Talk:Rational function

Irrational function
I presume an irrational function is one that is algabraic but not rational. Is this correct? Brianjd


 * I don't think irrational function is a standard term. Charles Matthews 09:13, 8 Dec 2004 (UTC)

There is a problem here (and in several other places). This article confuses a rational function and a rational expression. This is similar to the common confusion between a polynomial function and a polynomial. I propose moving this article to "rational expression", with links to and from rational function, and fixing other articles that fail to make this distinction. Comments? Rick Norwood 00:19, 22 December 2005 (UTC)


 * No one says 'rational expression'. Rational function is the standard term, and should be the article name. We should not mess with traditional terminology, even though there is the point you make, and also the lack of definition at values for which the denominator vanishes. Charles Matthews 08:30, 22 December 2005 (UTC)

Many people say "rational expression". It depends on the setting. In ring theory, the rational expressions are the quotient field of the integral domain of polynomials. They are not in any sense "functions" since they are not mappings -- no input or output. They are formal expressions, analogous to the rational numbers, which are the quotient field of the integers.

This is a distinction that was driven home to me in Freshman calculus at M.I.T. The expression x2 - 4 is a polynomial, x2 - 4 = 0 is an equation, and f(x) = x2 - 4 is a function. If the polynomial x - a is a factor of the polynomial, then a is a root of the corresponding polynomial equation, and a is an x-intercept of the corresponding polynomial function.

It still bugs me when people jumble all of these words together.

If you don't like the idea of a name change for this article, can we at least write f(x) = P(x)/Q(x), instead of just the bare expression? Rick Norwood 15:55, 22 December 2005 (UTC)


 * No disagreement from me that 'function' here is a misnomer. Policy is to use the common name. "Rational function" has five times the Google hits. I'm quiter happy for the article to have a full discussion of all those points. But in a sense the functional notation runs counter to the argument that this is not a function ... Charles Matthews 16:10, 22 December 2005 (UTC)

I appreciate the changes you've made. I'm going to look into how this works with various links to this page, especially from the page fraction. Rick Norwood 17:24, 22 December 2005 (UTC)

I note that the title of the article on polynomial functions is "polynomial", though the article begins by identifying polynomials and polynomial functions. (It makes a distinction a few sentences later.) I would like to see the distinction between the various kinds of mathematical objects consistant throughout wikipedia, but it would be a big job and also apt to ruffle feathers. I'm going to make a tentative start, and see what happens. Rick Norwood 23:35, 22 December 2005 (UTC)

To return to the subject after that a bit off topic conversation: Even if irrational function may not be a standard term, some people seem to know rather well what it should be, for example in wikipedia: List of integrals of irrational functions. I think that irrational function deserves to be defined here or maybe in it's on article. Currently irrational function redirects here, but the article doesn't even mention it.85.156.185.105 21:53, 5 November 2006 (UTC)

I'm not seeing any examples of functions which are not rational. All the examples suggest that all functions are rational - which is clearly not the case. 121.45.172.38 (talk) 05:02, 19 January 2010 (UTC)


 * All functions that cannot be written in the form f(x) = P(x)/Q(x) are not rational. If this isn't clear, maybe we need to say it explicitly.  Rick Norwood (talk) 14:50, 19 January 2010 (UTC)

Form and function.
polynomial function redirects to polynomial. But Rational expression redirects to rational function. I don't care which way the redirects go, but it should be consistent. My personal preference is for the object to be the title of the article and the function to be a major subtopic, but I can go either way. I'm placing this comment in the talk pages of both polynomial and rational function in hopes of finding a consensus. Rick Norwood 16:14, 24 December 2005 (UTC)


 * I did a more refined Google search: "rational expression"+"partial fractions", versus "rational function"+"partial fractions". This time 'function' predominated by a factor of 20. This supports my hunch that, important as the distinction may be pedagogically, 'expression' gets dropped as soon as the level of the material becomes more technical.


 * I then replaced "Partial fractions" by "computer algebra": also more than 10:1 in favour of 'function'. "Computer algebra system" was somewhat less conclusive. One could go on; but in no context have I yet found 'expression' preferred.


 * Charles Matthews 19:17, 24 December 2005 (UTC)

The point is that "rational expression" and "rational function" mean different things, not that one is a more popular and the other a less popular way of referring to the same thing. Am I just being pedantic when I say that 2x + 3, y = 2x + 3, and 2x + 3 = 0 are three different though similar mathematical objects? The first is a polynomial, the second a function, and the third an equation. Is it really ok to say that they are all functions, or that they are all equations, or that they are all polynomials?

If so, then it is the article "Polynomial" that should be retitled "Polynomial function", but we should be consistent. Rick Norwood 01:39, 25 December 2005 (UTC)


 * Disagree with any renaming. Polynomial and polynomial function are of course different things, but to see that you need to go to finite rings or rings of finite characteristic, and most people don't get there. Yes, I think it is being pedantic to ask for consistency where most people don't see a problem. This is an encyclopedia directed at the general public, if we were arguing about writing a book for PhD students, then I may be inclined to agree with you. Oleg Alexandrov (talk) 16:08, 30 December 2005 (UTC)


 * I did more research at Google. It turns out that RE+"College algebra" gets more hits than RF+"College algebra". This explains something to me: since 'college algebra' doesn't exist as a subject in the UK, we are talking here about a US-centric usage, of mainly pedagogic value. Charles Matthews 17:01, 30 December 2005 (UTC)


 * Try "algebraic fraction". That may be the UK term.  But, again, this is not a question of two different names for the same thing, but two different things.  x/y is something, and that something isn't a function.  I've heard these things called algebraic fractions; I've heard them called rational expressions.  They may have other names -- but rational function isn't one of them.  I run into them when I have to teach "college algebra" (now almost always called "precalculus"), and again when I teach the graduate course in ring theory. Rick Norwood 00:47, 31 December 2005 (UTC)


 * Algebraic fraction redirects here. The term is not used at all in the article, but in the section Definitions, $$\frac{P(x)}{Q(x)}$$ is referred to as a fraction. In view of the intense discussion on Talk:Fraction (mathematics), a clarification would be appreciated. Isheden (talk) 22:03, 2 October 2011 (UTC)

I have allowed some time to pass since editing polynomial. I am now going to begin editing this article. Rick Norwood 23:01, 8 January 2006 (UTC)

I feel it is hard to understand how one comes from the equations separated by: "Since this holds true for all x in the radius of convergence of the original Taylor series, it follows that". I feel this should be explained a bit further. Anze Vodovnik 23:48, 1 June 2006 (GMT+1)


 * I'll see what I can do. Rick Norwood 17:09, 2 June 2006 (UTC)

New Images
I've just created these three images for this article

http://en.wikipedia.org/wiki/Image:RationalDegree2byXedi.gif

http://en.wikipedia.org/wiki/Image:RationalDegree3byXedi.gif

http://en.wikipedia.org/wiki/Image:RationalDegree4byXedi.gif

Feel free to add them, as I do not know how to.

Xedi 17:30, 30 August 2006 (UTC)


 * Just added two of them, don't really know where to fit the third so didn't put it in. Xedi 18:08, 30 August 2006 (UTC)

Application
Just moved the paragraph beginning "These objects are first encountered in school algebra" to this section where it more properly belongs. I am also removing the reference to an alternative construction for hyperreal numbers, a reflection of my own original research that does not belong in Wikipedia, Alan R. Fisher 01:01, 5 July 2007 (UTC)

Shouldn't the sections on Taylor series, Complex analysis, and Abstract algebra be subsecions of this section? I will leave that for more experienced editors to decide. Alan R. Fisher 01:01, 5 July 2007 (UTC)

Examples
The examples section states that the limit of a certain function as x goes to infinity is... another function. Instead of limit, it should say asymptote, I guess. Also, there was something about the "negation" of the imaginary unit. I took the liberty of changing "negation" to "negative". —Preceding unsigned comment added by 24.232.19.210 (talk) 12:39, 21 August 2008 (UTC)

Algebraic geometry
This article doesn't address rational functions as defined in algebraic geometry, as elements of the function field of a variety... anyone care to aa —Preceding unsigned comment added by 99.231.110.182 (talk) 01:05, 15 November 2009 (UTC)


 * I've added a brief paragraph, based on the Springer reference. Charles Matthews (talk) 09:43, 15 November 2009 (UTC)


 * Rational functions are not morphisms to the projective line, since the numerator and the denominator can vanish simultaneously. —Preceding unsigned comment added by 79.119.109.101 (talk) 16:38, 21 June 2010 (UTC)

Clarification sought
I don't understand the meaning of the sentence below (which appears under "Definitions"). Can someone explain?

"... where one assumes that the fraction is written in its lower degree terms, that is, \textstyle P and \textstyle Q have several factors of the positive degree."

Thanks

Lead
The recent edits have clarified the distinction between rational function, rational fraction, and rational expression. This is very welcome, but I'd prefer the lead sentence to define rational function rather than rational fraction:
 * In mathematics, a rational function is any function which can be defined by a rational fraction, i.e. an algebraic fraction such that both the numerator and the denominator are polynomials.

Note that "rational fraction" is not written in bold here since rational fraction does not redirect to this article. Moreover, I'm not sure whether "rational expression" should redirect to this article. Wouldn't algebraic expression be a more straightforward target in analogy with rational fraction redirecting to algebraic fraction? Isheden (talk) 21:24, 1 June 2013 (UTC)


 * I can accept these modifications. My only concern is that the reader that arrive to any of these pages would be provided with definitions that are accurately correct. D.Lazard (talk) 21:42, 1 June 2013 (UTC)

total order
A section is needed on ordering and total orders, since I believe that, for example, the Levi-Civita field is a totally ordered subfield of the rational functions, but I'm not entirely clear on the details. Clearly, the power series $$F^G$$ can be totally ordered whenever F is a field, and G is an abelian group, and the coefficients of the power series are totally ordered. That is, when the series $$\sum_g a_g X^g$$ has the property that the set of non-zero coefficients $$\{a_g\}$$ can be totally ordered; one can then impose a lexicographic order on such series (that is, X behaves as if it were infinitessimal). However, how to extend this to a rational series is not obvious: am I supposed to "divide things out", to get an ordinary series, and then order things?? Is there some other ordering?

Also, the lead to the article surreal number suggests that rational functions can be totally ordered; clicking through to here provides no such discussion.67.198.37.16 (talk) 17:29, 5 July 2016 (UTC)

Ring
I believe there's a mistake in the first paragraph: a rational function can be a fractional of polynomials defined over rings and not only fields. — Preceding unsigned comment added by 79.179.6.85 (talk) 19:32, 8 August 2016 (UTC)
 * No, there is no mistake. Probably some authors generalize the definition for coefficients in a ring, but this is too marginal to be mentioned in the lead. In fact, if the ring of coefficients is an integral domain R, then the field of fractions of the polynomials over R (see the new second paragraph of the lead) is the field of the rational fractions over the field of fractions of R. So, in this case, there is no need for a definition of rational functions over R. For other rings of coefficients, there is no definition, which is widely accepted in the literature. D.Lazard (talk) 08:10, 9 August 2016 (UTC)

Requirement on the denominator?
If
 * $$f(x)=\frac{a_m x^m+a_{m-1} x^{m-1}+\dotsb +a_1x+a_0}{b_n x^n+b_{n-1} x^{n-1}+\dotsb +b_1x+b_0}$$

then, isn't it a requirement that bn is != 0 and n > 0? Svjo (talk) 15:30, 17 November 2016 (UTC)


 * The requirement is that the denominator should not be the zero polynomial. The conditions you mentioned will guarantee that, but they are not necessary since just having $b_{0} ≠ 0$ will also work. --Bill Cherowitzo (talk) 17:39, 17 November 2016 (UTC)
 * If bn = 0, this means that the degree of the denominator is less than n. This is not a requirement, although many reasonings are easier is one supposes that n is the degree of the denominator, that is if one removes the zero terms.
 * If the degree of the denominator is zero, that is if n = 0 and bn = b0 ≠ 0, this means that the denominator is a constant, and that the fraction is a polynomial. Thus, and this is the usual convention, a polynomial is a rational function; similarly, integers are rational numbers. This convention is necessary for the rational functions forming a field. D.Lazard (talk) 17:45, 17 November 2016 (UTC)

Proper rational function
User:D.Lazard you may think that this term has a more common meaning (and I doubted myself at first because I got it from a control systems source, not a maths source) but the first page of gbooks results agree with my definition, and many of those are books on some field of mathematics. So it is for you to provide a source. SpinningSpark 21:11, 24 December 2016 (UTC) ...and it is a bit rich demanding a source for a minor addition that a WP:BEFORE would easily have showm to be correct in an article that is entirely uncited. SpinningSpark 10:57, 25 December 2016 (UTC)

Biquadratic function
The use of biquadratic function in the network theory sense may have first been used by mathematician Ronald M. Foster who did some of the early work in the fied in the 1920s. See this 1963 paper for instance. SpinningSpark 23:07, 9 April 2020 (UTC)