Talk:Algebraic fraction

Problem with definition

I've just recently come across this article (not surprisingly given that it was only just created) and am struck by the final part of the opening sentence "and the dividend is not a multiple of the divisor". The venerable reference given notwithstanding (which I have no access to), such a condition seems to open all kinds of problems, while I can see absolutely no advantages. For one thing what does being a multiple of mean in the general setting of algebraic expressions; does multiplying by an algebraic fraction count as producing a multiple? Also this condition is not part of the definition of rational functions, making it problematic to consider those as a special case. Also such a restriction would make it impossible to state that the sum of two algebraic fractions can be rewritten as an algebraic fraction, not something the article currently states (actually it doesn't state very much, just introduces a lot of terminology) but something that seems desirable nonetheless. Couldn't (and shouldn't) the restriction be simply dropped? Marc van Leeuwen (talk) 13:16, 5 November 2011 (UTC)

You're right that the article does not state much yet. The main motivation for creating a new article in addition to rational function was that there are algebraic fractions that are not quotients of polynomials (known as irrational fractions). When considering the definition proposed in said reference, it occurred to me that there was no satisfactory definition of algebraic expression in the article Expression (mathematics), so I created an article algebraic expression as well. Both articles need futher work. Regarding the issue with the definition in this article: when the dividend is a multiple of the divisor the quotient is an integer, except at the points where the divisor is zero. It is probably a matter of taste whether an expression such as ${\tfrac {2x+4}{x+2}}$ should be regarded as an algebraic fraction or not. I think the restriction could be removed also, especially if some other source defines an algebraic fraction without the restriction. Isheden (talk) 15:38, 5 November 2011 (UTC)

I have now performed a search in the literature and found out that most other sources do not exclude the case where the dividend is a multiple of the divisor. Because of this, I changed the definition and provided a different reference. Isheden (talk) 18:32, 5 November 2011 (UTC)

First line of the lead

The first line of the line uses the wording "indicated quotient". This is a mathematical non sense: "indicated" has no meaning in mathematics, and its meaning here is totally obscure. The article is about "fractions" which are operators in mathematical expressions and are not the same thing as "quotients", which are results of divisions. Thus the lead is mathematically incorrect. I have corrected it and have been reverted. As WP requires mathematically correct definitions, I'll restore my version. D.Lazard (talk) 20:42, 2 March 2013 (UTC)

OK, but please remove the current reference and if possible replace it by one that supports the proposed wording. There has been a lot of discussion on Talk:Fraction (mathematics) in the past regarding the proper definition of the term fraction. Therefore it may be circular to define an algebraic fraction as a "fraction". Isheden (talk) 08:35, 3 March 2013 (UTC)

Specifically, do algebraic expressions form an integral domain and is there a corresponding field of fractions when algebraic fractions are defined this way? Or should algebraic fractions be viewed as a broader concept? Note that many authors use the term to mean rational expressions, in which case the situation is clear. Isheden (talk) 22:18, 3 March 2013 (UTC)

About the main topic of the article: It should be noted that "expression" were never been formally defined in older texts. The need of such a formal definition appeared only with the rise of computer algebra (this assertion is WP:OR). An expression is simply a well formed formula constructed from variables and numbers and a specific list of operators, which, for algebraic expressions are +, -, ×, / and the exponentiation with a rational number exponent (which includes square root). Two expressions are equal only if there are the same. Thus a+b and b+a are different expressions, as well as 2+2 and 4. Algebraic expressions are very difficult to manipulate, because, for example, ${\sqrt {1+{\sqrt {-3}}}}$ does not represent one complex number, but two (or even four) different ones. I do not see any other way of formally defining "algebraic fraction" than as an algebraic expression with the division operator as leading operator. The lack of good properties of algebraic expressions and thus of algebraic fractions, make that you will rarely encounter these terms in modern texts. IMHO, "algebraic fraction" does not deserve its own article and the part of this article about non-rational fractions would better be merged into algebraic expression.

Rational fraction: Things are different for rational fractions, and this is because rational fraction redirects here that I have edited this article: In fact, there are three related notions that should be distinguished and are not always in WP:

• Rational expression: an expression made with +, -, ×, / and integer exponents (apparently not defined in WP, "rational expression" is a redirect to rational function). A rational expression represent a element of the field of fractions of the ring of polynomials. An element of this field may be represented by many rational expressions.
• Rational fraction: a fraction of two polynomials. They constitute the field of rational fractions, which is constructed from the ring of polynomials exactly as the field of rational numbers is constructed from the integers.
• Rational function: the function defined by a rational expression or equivalently by a rational fraction.

The same distinctions occur when talking about polynomials. Thus, formally, $(x+1)(x-1)$ is not a polynomial, but a polynomial expression.

The confusion of WP about these distinctions make difficult to establish convenient links when editing WP articles related to these questions (there are a lot). This confusion also generates a lot of misuse of computer algebra systems like Maple (software) and Mathematica, for which expressions and functions are very different things. D.Lazard (talk) 10:07, 4 March 2013 (UTC)

About citations: You ask for citations. In this case, this is rather difficult, because many authors do not care for such an accuracy in wordings. Those who care for rarely make these distinction explicit, either because they write for experimented readers who are supposed to understand the distinction, or they write for beginners or students, and too much formal definitions would be counterproductive. Moreover, these distinctions, that rely on the distinction between an object ans its possible names (signifier and signified), are clearly non-elementary. Nevertheless, as far as the distinctions are not made explicit, I doubt that you will find any mathematician that would say that above definitions are wrong or even controversial. Thus one may correct inaccurate wordings everywhere in WP without explicit citation. Clearly a relevant citation would be better here (for both definitions of algebraic and rational fractions), and some certainly exist, but it is less urgent than an accurate wording. D.Lazard (talk) 11:40, 4 March 2013 (UTC)

My impression based on some literature search is similar to yours. The term algebraic fraction seems to be used either in modern books, often without definition but actually assuming rational algebraic fractions, or in very old text books that might have ignored the modern definition of a fraction as an element of a field of fractions with the associated good properties. If this is the case, then I agree that the term does not merit an article of its own. Instead, I think "rational fraction" and "algebraic fraction" should redirect to Fraction (mathematics) and rational expression to Algebraic expression. What's your opinion? Isheden (talk) 15:43, 4 March 2013 (UTC)

My opinion is that "algebraic fraction" should be a redirect to "rational fraction". It is not clear if "Rational fraction" should be or not the same article as Rational function. For many people, the distinction is thin. Therefore, in case of two articles the distinction must be clearly explained (hatnotes and interlinking of the two articles). For the moment, Rational function is clearly oriented toward the functional aspect of the question and says nothing about the algebraic properties. In any case a redirect to Fraction (mathematics) does not seem a good idea, because the rational fractions have many important properties that are not a direct consequence of being a field of fractions. Just now I think of partial fractions decomposition (see also partial fractions in integration) and the unicity of the Padé approximants; I think also of Lüroth theorem, asserting that every subfield (containing the field of coefficients) of the field of univariate rational fractions is isomorphic to it; strange that this important theorem has only one line in another article about a different topic. Thus my opinion is that we should tend to two different articles "rational fraction" and "Rational function". This needs to rewrite partially Rational function and to create, almost from scratch, a new article. Having other priorities, I will not do that myself in a near future. This is to manage the transition that I have introduced the definition of "rational fraction" in the lead of this article. D.Lazard (talk) 17:20, 4 March 2013 (UTC)

There is still lots of work to do in this article IMHO. For example, the article says that algebraic fractions are subject to the same laws as arithmetic fractions. Well, not necessarily! Take the sample fraction given in the article: ${\frac {\sqrt {x+2}}{x^{2}-3}}.$. It is *NOT* subject to all laws applicable on arithmetic fractions! An arithmetic fraction would be free of any restrictions. BUT with this one, there are exceptions that must be respected: especially the "gaps" in definition. There is no definite result for $x\in \mathbb {R}$ and $x<-2$ for the term's numerator; nor any definite result for $x\in \mathbb {R}$ and $|x|={\sqrt {3}}$ for its denominator. That's why I'd take the expression "subject to all laws" with a pinch of salt... maybe we had better restrict it to "subject to all laws within its domain". Since the domain of an algebraic fraction may or may not only be a subset of the domain of an arithmetic fraction. -andy 77.190.37.140 (talk) 18:47, 16 March 2013 (UTC)

As far as I understand, you objections concern the functions that are defined by the fractions. But the article is about expressions, which involve indeterminates, like x, that have not and are not supposed to have any numerical value. Such an expression defines a function (and even several functions, depending the range and the target of the function, for example real or complex). All your concerns are about functions, not expressions. D.Lazard (talk) 19:03, 16 March 2013 (UTC)

Multiplication and division of algebraic fractions

remember the rules of operation with basic fractions.

For multiplication: . Cancel factors common to the numerator and denominator. . Multiply the remaining numerators and denominators For Division

. x+3 /4 ÷ x+3/6

= 3/4 × 6/3

1/2 × 3/1 =3/2 Menqe (talk) 12:32, 5 July 2023 (UTC)