, it's not so much a matter of complex numbers having an imaginary part as that they have *two* parts, both real numbers representing something in the physical world, and it's just that there are systems like electromagnetic fields whose behavior looks simpler if you do the bookkeeping by labeling the electric field as the first component of a complex number and the magnetic as the second, and hiding the fact that the "multiplication" you do is, according to the rules of complex arithmetic, more complicated than the multiplication table you learned in grade school.
So where's this "retronym" thing coming from; is it more than someone's conjecture? --Jerzy 06:41, 30 Sep 2003 (UTC)
I don't think there was a monolithic nomenclature that all mathematicians subscribed to at the time when negative numbers, imaginary numbers, and real numbers were being investigated and systematized (15th century through 19 century-ish). History is messy like that. It is well known that negative numbers (first) and imaginary numbers (later) inspired censure (and sometimes even disgust) in the European mathematicians of the day, so the retronym thing is at least plausible, e.g. "The square root of -1? That's not a real number, it's some kind of imaginary thing!" I'd like to see a source too though. -- Cyan 07:06, 30 Sep 2003 (UTC)
- Just for information, the first reference to imaginary numbers in the Oxford English Dictionary is to Descartes (in French) in 1637, and the first reference in English is in 1706: W. JONES Syn. Palmar. Matheseos 127 The Original Components or Roots of all Equations, may be either Affirmative, Negative, Mix'd, or Imaginary. In contrast, the first reference to a real number that they give is in a 1727 encyclopedia: CHAMBERS Cycl. s.v. Root, If the value of x be positive, i.e. if x be a positive quantity,..the root [of an equation] is called a real or true root. (These are actually citations for the use of real or imaginary as adjectives with the modern meaning, so they weren't looking specifically for the phrase real number, which they don't cite until the 1910 Encyc. Brit.) Although I'm not sure the OED is as good about backdating mathematical terms as they are in general, this gives some indication at least. Anyway, since a rigorous definition and theory of real numbers wasn't really developed until the 19th century, it's hard to call it a retronym (what people had looked at in the past were real numbers but didn't define them; e.g. if you look at just the roots of polynomials, those are the algebraic numbers which are a countable subset of the reals). Steven G. Johnson 02:02, 12 Oct 2003 (UTC)
The article begins with:
- The real numbers are practically any numbers that can be expressed.
I have to say I find this unsatisfying, especially since most (i.e. all but a countable subset of) real numbers arguably cannot be specifically expressed. That is, if you take the meaning of expressed in a natural way: i.e. to be uniquely defined by a finite-length description (such as 4.73, √2, sin(1), ...). And only a strict subset of these expressible numbers are computable (i.e. to an arbitrary precision in a finite time). Steven G. Johnson 01:34, 12 Oct 2003 (UTC)
- It's impossible to explicitly specify a non-recursive number;
That depends on what "explicitly specify" means. Consider, for example, Chaitin's constant. This is non-recursive, but in some sense can be specified. Josh Cherry 23:31, 19 Oct 2003 (UTC)
And what is this Russian school that assumes all numbers are recursive? This sounds like a provably false statement to me, unless something nonobvious is meant by "recursive algorithm". It's provably impossible to specify every real numbers uniquely with a recursive algorithm in the computer-science sense, i.e. a finite-size finite-state machine, or equivalently a finite-length program in any Turing-complete computer language. (There are only a countable number of finite-length computer programs.) (One can even explicitly define a real number, whose digits are e.g. based on the halting problem, that is uniquely specified by a finite-length description but which is not computable in finite time by any program.) Steven G. Johnson 20:00, 21 Oct 2003 (UTC)
Certainly one can explicitly specify particular non-computable numbers. I haven't read everything above, so I'm not certain which "Russian school" is referred to, but it's probably something about constructivism, which is a philosophy that holds that an existence proof is not valid unless it "constructs" the object whose existence is to be proved. For example, if you were to deduce a contradiction from the proposition that every even number greater than 2 is a sum of two primes, that would not be taken by constructivists to be a proof of the existence of a counterexample. Michael Hardy 00:03, 22 Oct 2003 (UTC)
- However, most (all but a countable subset of) real numbers cannot be uniquely specified by a finite description of any sort (whether by a computer algorithm or otherwise). The current Wikipedia statement about the Russian school needs clarification (or deletion), because on its face it seems to imply the contrary. Steven G. Johnson 04:11, 22 Oct 2003 (UTC)
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Quantity Box
This has been removed several times by one person. I am among those who sees it as an enhancement of the article and the "explanations" for its removal wrong -- unless the word is just "silly". It's time to stop deleting and start defending the preceding deletions on this page: convince someone if you're so sure. --Jerzy(t) 01:56, 2004 Mar 28 (UTC)
- The box is not silly but unnecessary and distracting. The box is already too long (you have to scroll to see the whole list and it has a potential to get longer certainly. This kind of the box is only useful and necessary in cases like 1) the article is separated into several pages because the article would be too long if those were in one page. 2) The article is about one entity among several ones. There is no much relationship between real and natural numbers except they are numbers. You want me to convince you then can I also ask you to convince me first? To my knowledge, there was no consensus about having such a box. In other words, I am just reverting the article to what it used to be and see if we can have concensus or something. -- Taku 03:04, Mar 28, 2004 (UTC)
- I've just found this page. Wikipedia_talk:Article_series. It would help us to see whether to have a box or not. -- Taku 16:08, Mar 28, 2004 (UTC)
Negatives are that recent?
From the History section: "Negative numbers began to be generally accepted in the 1600s and were invented by Muslim mathematicians."
I find it hard to believe that negative numbers were just being accepted in Newton's time. I'm no history geek, but I'd like to get that confirmed.
- Same here. Newton obviously used negative numbers, and what about Napier? I had thought that negative numbers were introduced by medieval Italian accountants. Michael Hardy 20:59, 6 Dec 2004 (UTC)
- A serious mistake here. Chinese mathematicians (indeed they maybe more "accountants" than "mathematicians") developed similar concept long time ago. I believe that negative numbers have been reinvented many times, as surplus and deficiency are fundamental concepts in commerce. -wshun 11:45, 10 Dec 2004 (UTC)
Packing real numbers together
You can "pack" any finite number of integers or rational numbers into one (e.g. by alternating digits), but is this legal for real numbers? Fredrik | talk 02:47, 20 Feb 2005 (UTC)
- Yes you can do it with real numbers too, at least when they are all in the interval [0, 1).
- Start by writing each of those numbers in the form math>0.a_1a_2a_3\dots</math>, and for uniqueness, suppose that none of your numbers after a while contain only 9's.
- Then, write the first digit of the first number, then the first digit of the second number, all the way to the first digit of the last number. Then work on the second digit of each number and so on.
- Now, presumably you could do same thing without the assumption that the numbers are in [0, 1), but then you need to worry in addition about the exponent. Oleg Alexandrov 19:12, 20 Feb 2005 (UTC)
"Almost all" means...?
- Since only a countable set of real numbers can be algebraic, almost all real numbers are transcendental.
This is somewhat puzzling, because the link to the "almost all" article says
and the other meanings given don't apply. Surely there are not just finitely many algebraic numbers? -- KittySaturn 08:47, 2005 May 23 (UTC)
- The meaning is actually more in line with the description at almost everywhere. That is, the set of algebraic numbers have measure zero. -- Fropuff 15:26, 2005 May 23 (UTC)
- I've taken no number theory classes, so I may very well be wrong about this, but if i'm reading http://mathworld.wolfram.com/AlmostAll.html correctly then "almost all" can also mean "all of an uncountable set but countable number". —Miles←☎ 19:50, May 23, 2005 (UTC)