Talk:Fraction/Archive 2

Uruiamme's edit
Sorry, Uruiamme, but I can't make heads or tails of your recent edit, and am reverting it. You write, " Another way of describing a fraction is by saying that "numerous parts denominate the whole,"" I have no idea what this means, nor who you are quoting with the words inside quotes. There are similar problems with your other edits. I suggest discussing things here before making changes in the article. Rick Norwood (talk) 12:52, 30 October 2011 (UTC)


 * I fixed a number of different errors in the article. Do not revert what is indisputably better. I have boiled down what a fraction is in the first paragraph, introducing the fact that it is made up of some key components. Also, why revert the edits to the word usage of "indicating" for "telling"?


 * As to "numerous parts denominate the whole," which word do you not understand? Each of these words appear in the definition of different parts of a fraction, albeit in a different form.
 * numerous -> number. From "any number of equal parts"
 * parts --> parts. From "any number of equal parts" and "A fraction represents a part of a whole."
 * denominate --> denominator. From "the denominator indicating how many of those parts make up a whole"
 * the whole. --> whole. From "A fraction represents a part of a whole."
 * This sentence therefore succinctly abbreviates what a fraction is, by its definition, and serves as a potent mnemonic for its composition or structure. You now have an idea what this sentence means, so I am sure you can see its usefulness.


 * I fixed an erroneous 3/4 = 3:4 ratio. If I have 3 parts salt and one part water, it is 3/4 salt. It is also 3:1 salt to water, not 3:4. Um, correct me if I am wrong, but this was an egregious concept error in the body which I fixed without comment up till now. I don't want to be at odds (q.v.) to the subject of ratios or of fractions.


 * I also improved a part of the article regarding the pronunciation of 1/4. It has at least two pronunciations and multiple spellings. These facts are not obvious to all readers of the article (i.e. quarter means fourth). I tied the pie picture into the body when 3/4 was being discussed, as there is 3/4 shown. This is not an edit for reverting, as there is no controversy as to the content, it is not garbage characters, and not POV or something. Save your revert button for the noobs. I like to saw logs! (talk) 06:25, 1 November 2011 (UTC)


 * I did not remove your edits, but I did tag parts. Your mnemonic is interesting, but it is unsourced and thus questionable that it belongs in the article at all, much less in the first paragraph. An encyclopedia is not a place to publish original research. It is also unsourced that the terms numerator and denominator are derived from that mnemonic, as your edits seemed to affirm. --JimWae (talk) 06:35, 1 November 2011 (UTC)
 * One thing we do need to determine is if a ratio can only be part to part, or can also be part to whole--JimWae (talk) 06:38, 1 November 2011 (UTC)

Uruiamme: If your edit was "indisputably better" I would not have disputed that it was better. Very well, I'll do a line edit instead of reverting. I've left the parts of your addition to the caption of the picture that made sense, though I still think they were unnecessary. Putting the Latin word in italics is good. And I've left your use of "indicating" instead of "telling", since it really doesn't matter. I agree that a fraction is not a ratio. Rick Norwood (talk) 13:56, 1 November 2011 (UTC)

JimWae: A ratio compares two or more numbers. You can compare part to part. You can compare part to whole. You can compare one whole to another whole. But a ratio is not a fraction.

Rick Norwood (talk) 14:16, 1 November 2011 (UTC)


 * That is what I thought, thus 3/4 does not have to express a ratio of 3:1 (as Uruiamme had "corrected" the text). I will point out though, per the old lede, that while a ratio is not a fraction, it is quite a different thing to say that a fraction can express a ratio. Given that fractions can be irrational though, I am not going to argue for the return of ratio to the def. I do still maintain that the lede needs to say that a fraction can be understood as a number and as a numeral.--JimWae (talk) 17:40, 1 November 2011 (UTC)

The Lead
Ok, then, here is the lead you reverted. Would you state your objections to this lead:

A fraction (from fractus, "broken") represents a part of a whole or, more generally, any number of equal parts.

The earliest fractions were reciprocals of integers: ancient symbols representing one part of two, one part of three, one part of four, and so on. A much later development were the common or "vulgar" fractions, still used today, such as $1/2$, $5/8$, $3/4$, etc., which consist of a numerator and a denominator&mdash;the numerator representing a number of equal parts and the denominator telling how many of those parts make up a whole. An example is $3/4$, in which the numerator, 3, tells us that the fraction represents 3 equal parts, and the denominator, 4, tells us that 4 parts make up a whole.

A still later development was the decimal fraction, now called simply a decimal, in which the denominator is a power of ten, determined by the number of digits to the right of a decimal separator, the appearance of which (e.g., a period, a raised period (•), a comma) depends on the locale (for examples, see decimal separator). Thus for 0.75 the numerator is 75 and the denominator is 10 to the second power, viz. 100, because there are two digits to the right of the decimal separator.

A third kind of fraction still in common use is the percentage, in which the denominator is always 100. Thus 75% means $75/100$.

Fractional numbers can also be expressed using negative exponents, such as 10-1 which represents $1/undefined$, 7&times;5-2 which represents $7/25$, and 6.023&times;10-7 which represents 0.0000006023.

Other uses for fractions are to represent ratios, and to represent division. Thus the fraction $3/4$ is also used to represent the ratio 3:4 (three to four) and the division 3 ÷ 4 (three divided by four).

In mathematics, the set of all numbers which can be expressed as a fraction $m/n$, where m and n are integers and n is not zero, is called the set of rational numbers. This set is represented by the symbol Q. Sometimes the word fraction is also applied to numbers that are not rational when they have the same form as a fraction, for example $√2/2$ (see Square root of 2) and $&pi;/2$ (see Proof that π is irrational). Rick Norwood (talk) 14:08, 4 October 2011 (UTC)


 * Is this question directed at me? (And would you like to go back and sign your post?)  My objection is principally to the first sentence, as already explained above.  On reflection, I think the best option is to start the article simply with "A fraction (from Latin: fractus, "broken") is a number that can be expressed as the ratio of two numbers".   The issue of comparison of parts to a whole can be mentioned later.  I also rephrased the last sentence more concisely and removed the commas in the penultimate sentence (the fact that m and n are integers constitutes an essential clause as described at Dependent clause), but these things aren't a big deal. Jowa fan (talk) 00:24, 4 October 2011 (UTC)

Sorry about forgetting to sign my post. I've signed it now.

There are two problems with defining a fraction as a ratio, both of which I've mentioned before. The first, and most important, is that a fraction isn't a ratio. The second is that the word "fraction" is more familiar than the word "ratio", and familiar words should not be defined in terms of less familiar words.

Your other changes are fine, and I'll incorporate them. I'm going to put everything except the first sentence into the article, and we can continue to discuss the first sentence. Rick Norwood (talk) 14:08, 4 October 2011 (UTC)


 * As I've suggested several times previously, go and ask someone non-technical (and non-mathematical), whether they can guess what word is described as "a number that can be expressed as the ratio of two numbers". While fractions can be expressed as ratios, it is not obvious that the ratio 3:4 is represented by the fractions 3/7 and 4/7. Using the technical term "ratio" is bad enough, but expected people to understand the relationship with fractions is something that should be relegated to a section later in the article. --Iantresman (talk) 15:04, 4 October 2011 (UTC)

In the articles of Wikipedia and its talk pages, sometimes an adjective preceeds the noun “fraction” to compose one of the following expressions: “algebraic fraction”, “binary fraction”, “common fraction”, “continued fraction”, “decimal fraction”, “equivalent fractions”, “improper fraction”, “irrational fraction”, “numerical fraction”, “partial fraction”, “proper fraction”, “simple fraction”, “vulgar fraction”. For example, to define an algebraic fraction, I should not begin with something like “an algebraic fraction is a fraction…”  First I should give the context of use of the expression “algebraic fraction”.

Often the context is obvious when we are talking. For example, when everyone in an assembly knows that the context is the rational functions, then the word “fraction” will be pronounced without disturbing the minds. But we are reading and writing. In the title “Fraction_(mathematics)”, the word “mathematics” place us in a very wide context. And there are many disputes about the meaning of an expression, and how to write this or that.

Most of the messages of this talk page are relative to problems of terminology. Below I give you a few passages. “These terms are not equivalent at all. Rational numbers are elements of the field of rational numbers, they are numbers. Fractions are expressions…”
 * At 18:10, 7 April 2005 “I disagree. One often hears talk of irrational fractions even sometimes amongst mathematicians.”
 * At 18:03, 26 May 2005 “Concerning Revolvers last edit, I think it was a good idea to distinguish clearly between numerical fractions (which are just complicated ways of writing a simple number) and the other quite different objects: rational functions, partial fractions.”
 * At 14:59, 28 October 2005 “The definitions of proper and improper fractions do not correspond to common usage. Specifically, a fraction equal to one is considered improper.  Ref http://mathworld.wolfram.com/ProperFraction.html, for example.”
 * At 20:35, 27 December 2005 “I agree with Histrion that Vulger fractions should merge here. The term is now archaic.”
 * At 03:15, 26 May 2006 “I've seen the usage of the word "fraction" to mean "a value less than one" (which includes decimal values).”
 * At 07:16, 4 July 2007 “A vulgar fraction (or common fraction) is a rational number written as one integer (the numerator) divided by a non-zero integer (the denominator)”
 * Revision as of 15:49, 13 January 2009 “Which of the following are NOT vulgar fractions?”
 * At 15:23, 16 March 2010 “So I would assume you think all instaces of the word "half" should be replaced by "one second"?”
 * At 11:29, 11 January 2011

In the last passage, I agree with the necessary distinction between a fraction and a number represented by a fraction. The context may be given in one word:  arithmetic.  There are neither irrational numbers, nor complex numbers in arithmetic. Notice that the Wiktionary defines “fraction” in , because the ordinary context is “the field of rational numbers”, as said above. “Rational numbers”,  that expression is not for kids.

Either someone creates a new page entitled "Fraction (arithmetic)", or this lead section clearly defines the word "fraction" in , because it makes sense to go from simple to more complicated. The four operations of arithmetic are addition, subtraction, multiplication, division. No square root, for example.

No adjective before the noun in the title, no preposition “of” after the noun. So the first sentence of the article must define the single and isolated word “fraction”. Actually, a discussion about the introduction is at the same time a discussion about the content and the plan of the article. Remember what says Wikipedia:  “the appropriate length of the lead section depends on the total length of the article. As a general guideline, the lead should be no longer than four paragraphs.”

If a new section is entitled “Algebraic fraction”, certainly the section should contain a link to the page “Rational function”. If “Continued fraction” is the title of a new section, certainly the section should contain a link to the article “Continued fraction”. The best rational  approximations of a real number are rational numbers, while a value of an algebraic fraction is not necessarly a rational number. "Etymology and history" should be the title of a new section, for a few people interested by the fractions of which the numerator is less than the denominator, for example.

"Fraction of a quantity" will be probably the title of a new section. No idea of calculation of any quantity of cake in the introductory image, sure. That image should announce the theme of other images, that help understand calculus with fractions, 2 + 3&thinsp;/&thinsp;4 for example. A theme is a brackground of an article, like some wallpapers in another article — I give an example at random, of course —.

That article certainly will be soon consistent from all points of view. Remember that the current article defines a fraction as any ratio that represents any number, so why not a redirection to the page "Ratio"? Remember what I wrote in the article:  “a fraction (from fractus, "broken") is a writing:  a ratio of two integers that represents a number.”  And nobody convinced me that my “suggestion 2”  is so bad: “a fraction is a mathematical writing:  a ratio of two integers that represents their quotient. For example, the fraction 3&thinsp;/&thinsp;4 equals  0.75. The line that separates the integers is often horizontal:  a vinculum.  The two integers are called the numerator and denominator of the fraction.”

Thank you for your attention to my message. And I wish that nobody touches up my anterior messages, to oblige me to verify again the meaning of what I wrote. — Aughost (talk) 18:11, 4 October 2011 (UTC)


 * I think it is a very good idea to limit the definition of fraction to the meaning in arithmetics. However, for several reasons it is not appropriate to define a fraction in terms of a ratio. For example, if you have 2 apples and 3 oranges, the fraction of apples and oranges to the total pieces of fruit is $$\frac{2}{5}$$ and $$\frac{3}{5}$$, respectively, whereas the ratio of apples to oranges is 2:3, which has the same value as the fraction $$\frac{2}{3}$$. I would suggest instead a lead along the following line, based on the definition in a book by George Lees entitled "Elements of arithmetic, algebra and geometry": In arithmetic, a fraction is a quantity which represents a part or parts of a unit or a whole. A fraction is expressed by two terms, called the numerator and the denominator. The denominator indicates the number of equal parts into which the whole is divided, and the numerator indicates the number of these equal parts contained in the fraction. A fraction may be considered as the quotient arising from the division of the numerator by the denominator. Isheden (talk) 22:10, 4 October 2011 (UTC)


 * Tell me if I understand or not your example with 2&thinsp;/&thinsp;5 and  3&thinsp;/&thinsp;5. You are talking about proportions (fractions)   fruits that are apples and oranges.  In that example, “two fifths   five fruits” represents the number of fruits that are apples.  In mathematics, the symbol × replaces the preposition “of”, and we write:  2&thinsp;/&thinsp;5 × 5 = 2 apples. If you like that example, it might be exposed in a section entitled "About fractions of quantities".  Now I quote a definition of “proportion”  in the Wiktionary:  “a quantity of something that is part of the whole amount or number.”  To be accurate, “a fraction   a given quantity” is not a mathematical writing called a ratio.  What do you think about these details? — Aughost (talk) 04:46, 5 October 2011 (UTC)
 * I think we should not use the words "a writing" when discussing fraction because it is not a commonly understood english phrase. Cliff (talk) 05:23, 5 October 2011 (UTC)
 * My point is that the related concepts of ratio, proportion, and fraction should not be mixed up. The basic meaning of fraction is a part of a whole. You are right that the preposition of corresponds to multiplication. In this case, the fraction of apples is 2/5, which can also be seen as a compound fraction (fraction of a fraction) $$\frac{2}{5} \times \frac{1}{1}$$. But it is simpler to say "the fraction of apples is 2/5". The term ratio is broader than the term fraction. For example, ratios can be part-part comparisons. Thus, in this sample the ratio of apples to pears is 2:0, which cannot be expressed as a fraction because the denominator in a fraction must not be 0. You could also say the quantities of apples, oranges and pears are as 2:3:0. As another example, you can without doubt say that the ratio of circumference to radius of a circle is $$2\pi:1$$, but it is unclear if $$\frac{2\pi}{1}$$ is a fraction since the quotient is irrational. I also agree with user:Rick Norwood that we should not define the more common word fraction in terms of the less known word ratio. Presently, the distinction between the three concepts ratio, proportion, and fraction are discussed in the article Ratio. Isheden (talk) 09:37, 5 October 2011 (UTC)


 * Nobody has advocated saying "all ratios are fractions", nor that "fractions are any ratio".--JimWae (talk) 10:32, 5 October 2011 (UTC)


 * Isheden wrote: "The basic meaning of fraction is a part of a whole."
 * A great description. No technical jargon, no need to define other words. Since this is the basic meaning, that is the definition we should describe first, and then go on to describing how fractions are written, different kinds of fractions, etc. --Iantresman (talk) 11:11, 5 October 2011 (UTC)

I think we have a consensus to avoid the word "ratio" in the first sentence, so I'm going to make that change. Rick Norwood (talk) 14:12, 5 October 2011 (UTC)


 * I would agree with that, and then we have to ensure that the points made by Aughost are addressed subsequently in the article. --Iantresman (talk) 15:12, 5 October 2011 (UTC)

The same word may denote several different things. For example, the entry “proportion”  in the Wiktionary gives to the word different meanings: an equality between two ratios, or a part of a quantity. A word is not a concept. We must explain different meanings in different contexts, but all the explanations will not be in the introduction, of course.

The abuses of language are even more common in spoken language than in written language. It would be crazy to want an entire article without abuse of language. But it is reasonable to avoid any abuse of language in the first sentence of the article, because of all the disputes on the talk page over the years. Like a figure is not a number, a fraction is not a rational number. The following sentence might announce a content of the article somewhere in its introduction:  “a fraction is a way to write a rational number”.

We will not write this example in the introduction, sure:  there are fruits that are apples and oranges, their number is one hundred and twenty, the proportion (fraction) of apples is 2&thinsp;/&thinsp;5 or  40%. Different mathematical writings may denote the same value, hence the symbol =. For example, the number of apples is: 2&thinsp;/&thinsp;5 × 120 = 40&thinsp;/&thinsp;100 × 120 = 0.40 × 120 = 48 apples.

I have read the last changes of the article, is it a joke?  We are not in mathematics with that first sentence.  Maybe Rick Norwood does not read all the disputes on the talk page over the years. Aughost (talk) 17:32, 5 October 2011 (UTC)


 * Aughost, Why do you think that in order to be discussing mathematics we must use terms like ratio, integer, rational number, etc? What intro would be both accessible to persons not familiar with mathematics, and yet also be "in mathematics"? I don't think it can be done to be "in mathematics" as you say, and yet also remain understandable to those who do not know mathematics. We have to abandon one of these. I say that we should abandon being "in mathematics", at least for the lead. Cliff (talk) 19:41, 5 October 2011 (UTC)
 * Someone that is scared by the only word “mathematics”, that is confronted to calculation exercises such as calculating the number of apples, above, will be happy if the article is an efficient help, and will be reconciled a little with mathematics. Aughost (talk) 20:22, 5 October 2011 (UTC)


 * Correct, the first sentence is deliberately not mathematical, because the word "fraction" also has a more general, non-mathematical meaning. It makes sense that we describe the more general definition first, in a non-technical way as suggested by the "Wikipedia manual of style for mathematics: Introduction" which suggests that we start with:
 * "A general approach is to start simple, then move toward more abstract and technical statements as the article proceeds."
 * "The lead should as far as possible be accessible to a general reader, so specialized terminology and symbols should be avoided as much as possible"
 * (My bold). We do not exclude the complicated, more rigorous, more technical and mathematical definition, but save it for later, because the average person will not understand it. --Iantresman (talk) 21:28, 5 October 2011 (UTC)You understand yourself, sure,  and no doubt that some friends are understanding you.  Wrongly  you believe that the current first sentence  is free of ambiguity for the “average person”,  because of your current fixed idea:  some proportions of some quantities, expressed by  .— Aughost (talk) 05:23, 6 October 2011 (UTC)

Aughost: There is no need to highlight your posts. It is a form of shouting. Also, you are becoming rude. Please confine your comments to the subject at hand, and do not tell other people what they "believe" or that they have a "fixed idea". Rick Norwood (talk) 12:04, 6 October 2011 (UTC)


 * Aughost, of course the simpler sentence is more ambiguous, because it is more general. It is not a mathematical definition, and is not intended to be. As I have suggested previously, take your proposed definitions, and ask some non-technical people, what they understand by it. Your definitions are more rigorous, and more technical, and more than welcome, later in the article. --Iantresman (talk) 14:21, 6 October 2011 (UTC)


 * By forgetting the present message, I have counted 19 instances of the expression “part of a whole” in this page, in the period from 23:59, 20 March 2011 (UTC)  to my message today, including one instance below in the section "Aughost's concerns".  How can you ignore the problems with that expression “part of a whole”? — Aughost (talk) 09:37, 7 October 2011 (UTC)


 * I accept your criticism regarding "part of a whole", it can be re-worded. My criticism is just that the introductory sentence should not include any technical jargon, per "Wikipedia manual of style for mathematics: Introduction" --Iantresman (talk) 12:07, 7 October 2011 (UTC)


 * For example at 15:09, 24 August 2005,  you replace  the word “quotient” by an expression that discourages any normal human being.  Is it so difficult to write  0.75 × 4 = 3, for example, to explain that 0.75  equals the ratio  3&thinsp;/&thinsp;4&thinsp;? The word “quotient” belongs to elementary arithmetic, why so much effort to avoid it? — Aughost (talk) 11:16, 8 October 2011 (UTC)


 * Since neither formulation is current, why are you reopening an issue that is now six years out of date? Rick Norwood (talk) 12:13, 8 October 2011 (UTC)

I don't think anyone is ignoring problems. The original and most common meaning of fraction is "part of a whole". In mathematics, that meaning has been extended, but outside mathematics that is still the most common meaning. The article, in the first sentence, says that there are extensions of that meaning. For example, 2 can be written as a fraction, 6/3. But I don't think anyone would say that 2 "is" a fraction. Mathematicians would say that 2 is a rational number, but wouldn't say that 2 is a fraction. Everyone would agree that 1/2 is a fraction. Thus, we give the more common usage before the less common, but do not neglect to mention all mathematical uses, and have disambiguation for non-mathematical uses of the word. Rick Norwood (talk) 12:11, 7 October 2011 (UTC)

After reading the section below entitled "Aughost's concerns",  I imagine a possible solution. Confronted with the paralysis of some pages during several years, a solution could be presentations of some appraisals, with statistics about some signatures. For example, assume that a contributor is named "Sympathetic", assume that he destroyed systematically the sentences that contain the word "quotient", assume that to replace these sentences, the Sympathetic’s expressions are so incredible that the average author is right away discouraged, so it does not matter neither that the pseudonym is "Sympathetic", nor that the contributor is aware or not of consequences of his actions, we could deduce from statistics that the user "Sympathetic" really prevents from improve Wikipedia.

At 20:52, 21 January 2011, Cliff suggested to merge  the page "Fraction_(mathematics)" into the page "Rational_number", when the introductory sentence was:  “A fraction (…) is a number that can represent part of a whole.” Notice that the current beginning  of the introductory sentence is a little better:  “A fraction (…) represents a part of a whole”, because there is immediately the verb “represent”. Yet most of pupils in elementary school don’t understand the expression “part of a whole”, because what is taught in elementary school is a set of rules to write equalities in arithmetic, with fractions.

In the present section I read above:  “I think we should not use the words "a writing" when discussing fraction because it is not a commonly understood english phrase.”  The date of that message is “5 October 2011”, the author is Cliff.  The initial cause of that message is my changes in the introductory paragraph  of the article:  “A fraction (…) is a writing…”, as explained in the section "In_order_to_define_correctly_the_word_"fraction".  Maybe someone will soon propose to merge the page "Fraction_(mathematics)" into the page "Rational_number", and nobody will dare explain why the proposal is bad, because the word “writing” was strictly forbidden by Cliff.

We read in the "Manual_of_Style":  “in general, specialized terminology and symbols should be avoided in an introduction. Mathematical equations and formulas should not be used except in mathematics articles.”  I insist:  “except in mathematics articles.”  Many people can understand that two statements like “0.75 = 3&thinsp;/&thinsp;4” and “0.75 × 4 = 3” are equivalent. — Aughost (talk) 20:15, 10 October 2011 (UTC)


 * The Wikipedia Manual of Style indeed mentions that "Mathematical equations and formulas should not be used except in mathematics articles". It also tells us "specialized terminology and symbols should be avoided in an introduction" (my emphasis). I agree. We should provide a general non-technical non-mathematical description of fractions first, and then gradually introduce a mathematical treatment. --Iantresman (talk) 21:43, 10 October 2011 (UTC)


 * You understand the recommendation, sure.  The symbol =  is elementary, it will be present in the article.  Put the equals sign in the introductory sentence will also indicate that an elementary symbol is not a scare to anyone. — Aughost (talk) 03:42, 11 October 2011 (UTC)


 * In the current introductory sentence,  “part of a whole” is a phrase in jargon of pedagogy, most of kids don't understand when they read the expression.  Like me, many people are children who have grown up.  Like me, many people have several personalities.  When I am a modest mathematician, after reading the current introduction I could ask you, as a joke:  the ratio of volumes of three quarters and the whole cake may it be rational, if the ratio of their masses is rational?  Remember that the three other paragraphs of the introduction will announce the content of the article, certainly with the adjective “rational” somewhere. — Aughost (talk) 15:33, 11 October 2011 (UTC)


 * Aughost, Please do not assume my intentions when I requested a merge, I resent your claiming to know my intentions and using these assumed intentions to make a point for yourself (a point with which I disagree). It is rude. Also, others have asked you to please stop highlighting your posts. We can read what you write, and we will. You do not need to bring extra attention to your words. Further, the word "write" in english is a verb. "writing" is the tense that indicates that an action is currently happening. "A writing" indicates that "writing" is a noun, which it is not. That is why we should not use that term, simply because it does not make sense. Cliff (talk) 18:51, 11 October 2011 (UTC)


 * In the entry “scripture”  of the Wiktionary, we read:  “a sacred writing or book”.  I am not saying that a fraction is a book, of course. — Aughost (talk) 19:30, 11 October 2011 (UTC)

A verb used as a noun is a gerund. HOWEVER, fractions are not merely symbols/numerals. They also ARE something - that part of a number that is not an integer (ex: 0.333 and 0.333 in 1.333). Numbers do not "represent" anything - rather numerals (and numeric expressions) represent numbers. Written nnumerals and spoken words can refer to fractions - which are numbers (and do not *represent* anything else). Even if we do not assume that the reader can explain the difference between a number and a numeral, we ought not misrepresent numbers AS numerals. That difference is usually introduced in grade school. This is not simple wiki and we are permitted to assume roughly a grade 9 intelligence. There is no reason NOT to use numeral and number when defining fraction - to cover the dual meaning that makes 9/1 and 0/1 fractions as numerals but not as numbers. --JimWae (talk) 19:53, 11 October 2011 (UTC)


 * Some messages are diversions. — Aughost (talk) 07:30, 13 October 2011 (UTC)


 * After several years of disputes to define the topic of the article, after many inappropriate answers, now it seems that JimWae criticises a definition of the title word: “fraction”.  Apparently Rick Norwood is informed on that definition, since Rick Norwood creates the section entitled "JimWae's_edit"  to congratulate JimWae.  I don't know what is that definition of fraction.  The difficulty to understand this talk page is obvious, because of some messages that are deliberately dispersed, for example. — Aughost (talk) 15:34, 14 October 2011 (UTC)


 * I think Rick Norwood was referring to JimWae's edit in the section "Comparing fractions" of the article, which was done around two hours later. Isheden (talk) 18:52, 14 October 2011 (UTC)


 * I think that writing used as Aughost has done is actually a deverbal noun, not a gerund. Either way I don't think it sounds good and think it should be avoided. Cliff (talk) 03:02, 20 October 2011 (UTC)

Why “a writing”  instead of “an expression”?  In order to be immediately understood by most of kids. It would be easy to replace “a writing” with “an expression” in my "suggestion 2",  for example. Yet I read Cliff's message above, with the question:  "why do you think that in order to be discussing mathematics we must use terms like ratio, integer, rational number, etc?" So I am very surprised. And I doubt that a contributor like Cliff is working to improve "Fraction_(mathematics)".

The current first sentence of the article is obscure:  “A fraction (from Latin: fractus, "broken") represents a part of a whole or, more generally, any number of equal parts.”  We read 1/4 within with character /  that corresponds to ÷  on a key of many calculators. “Quotient” is it a swear word? The two sentences below could be the start of the first paragraph of the introduction.

Suggestion 3 In arithmetic, a fraction is an expression that represents a «quotient» of two integers. For example,  0.75 × 4 = 3, so fraction  3&thinsp;/&thinsp;4  represents 0.75, that is the quotient of 3 by 4.

— Aughost (talk) 10:27, 20 October 2011 (UTC)

I think your Suggestion 3 is good, later on on the introduction. I think the very first description of fraction should (a) describe the general use of fractions (eg. I blink in a fraction of a second) (b) A non-mathematical, non-technical description, with no technical terms. Then we should describe the more specific mathematical (arithmetical) use of the word fraction, noting the many people may not understand what a "quotient" is, and perhaps we should define it too, before we use it.

I recall that the Mathematical manual of style (which I have mention several times now), tells us that "specialized terminology and symbols should be avoided in an introduction". The best way to check whether you have used specialize terminology (which I have also mentioned several times), is to ask a couple of non-technical people whether they understand the description. When I did this, they ALL said that they did not understand the word "quotient", although one of them thought they had heard of it. --Iantresman (talk) 11:24, 20 October 2011 (UTC)


 * I have difficulty working with editors who assume others edit in bad faith. I feel attacked and will refrain from discussion involving this editor to refrain from attacking defensively. Good day to all. Cliff (talk) 14:27, 20 October 2011 (UTC)


 * Iantresman repeats himself and he forgets that I have already answered.  His truncation of the recommendation  is tendentious when he forgets the end of the second sentence of the second paragraph:  “except in mathematics articles”.  When he says that he blinks in a flash, he is not in mathematics.  Yet we must write below the title "Fraction_(mathematics)". — Aughost (talk) 14:54, 20 October 2011 (UTC)


 * Correct. We can use "specialized terminology and symbols" in mathematical articles, but they "should be avoided in an introduction" (Wikipedia manual of style). There is a better guide that confirms this, the "Wikipedia:Manual of Style/Mathematics:Article Introduction" which is a the style guide specifically for maths. The end of the first paragraph reads: "The lead should as far as possible be accessible to a general reader, so specialized terminology and symbols should be avoided as much as possible." (my emphasis). I repeat myself because the Manual of Style for mathematics suggests that technical jargon should be avoided from the introduction.
 * "Blinks in a flash" is a metaphor. I did not suggest this. When I blink in a fraction of a second, I am indeed using a concept which can best be described as mathematical. It does not require that we offer only a mathematical treatment. Indeed, we MUST provide a mathematical treatment, I FULLY support that we provide a mathematical treatment, that we use as much jargon, technical language, and equations as you like... just not in the introduction, otherwise we lose the general audience.
 * The introduction should be aimed at children, they should be able to understand the description as well as anyone else. --Iantresman (talk) 20:04, 20 October 2011 (UTC)

After my "suggestion 3", Iantresman began with these words:  “I think your Suggestion 3 is good”. And I was very glad, I am still very glad. Later, he wrote:  “the introduction should be aimed at children, they should be able to understand the description as well as anyone else.”  Since everybody can understand something that children understand, Iantresman and me can conclude:  my "suggestion 3"  would be better by replacing “an expression” with “a writing”.

However, Ianstresman thinks that specialized terminology and symbols should be avoided in an introduction, according to the recommendations of Wikipedia. So I have to explain the meaning of any title. When the title is "Manual_of_Style/Lead_section",  the whole page below the title is relative to the introduction of an article. So Ianstresman makes a mistake when he construes this passage:  “mathematical equations and formulas should not be used except in mathematics articles.”  We may write equations or formulas in the introduction of an article that deals with mathematics.

“I blink in a fraction of a second” is a bad example, because the exact duration of the action does not matter in the ordinary meaning of that phrase.

— Aughost (talk) 08:55, 23 October 2011 (UTC)

Aughost, firstly, can I respectfully request that your indent your replies in the standard way (see Guidelines here), and without adding colourful horizontal border which give the impression of demarkating the thread.

We obviously disagree on the Manual of Style/Lead section. This is why I brought up the Wikipedia:Manual of Style/Mathematics:Article Introduction. It confirms: "The lead should as far as possible be accessible to a general reader, so specialized terminology and symbols should be avoided as much as possible." This is not just what I think, but the guidelines for the whole maths project.

This is why I do not support your suggestion 3 as the introductory sentence. I have already asked my family (which includes a 17-year-old who has taken A-level maths), and they do not understand the terminology "expression", "quotient", and other family members were not sure about "integer".

I also do not think that replacing "expression" with "writing" solves the problem. Fractions are not always written. The concept of splitting a cake into fractions does not requires a writing system. But the way we writen fractions is very important in maths, and MUST be described after the introduction.

Regarding "blinking in a fraction of a second" being a bad example, because the exact duration of the action does not matter", should not matter. A fraction is a fraction regardless of the duration. The concept is identical. --Iantresman (talk) 12:19, 23 October 2011 (UTC)


 * This article is about the mathematical usage of the term fraction, not the arithmetical
 * restricting the definition to integer numerators and denominators is not comprehensive - not encyclopedic
 * terms taught in basic arithmetic (before algebra) are not "specialized terminology". It is simply not possible to define every term with simpler terms. The following terms are some that are not among specialized terms: numerator, denominator, number, numeral, ratio, quotient
 * far too many posts here are directed at individuals rather than the content of their posts
 * there are TWO meaning of "fraction" - one as applied to numbers and one as applied to numerals/symbols. A comprehensive article needs to present BOTH in the first paragraph.
 * This is not Simple Wikipedia - the lede needs to be both comprehensive and technically accurate
 * The present lede is insufficient, but Suggestion 3 (above) is worse
 * --JimWae (talk) 19:25, 20 October 2011 (UTC)

JimWae’s message is a list of seven affirmations, where some sentences are negative, for example:  «terms taught in basic arithmetic (before algebra) are not "specialized terminology"». And I try find which contribution or which message corresponds to each assertion, except in the last point:  “the present lede is insufficient, but Suggestion 3 (above) is worse.”

Assume that the very first words of "suggestion 3"  are bad in JimWae’s opinion, “in arithmetic” would be a fault according to his first affirmation:  “this article is about the mathematical usage of the term fraction, not the arithmetical”. So I reply to JimWae by quoting anterior passages: Isheden started his answer to this anterior message  by this sentence:  “I think it is a very good idea to limit the definition of fraction to the meaning in arithmetics.”  Moreover, remember the following page titles:  "Algebraic_fraction",  "Binary_fraction",  "Continued_fraction",  etc. Several links toward that pages might appear in "Fraction_(mathematics)", of which the main topic will be:  fractions in arithmetic. Why? The ordinary context of the title word is arithmetic. How to prove it?
 * “The more complicated we make the lead, the worse it gets” (Rick Norwood)
 * “We need to make the lead simpler, not more complicated” (Rick Norwood)
 * “it makes sense to go from simple to more complicated” (Aughost — I am Aughost —)

In this talk page, some people complain about the use of outdated terms like “vulgar fraction” or “common fraction”, that obscures the plain meaning of "fraction". For example, Cliff wrote this message, January 10, 2011:  «I am disturbed by the fact that wikipedia continues to use the term “vulgar fraction.” Until today I never heard the term, and am surprised to see it so vigorously defended except in a history section. I have a masters degree in mathematics and live in the United States. Is it just the states that have stopped using this term? Just my state? Is this term still used or is it a term referenced from the 1940s?»  The trick of outdated terms cannot be perpetual, please JimWae, could you stop?

The current introduction is not in conformity with the recommendations of Wikipedia, because there are more than four paragraphs, for example. The current introduction does not announce the content of the page, and its beginning is wrong. So the current introduction is worse than insufficient. How the two sentences of "suggestion 3" could be worse than the current introduction, I don't know, I cannot imagine.

For example, a price equals  120 / 100 of its previous value after a 20% increase. A writing like 120&thinsp;/&thinsp;100 is it a fraction? Yes. But saying that 120&thinsp;/&thinsp;100 “represents a part of a whole” is meaningless. Therefore, the current first sentence  of the article is wrong.

With “a writing” instead of “an expression”, the "suggestion 3"  is immediately understood by most of kids. Therefore, most of people can understand this beginning of the article:“In arithmetic, a fraction is a writing that represents a «quotient» of two integers. For example,  0.75 × 4 = 3, so fraction  3&thinsp;/&thinsp;4  represents 0.75, that is the quotient of 3 by 4.”

— Aughost (talk) 10:27, 23 October 2011 (UTC)

What a waste of time.......... --JimWae (talk) 06:25, 29 October 2011 (UTC)
 * The above comment was inserted after an edit war over the use of the wall around a Aughost's text. It was also due, among other things, to his responding to my comments by copying the comments of other people from previous discussions, rather than addressing my points directly. --JimWae (talk) 20:25, 8 November 2011 (UTC)

What a waste of time..........--JimWae (talk) 21:12, 3 November 2011 (UTC)
 * The above comment was inserted after Aughost repeatedly removed my comments from this section--JimWae (talk) 20:25, 8 November 2011 (UTC)

Alarming discourse

 * Proposal. Any section with the bar edits gets immediately archived.  No one can read it, anyway, with all those nowraps which are more likely to crash the small page width which Aughost is worried about than to keep concepts together.  — Arthur Rubin  (talk) 07:25, 29 October 2011 (UTC)


 * Among many ways to cheat, see here. — Aughost (talk) 11:15, 3 November 2011 (UTC)

What a waste of time..........--JimWae (talk) 21:12, 3 November 2011 (UTC)
 * The above comment was inserted after Aughost repeatedly removed my comments to a new section rather than to this section--JimWae (talk) 20:25, 8 November 2011 (UTC)

Arithmetic vs. algebraic fractions
Intrigued by the intense discussion about the definition in the lead of this article, I have done some literature studies to find out what is actually meant by the term fraction in mathematics. The results are summarized below. I think much of the confusion arises because there are two quite different meanings in the realms of arithmetic and algebra, respectively.

In arithmetic, a fraction is a quantity which represents a part or parts of a unit or a whole. A fraction is expressed by two terms, called the numerator and the denominator. The denominator indicates the number of equal parts into which the whole is divided, and the numerator indicates the number of these equal parts contained in the fraction. A fraction may be considered as the quotient arising from the division of the numerator by the denominator, and is a rational number.

In elementary algebra, a fraction (known as an algebraic fraction) is an indicated division in which the divisor, or both dividend and divisor, are algebraic expressions, and the dividend is not a multiple of the divisor. Two examples of algebraic fractions are $$\frac{3x}{x^2+2x-3}$$ and $$\frac{\sqrt{x+2}}{x^2-3}$$. If the numerator and the denominator are polynomials, as in example 1, the algebraic fraction is called a rational algebraic fraction or simply rational fraction (or rational expression). An irrational fraction is one that contains the variable under a fractional exponent, as in example 2. Since any coefficient is a polynomial of degree zero, a radical expression such as √2/2 is considered a rational fraction, whereas $$\frac{10}{\sqrt{a}}$$ is considered an irrational fraction.

Algebraic fractions are subject to the same laws as arithmetic fractions. Rational fractions can be characterized in a similar way as arithmetic fractions, for example as proper or improper, depending on the degrees of the polynomials in the numeratator and the denominator. An improper rational fraction can be expressed as the sum of a polynomial and a proper rational fraction. A proper rational fraction can, just like a proper arithmetic fraction, be decomposed into partial fractions.

I am looking forward to a discussion about these two meanings of the term fraction. Should we try to cover both in the article or limit this article to the definition in arithmetic? Isheden (talk) 21:26, 31 October 2011 (UTC)


 * This is very good, with two technical points. $$\frac{x^2-4}{x-2}$$ is usually considered a rational expression, even though it is equal to a polynomial.  See, for example, Brown and Churchill, Complex Analysis, Eighth Edition, p. 36-37, "Quotients P(z)/Q(z) of polynomials are called rational fractions"  Since Q(z) may be equal to 1, which is a polynomial of degree zero, the polynomials are a subset of the rational fractions just as the integers are a subset of the rational numbers.  Second, √2/2 is not considered a rational fraction, though it is considered a rational fraction.  The distinction is that in the first case "rational" is an adjective modifying "fraction", while in the second case "rational fraction" is a two-word noun. Rick Norwood (talk) 14:30, 1 November 2011 (UTC)


 * Yes, you're right. $$\frac{x^2-4}{x-2}$$ can be considered an improper rational fraction, which can be expressed simply as the polynomial $$x+2$$ (except at x=2). You're also right that the italics should be removed for the examples: √2/2 is a rational fraction and $$\frac{10}{\sqrt{a}}$$ is an irrational fraction. Isheden (talk) 17:58, 1 November 2011 (UTC)


 * The first words of this section are:  “Intrigued by the intense discussion about the definition…”.  I too am intrigued, because Isheden and me were already in this discussion when the definition came back again to any ratio.  Now a fraction might be a part “of a whole”!  How many occurrences of this phrase in the present page: “of a whole”?  A phrase like a first bite of pudding,  when the remainder may make you sick.  After the two sentences of "suggestion 3",  it would be nice to give the meaning of a line of fraction.  What is the meaning of such a line? — Aughost (talk) 12:03, 3 November 2011 (UTC)
 * I'm having difficulties following your argumentation. How can you be so upset about a description of an arithmetic fraction that you will find in any reference? Why is it important how many times the phrase "of a whole" occurs in this article? What is "line of fraction"? If you can provide a reference that supports your suggestion 3, we can include it as well. Otherwise, it's original research which must not be contained in Wikipedia articles, see WP:NOR.
 * My point is that there is a much broader definition of fraction in algebra, and if the article has the title "Fraction (mathematics)", both arithmetic and algebraic fractions should be treated. If the article is limited to the meaning in arithmetic, then expressions such as √2/2 should be removed. Of course, the corresponding references for the definitions above will be cited in the article. Isheden (talk) 13:23, 3 November 2011 (UTC)


 * We have not to provide a source to justify the meaning of a line of fraction.  We must give a mathematical definition of fraction, yet a part of a whole  is a distortion of the meaning of a  some quantity.  For example,  120&thinsp;/&thinsp;100  is a fraction, but saying that 120&thinsp;/&thinsp;100  represents a part of a whole would be meaningless.  That explanation was already given. — Aughost (talk) 20:08, 3 November 2011 (UTC)


 * 120/100 can be interpreted as 120 of 100 equal parts of a unit. For example, a meter is subdivided into hundredths (called centimeters) and a distance of 1.2 meters is equal to 120 hundredths of a meter (or 120 cm). Do you mean that since a fraction is unitless, it would be misleading to define a fraction as representing parts of a unit? I still don't understand what a line of fraction is. Wikipedia is not the right place for coining new terms or publishing original results. Isheden (talk) 09:33, 4 November 2011 (UTC)


 * In the example of "suggestion 3",  3 and 4 are the terms of the fraction.  They are separated by a line, that represents an operation. — Aughost (talk) 01:14, 5 November 2011 (UTC)


 * I take from your argumentation that you would be happier with a definition in line with I think such a definition would be fine for a fraction in arithmetic. The description with parts of a unit or a whole can be added afterwards. Presently this is discussed in the section "Writing fractions". Isheden (talk) 09:39, 5 November 2011 (UTC)

The vinculum in a fraction does not, in general, represent an operation any more than the point in a decimal represents an operation. The confusion arises because the same symbol is also sometimes used to represent the operation of division. This is one of many cases in mathematics where the same symbol is used with different meanings. Another example is the use of a superscript −1 to represent both reciprocals of numbers and inverses of functions. Beginners are often confused by this but, as I tell my students, historically mathematical notation was designed to make things easy for professionals, not for students.

This article must (and does) begin with the simple and proceed to the more complex. Thus a fraction is, most importantly, a part of a whole. That is the meaning of 1/2. It is not a ratio, not a division problem, it is a number. The article then goes on to more advanced topics, including ratios, division, and algebraic fractions. These topics should be (and are) covered, but the most common meaning is covered first.

Rick Norwood (talk) 13:39, 5 November 2011 (UTC)


 * All these digressions are bothering us.  To be simple, Rick Norwood might choose the same terms than in "suggestion 3",  to give us an example where the line of fraction does not represent division. — Aughost (talk) 06:45, 6 November 2011 (UTC)

The symbol you are calling "line of fraction" is called a vinculum. I just gave an example where it does not mean division: the primary meaning of 1/2 is the number describing one part, where two of those parts make up a whole. It is a number. To say it is a division problem, 1 divided by 2, whose answer is 1/2 leads to infinite regress. Further, 1/2 is an element of the set of rational numbers, not an element of the set of division "problems". And a division "problem", to the extent that "problem" means anything, means a binary operation, which again is not a number. Rick Norwood (talk) 11:54, 6 November 2011 (UTC)


 * “Line of fraction” is the best phrase to denote the symbol of division.  In "suggestion 3",  I gave an example where the line is a character slash, not a vinculum.  In Rick Norwood’s example too, the line is not a vinculum:  1/2. I don't understand the “problems” with  1/2. Maybe in Rick Norwood’s opinion, the article is written for some Ancients, its topic would be history of mathematics.  Anyway, in any fraction, the line represents division. — Aughost (talk) 11:21, 7 November 2011 (UTC)

Aughost: I would appreciate it if you stuck to the subject at hand, and avoided personal remarks. Googling the phrase "line of fraction" shows that it is used more in Low Pressure Liquid Chromatography than in mathematics. You are correct that the example I gave employs a solidus, not a vinculum. The fraction 1/2 still does not mean "one divided by two", since division is a binary operation while a fraction is a rational number. Suppose we were to define 1/2 as "one divided by two". A division problem has an answer, which is a number. What number is the answer? Rick Norwood (talk) 21:18, 7 November 2011 (UTC)


 * I believe that in chromatography, the phrase "line of fraction" refers to a product line, which produces a chemical fraction similar to fractional distillation. See example on this page. --Iantresman (talk) 09:16, 8 November 2011 (UTC)

simple fraction
I find the phrase "simple fraction" used on the web with three different meanings: 1) an easy fraction, 1/2 but not 37/85. 2) a fraction reduced to lowest terms, and 3) a fraction with an integer numerator and denominator.  I've never seen "simple fraction" defined in a math book, only on the web.  References? Rick Norwood (talk) 21:08, 7 November 2011 (UTC)

A quick search gave these with the only exception being a "workbook " page that gave the meaning the same as that for "proper fraction" -- but never mentioned proper v improper. The term has also been used un-self-consciously throughout the article by others. Its usage seems significant enough to have it included in the article.--JimWae (talk) 21:37, 7 November 2011 (UTC)
 * http://www.sosmath.com/algebra/fraction/frac3/frac31/frac31.html
 * http://www.thefreedictionary.com/simple+fraction
 * http://www.merriam-webster.com/dictionary/simple%20fraction
 * http://www.answers.com/topic/simple-fraction
 * http://dictionary.reference.com/browse/simple+fraction


 * I can provide references for the following (from different books):
 * "When both terms of a fraction are integral, the fraction is called a simple fraction. Simple fractions are classified as proper fractions and improper fractions."
 * "To distinguish a fraction of unity from a fraction of any number less or greater than unity, it is usual to term the former a simple and the latter a complex or compound fraction."
 * "A single expression, as $$\tfrac{1}{3}$$, $$\tfrac{a}{b}$$, or $$\tfrac{c}{d}$$ is called a simple fraction. It may be either proper or improper."
 * While the term "simple fraction" is often used to distinguish from complex or compound fractions, it seems that the definition resembles very closely that of a "common fraction". Isheden (talk) 21:54, 7 November 2011 (UTC)


 * Your second source does not use the term "complex fraction" in the normal way. Please provide a citation so that we can examine whether it is appropriate for the article. Your third source uses the term "single expression" which is vague and undefined. I think that most would agree that $$\tfrac{x^2-7}{y+2}-\tfrac{4}{z}$$ is a single expression, yet it is clearly not a "simple fraction". Definitions can vary, it is our job to choose the common definition. Cliff (talk) 07:33, 8 November 2011 (UTC)
 * Various sources are somewhat contradictory regarding the terms complex and compound fractions, but I've added sources for the most common usage, at least in older literature. We could add a source for simple fraction also, but it is really the same as a common fraction as pointed out by JimWae. Isheden (talk) 22:07, 9 November 2011 (UTC)

Also every source at http://onelook.com/?w=simple+fraction&ls=a gives only this one meaning for the term. --JimWae (talk) 22:00, 7 November 2011 (UTC)


 * As noted previously, it's no good looking to English language dictionaries for reliable definitions of mathematical terms. If we want to find sources for these things, the place to look is in school textbooks.  For what it's worth, my hazy memory from primary school is that a "simple" fraction is one that is not a "compound fraction", where a compound fraction is the same as a mixed number, e.g. $$1 \tfrac{1}{2}$$.  This is supported by, but I don't think we can view that site as a reliable source.  Currently the article defines compound fraction differently, without giving a source. Jowa fan (talk) 00:43, 9 November 2011 (UTC)
 * As mentioned above, I've added sources for the term "compound fraction". Isheden (talk) 22:07, 9 November 2011 (UTC)

also http://www.britannica.com/EBchecked/topic/215508/fraction --JimWae (talk) 01:02, 9 November 2011 (UTC)

Symbol of division
This fact must be written very soon in the introduction:  the line that separates the two terms of a fraction, like 3 and 4 in "suggestion 3",  represents division. The line may be horizontal or slanted, it does not matter. The current first sentence  of the article distorts the meaning of the title word: “fraction”. — Aughost (talk) 09:46, 10 November 2011 (UTC)


 * This fact is in the introduction. Quoting: "Other uses for fractions are to represent ratios and to represent division. Thus the fraction 3/4 is also used to represent the ratio 3:4 (the ratio of the part to the whole) and the division 3 ÷ 4 (three divided by four)." Cliff (talk) 15:58, 10 November 2011 (UTC)

A large number of rapid edits.
With so many edits made so rapidly, it is very hard to keep this article at an acceptable level of quality. I'll give just one example, and I don't know who wrote it:

"In mathematics, we typically write fractions as two numbers separated by a line, sometimes one above the other, sometimes at an angle such as &frac12;, $$\tfrac{5}{8},$$ 3/4, etc., consisting of a numerator (the top number) and a denominator (bottom number)."

It is the slanting line, not the two numbers, that is at an angle. Also, in 2/3, neither number is above the other, neither is the "top number", and neither is the "bottom number". I'm going to fix this sentence, but there are dozens more sentences to check that have been added since yesterday.

Rick Norwood (talk) 18:39, 10 November 2011 (UTC)


 * I'm happy to take responsibility, and more than happy to see it improved. --Iantresman (talk) 19:57, 10 November 2011 (UTC)


 * I thought we had agreed not to edit the beginning of the lead substantially without prior discussion. But apart from that, it's not very encouraging to learn that any changes to the article threatens its quality. The feedback from some representative of the WikiProject mathematics was that the article is not very good, and you write yourself that there is still a lot of work to be done. If you are concerned with any of my rather bold edits, please raise your concerns, improve them further, or revert if you should find them unacceptable. However, I cannot see that you have done any real substantial revisions to my last changes, except to the definition of a complex fraction, which was supported by two references. Can you provide a reference for your suggestion "includes a fraction"? To my understanding, this definition would apply also to an expression such as $$\frac{\tfrac{1}{2} \div \tfrac{3}{4}}{5 \times \tfrac{2}{3}}$$. Furthermore, if you're more happy with "steps" than "positions" so be it, but I think "positions" or "places" are at least as appropriate when discussing positional notation. Having said that, I appreciate your efforts towards a high-quality article. Isheden (talk) 21:26, 10 November 2011 (UTC)


 * Sorry, I missed that (agreeing not edit), and I've written nothing on complex fraction. --Iantresman (talk) 22:15, 10 November 2011 (UTC)


 * Apart from the first sentence, this was meant as a reply to Rick Norwood. Sorry for the confusion. Isheden (talk) 22:47, 10 November 2011 (UTC)


 * No problem. --Iantresman (talk) 23:24, 10 November 2011 (UTC)

Isheden: I was not writing about your edits or those of any one editor, specifically, but rather about hasty edits in general. You, yourself, saw the reason I changed "is a fraction or a mixed number" to "includes a fraction". You gave an example where "is a fraction" did not apply but includes a fraction did. My point is that elementary mathematics cannot be explained by someone who does not have a deep understanding of elementary mathematics. Just knowing how to manipulate fractions is not enough. Additional knowledge includes understanding fractions as the quotient field of ordered pairs of integers modulo an equivalence relation. You don't say that, of course, but you need to understand it. And you need to have observed how students, especially students in training to be teachers, misunderstand written instructions. One example of that is the assumption that, in a division problem, the "second" number is the divisor. There are too many different ways to write a division problem for "second" number to be a helpful description, and students often confuse which number is the divisor and which is the dividend, but are usually successful in identifying which number you are dividing "by" and which number you are dividing "into". Rick Norwood (talk) 13:09, 11 November 2011 (UTC)


 * Definitely it's valuable to have people with experience of teaching elementary mathematics as active editors. Other editors may have other strengths in terms of knowledge and experience and a high-quality article evolves when everyone brings in his or her specific knowledge. I don't see a big issue when minor errors slip into the edits because people will scrutinize the changes, correct obvious errors and/or highlight different points of view on the discussion page. Regarding the definitions of complex fraction and compound fraction, I have looked in many references and finally selected two sources as representative. If you think that "complex fraction" should be defined more broadly, you should provide a reference supporting the broader definition. It is pointless to provide those two references to support a definition in the article that is substantially different. Isheden (talk) 14:12, 11 November 2011 (UTC)

The term "term".
I would prefer to reserve "term" for the entries in a series or sequence (finite or infinite) but the use of "term" for the numerator and denominator of a fraction is well established, as in "reduce to lowest terms". However, as Isheden points out, "terms" is not a good word for the digits in an infinite decimal, even though they are understood as the terms in an infinite series, so I have replaced "terms" with "digits". Rick Norwood (talk) 13:14, 11 November 2011 (UTC)
 * In an article about fractions, I think it will be difficult to reserve "term" for anything else than the numerator and the denominator. Isheden (talk)

Afraid of negativity?
When I recently introduced a fraction that happened to be negative (-7/1) into the article, I noticed that some editor disliked negativity and changed it to the happier 7/1. Is there some stigma against negative fractions? It's like the editors of this article have a prejudice against any fraction less than zero. There could be discussions about them when discussing > and < operands. I am beginning to feel sorry for them, and I just wanted to be the Defender of Negativity. I feel that we need to have affirmative action on the part of editors to stick up for negatives of all stripes since they have been discriminated against since this article was created. It will be hard to undo what has been done, so I am asking for donations. Go to my talk page if you want to give a negative fraction a new vision for life and a chance to make a difference... before it's too late. Remember, negatives are just like the positives; they just look a little different on the outside. Don't be ashamed any longer; stick up for them when you edit articles and puppies will frolic on grass and stuff. I like to saw logs! (talk) 07:18, 19 November 2011 (UTC) (a.k.a. Defender of Negativity)
 * In my opinion, the example of an integer written like a fraction with a denominator of 1 does not belong in the lead at all. In the lead, it would be more appropriate to discuss that improper fractions can be written as a mixed number (a sum of an integer and a proper fraction). Also, since negative fractions are not mentioned in the article, I don't see the need for them in the lead. Isheden (talk) 11:14, 19 November 2011 (UTC)

Cute. But when explaining simple concepts, use simple examples. Rick Norwood (talk) 13:03, 19 November 2011 (UTC)

Ah, hah!!! We have all found you out, Isheden!! YOU ARE prejudiced against negative fractions, and you want to pretend they practically don't exist. You want to rewrite history about those poor, wretched negative fractions. I'll tell you what, Isheden, they work just as hard as positive fractions and the cabal is obviously against negative fractions because you admit that "don't see the need for them" and "negative fractions are not mentioned in the article." Those two points represent the vast majority of people out there who hate the little fractions who were simply born with a tiny "-" in front of them or have a "-" someplace in their bodies. You see no need for them, while I (and undoubtedly others who have donated to the cause) see negative fractions as the underprivileged and working-class fractions without which this world would cease to exist.

As the Defender of Negativity, I, Uruiamme declare that as long as negative fractions are shunned, that I will tirelessly defend them until they take their place in the annals of Fraction (mathematics), and that the anti-negative cabal must be stopped. I like to saw logs! (talk) 22:36, 20 November 2011 (UTC)


 * Just for the records, I was not the one who removed your negative integer. User:Rick Norwood did, and I think his answer above was directed to you. Correct me if I'm wrong. Isheden (talk) 08:14, 21 November 2011 (UTC)

Splitting proposal (Rationalization of denominators)
The section does not discuss rationalization of fractions, but only rationalization of the denominator. Therefore, I think the section as it is written now would fit much better in the context of simplified form of a radical expression, see nth root. If the examples were modified to include variables, for example $$\frac{3}{\sqrt{x}}$$ with x positive, then this algebraic fraction could be rationalized by the variable transformation $$z^2=x$$. In this case, the section would fit better in Algebraic fraction. Comments from other editors are appreciated. Isheden (talk) 15:26, 19 November 2011 (UTC)


 * This seems off-topic in an article on elementary mathematics. Change-of-variable techniques are not necessary in dealing with fractions, and are not unique to fractions. Rick Norwood (talk) 13:00, 20 November 2011 (UTC)


 * While it is not necessary to cover HOW to do the rationalization of the denominator in this article, it is also not necessary to show HOW to do many of the other operations -- as the article now does. For the article to be comprehensive, irrational fractions does need some coverage besides being in the lede. I see no reason to single out this section for removal, when other sections have the same issue. Something needs to be said in the body about irrational fractions, particularly since so many people think that since rational numbers can all be expressed as fractions that it somehow "follows" that all fractions must be rational numbers. --JimWae (talk) 20:58, 20 November 2011 (UTC)


 * The term "irrational fraction" occurs only seldom in the literature, and it does not refer to a quotient involving irrational numbers, see "Arithmetic vs. algebraic fractions" above. That's the reason I removed the claim 'An expression that has the form of a fraction but actually represents division by or into an irrational number is sometimes called an "irrational fraction".' Since this article is focusing on arithmetic fractions, I think rationalization of irrational denominators is out of scope. What other sections have the same issue in your view? Isheden (talk) 22:07, 20 November 2011 (UTC)
 * For the article to be comprehensive on broader definitions of the term fraction, I think the content of the articles algebraic fraction and rational function should be covered as well. Isheden (talk) 22:13, 20 November 2011 (UTC)

JimWae: Note that the article should, and does, discuss one method for rationalizing denominators. Isheden's proposal is to add a second method, change of variables, which is less appropriate to fractions.Rick Norwood (talk) 01:06, 21 November 2011 (UTC)


 * Let's forget the idea of changing variables. My proposal is to move the section as it is to Nth root because it fits better in that context. Why should rationalization of denominators be covered in this article? It is only applicable to roots anyway, not to a quotient of any two real numbers. Isheden (talk) 08:21, 21 November 2011 (UTC)


 * DO you mean "move" as in "remove" (cut from here and paste elsewhere) or copy (and paste elsewhere)? --JimWae (talk) 09:15, 21 November 2011 (UTC)


 * My suggestion is to cut it from this article and paste it in nth root. While expressions such as $$\frac{3}{3-2\sqrt{5}}$$ and $$\frac{3x}{x^2+2x-3}$$ are indeed fractions in the sense quotients of algebraic expressions, they are most commonly referred to as radical expression and rational expression, respectively, and should in my view rather be treated in detail in those other articles. The lead in this article defines fraction as a part of a whole or, more generally, any number of equal parts, which corresponds to a (positive) rational number. Isheden (talk) 16:41, 21 November 2011 (UTC)

In my view, rationalizing denominators should be covered briefly here, in a section on rational expressions, and the details left for other articles. The main point to be stressed in this article is that rational expressions obey the same rules as fractions. Rick Norwood (talk) 14:54, 22 November 2011 (UTC)


 * Since there is no consensus to split out the section, I edited it in line with the discussion above. Isheden (talk) 22:27, 9 December 2011 (UTC)

Fraction vs Rational number
Isn't this article making a confusion between a fraction and a rational number, when it discusses the decimal representation with Repeating decimal?

Maybe fraction have a different meaning in French (my mother tongue) but I would not consider 0.5 to be a fraction. I would say 1/2 or 5/10 are fractions. All of them are representation of the same rational number.

For your information the definition of fraction on Wikipedia.fr corresponds exactly to my original understanding:

(A fraction is a not-computed division made ​​between two relative integers) (Elements [from the field of fractions] are written a/b — Preceding unsigned comment added by Régis Décamps (talk • contribs) 17:33, 27 December 2011 (UTC)
 * Une fraction est une division non effectuée entre deux nombres entiers relatifs
 * Les éléments [du corps des fractions] se notent a/b


 * There has been an intense discussion regarding the definition of fraction on this discussion page, see also the archived discussion. The definition you're referring to describes a common or vulgar fraction (also known as simple fraction). 0.5 is not a common fraction, but it's known as a decimal fraction and corresponds to a common fraction whose denominator is a power of ten (in this case 5/10). The term fraction may also refer to for example a complex fraction. On the other hand, it might be a good idea to treat decimals separately as representations of common fractions rather than the way they are introduced now. Isheden (talk) 22:39, 27 December 2011 (UTC)


 * I think it would be better to talk about the decimal representation in the Rational number page. And on the contrary, take the "formal" definition in the Fraction (mathematics) page. What bothers me is that the "fraction" page does not formally defines a fraction. As a result, it seems impossible to talk about matrix fraction on this page. And I'm not sure I'm referring to a "vulgar fraction" any more. Régis Décamps (talk) 09:48, 28 December 2011 (UTC)


 * To bring it to the point, it has proved very difficult to find a formal definition that covers all cases listed in the French article under the section heading "Autres fractions". I don't see how the definition in the French article could be used to describe for example a rational expression such as $$\tfrac{x + 2}{x^2 + 1}$$ as a fraction. Isheden (talk) 11:22, 28 December 2011 (UTC)

Formulas for the arithmetic of fractions
You could replace a, b, c, and d by for example a polynomial or an expression containing radicals, but that is not the point that is made in this section. It would be fine to mention that the formulas are valid in more general cases further down in the article. Isheden (talk) 11:15, 28 December 2011 (UTC)

Probably the best way of defining fractions is to follow and define fraction as an ordered pair while letting a, b, c, and d be restricted to different fields. In the simplest case, they are positive integers and the resulting fractions are positive rationals, corresponding to a part of a whole. This can be extended to any integers (except for 0 in the case of the denominator), to algebraic numbers, to reals, to polynomials, to any algebraic expression, ... See field of fractions for more details. With this definition, continued fractions do not classify as fractions but are a generalization. Isheden (talk) 12:14, 28 December 2011 (UTC)

Negative common fractions?
I'm not convinced that the term common fraction includes negative rational numbers. According to what I've read in various sources, a common fraction a/b can be classified as either proper (when a < b) or improper (when a ≥ b). How do negative fractions fit into this? And how could a negative fraction be interpreted as a part of a whole? Can someone provide a reliable source that explicitly gives a negative fraction as an example of a common fraction and defines proper and improper fractions in terms of absolute values, as is done in the article at present? Isheden (talk) 22:16, 28 December 2011 (UTC)


 * As The Defender of Negativity, I would first point to a common situation. It is called, curiously enough, subtraction. Without the additive inverse of a fraction, we spiral back down to the first semester of first grade in elementary school. Without an additive inverse, we can't use fractions to solve even the simplest linear equation.


 * I would also point out the common use when measuring anything in a linear or 2 dimensional plot. We must have a way for fractions to play well on the most elementary of things common to mathematics, like the number line. If a fraction need not apply to the number line, then how can people measure things in science and engineering with negative values? Cannot gauge pressure be negative? Can't I measure -1.25 psig from such a gauge? Can't I measure -33 1/2 °F? Last I checked, those fractions are perfectly acceptable in engineering, so why the need for "various sources"? I would fathom that you must have neglected anything to do with practical or applied sciences, and must have casually looked at math books. And perhaps you have never had a negative balance, or you've never been in the red with a balance of -$40.34?


 * I hope, in all honesty, that I have boiled it down to the lowest common denominator for you. Negative fractions work in an identical way to positive fractions. You generally apply the negative sign to the numerator and go with it. Works every time. Fear them not. They are not the enemy. You were not thinking clearly. They have meaning. I like to saw logs! (talk) 02:54, 29 December 2011 (UTC)


 * Of course you need an additive inverse and if you actually read the definition section I inserted, you will see that I'm fine with negative fractions or with any other element of a quotient field for that matter. This however has more to do with how this subject developed historically andd how it is taught in elementary mathematics. My question is specific: Can someone provide a source that gives a negative fraction as an example of a common fraction or vulgar fraction or simple fraction and uses absolute values in the definition of proper fraction and improper fraction? Isheden (talk) 06:15, 29 December 2011 (UTC)
 * Put another way, is there a source that claims that the set of common fractions is closed under subtraction? Isheden (talk) 06:17, 29 December 2011 (UTC)

Here are 3 such sources None of those sources are indisputable, but at least we are not making it up. As the last source says, the definition is usually given to kids who do not work with negative numbers, and more advanced mathematics does not care too much about how it is defined. I think the reason we distinguish proper from improper is that it is important to realize that all improper fractions can be converted to mixed numerals. Negative improper fractions CAN be so converted, and so "functionally" they are as improper as positive ones. The alternative is to call all negative common fractions with negative numerators proper (per the simple def of the numerator being less than the denominator) and all negative common fractions with a negative denominator improper. Not only is there no functional reason to do so, but it results in equivalent fractions (eg -5/7 and 5/-7) being both proper and improper--JimWae (talk) 06:50, 29 December 2011 (UTC)
 * http://www.necompact.org/ea/gle_support/Math/resources_number/prop_fraction.htm
 * http://dictionary.reverso.net/english-definition/proper%20fraction
 * http://mathforum.org/library/drmath/view/65128.html
 * Thank you for this. I agree that the distinction between proper and improper fractions is that improper fractions can be written as a mixed numeral. This also applies to fractions in algebra, where improper fractions can be written as mixed expressions, see algebraic fraction. I agree also that in more advanced mathematics the distinction is unimportant. For the kids, however, it may be helpful to start defining proper/improper for positive fractions and subsequently discuss the case of negative fractions. Isheden (talk) 12:04, 29 December 2011 (UTC)

That relationship between improper fractions and mixed numerals also justifies calling 7/7 improper. The "tipping point" for a fraction being "improper" is thus whether or not it can also be written in a way that includes a (non-zero) integer --JimWae (talk) 07:08, 29 December 2011 (UTC)
 * OK, so then we agree on that point. Isheden (talk) 12:04, 29 December 2011 (UTC)

There are lots more sources if one searches for "negative IMproper fraction". Some use the term despite elsewhere giving the simple definition (num<den) that would not include them as improper (because the simple def does not even consider fractions that have negative values) --JimWae (talk) 07:37, 29 December 2011 (UTC)
 * http://dictionary.reverso.net/english-definition/improper%20fraction
 * http://mathforum.org/library/drmath/view/69479.html
 * http://mathsfirst.massey.ac.nz/Algebra/Fractions/MixFrac.htm
 * http://www.tpub.com/math1/5b.htm (by implication)
 * http://www.algebra.com/algebra/homework/NumericFractions/Numeric_Fractions.faq.question.118668.html
 * http://www.mathsisfun.com/mixed-fractions-multiply.html
 * http://www.mathleague.com/help/posandneg/posandneg.htm
 * http://mathsfirst.massey.ac.nz/Algebra/Fractions/MixFrac.htm
 * http://www.mathandchess.citymaker.com/f/Article_-_The_Minus_Sign.pdf
 * http://books.google.com/books?id=4YtB63kdK0AC&pg=PA840&lpg=PA840&dq=%22negative+improper%22+fraction&source=bl&ots=1IzQEjlsVw&sig=JCB07hnBIaGhKbMEN-j7xG9cn9A&hl=en&sa=X&ei=DRz8Tv2hIKuMigLO3r3MAQ&ved=0CDgQ6AEwBTgK#v=onepage&q=%22negative%20improper%22%20fraction&f=false
 * Since fractions describe one or more parts of a unit or a whole, people usually consider positive fractions. For positive fractions, the simple definition is fine. Perhaps the best is to discuss negative fractions in a separate subsection for pedagogical reasons? Isheden (talk) 12:04, 29 December 2011 (UTC)
 * Okay, some clarity again. Recall that a denominator with a zero in it makes the fraction's meaning undefined. So, too, could we call a negative fraction's place in the contrived world of "proper" and "improper." If you say -5/3 is improper, then our definition of improper should either cover absolute value or simply make the fractions with horizontal fraction bars (like they should be for performing mathematical operations) and place the negative sign in front of the fraction bar. So just show it right and it makes sense to the reader... $$-\tfrac{5}{3}$$ is a negative improper fraction and $$-\tfrac{2}{9}$$ is a negative proper fraction, and $$-3\tfrac{1}{3}$$ is a negative mixed fraction, and $$-12.5$$ is a negative decimal fraction,... etc. I like to saw logs! (talk) 08:45, 30 December 2011 (UTC)

There is no need to reinvent the wheel. Fractions and negative numbers have both been in common use for hundreds of years, widely discussed, covered in detail in textbooks, and a standard nomenclature has grown up around them. In particular, the phrase "common fraction" has to a large extent replaced the phrase "vulgar fraction" because the meaning of the word "vulgar" has changed. A "common fraction" is just what is now called a fraction, be it positive, negative, or zero. The distinction is between the a/b notation and decimal notation. It was useful when decimals were called "decimal fractions". It is not much use now.

Isheden: your objection in the first paragraph goes back to the fact that in grade school students learn fractions before they learn negative numbers. In grade school, we can say a/b is "proper" if a is less than b. In high school, we elaborate: a/b is proper if the absolute value of a is less than the absolute value of b.  The fraction -1/2 means, simply, that we are in debt one part of a whole, where if we were twice as deeply in debt we would be one whole unit in debt. There's nothing mysterious here!

And to answer your question about how fractions are taught in grade school, the answer is: badly.

Rick Norwood (talk) 23:03, 30 December 2011 (UTC)
 * I already carried out an edit before reading your post. Now the idea is to introduce common fractions as parts of a whole (which implies they are positive) and defining proper fractions as in grade school. In the subsection Negative fractions, the high school definition is mentioned. Then, a formal definition of fraction follows for readers who require more rigor. Isheden (talk) 23:35, 30 December 2011 (UTC)
 * I've done some rewriting. A negative fraction still represents one or more parts of a whole.  You can be a half dollar in debt! Rick Norwood (talk) 23:43, 30 December 2011 (UTC)


 * Comment to Rick: I guess the pie example leads people to think in terms of positive fractions. I think it would be nice with an illustration of a number line with a unit interval as another example. Isheden (talk) 00:00, 31 December 2011 (UTC)


 * Wikipedia articles are not written for grade school. They assume roughly a grade 10 education - though many people need to be reminded about what they were taught by grade 10--JimWae (talk) 23:46, 30 December 2011 (UTC)


 * I can assure you that, at least in the United States, many college students do not have what used to be called a tenth grade education. Many of them turn to Wikipedia for the information they did not get in school.  However, I agree with JimWae.  We don't help these benighted souls by dumbing things down for them.  The most important thing is to write carefully, and think long and hard before we write. Rick Norwood (talk) 23:54, 30 December 2011 (UTC)