Talk:Fraction

Definition of "improper fraction" not supported by cited source

This article states: "When the numerator and the denominator are both positive, the fraction is called proper if the numerator is less than the denominator, and improper otherwise". This is cited to Perry and Perry, "Mathematics I". By this definition, the fraction "2/2" would be an improper fraction.

However, following the cite to the preview at https://link.springer.com/chapter/10.1007/978-1-349-05230-1_2, what that source actually says is: "A proper fraction has the numerator less than the denominator and an improper fraction has the numerator greater than the denominator". Fractions like 2/2 don't seem to be classified as either proper or improper. (Unfortunate, because I really could've done with a citeable authority on this point and I thought I had one...)

Could somebody please edit this article to change "improper otherwise" to "improper if the numerator is greater than the denominator", in line with the source? 110.23.152.248 (talk) 14:01, 27 June 2023 (UTC)

2 is not less than 2 ("less" is different from "less or equal" So, the article is correct D.Lazard (talk) 14:28, 27 June 2023 (UTC)

Thanks, I am very well aware of the difference between strict and non-strict inequalities, but you seem to have misread what I wrote. The definitions given in Perry and Perry exclude a fraction like 2/2 from being either "proper" or "improper" (since the two concepts are defined independently, both on strict inequalities). But this article gives a definition which implies 2/2 to be an improper fraction, citing Perry and Perry as its source. 110.23.152.248 (talk) 14:55, 27 June 2023 (UTC)

With the modification that you suggest, every positive integer, for example 10 = 10/1, would be an improper fraction, with the unique exception 1 = 1/1 = 2/2 = ... This would be ridiculous. Note that Perry and Perry do not consider at all the fractions that can be reduced to an integer (at least in the linked page). Moreover, the explanation of the word "improper" that they give implies that 1 is an improper fraction. So, the article must not be changed, and a better reference must be provided. There are certainly plenty of such references. D.Lazard (talk) 15:17, 27 June 2023 (UTC)

What part of P&P's explanation do you interpret as implying that 1 is an improper fraction? The only statement I see in that excerpt which addresses the question is the statement that "an improper fraction has the numerator greater than the denominator", which clearly implies the contrary.

Personally I would much prefer if the definition of "improper fraction" included 1. As you note, that would make it consistent over the natural numbers; it would also be much cleaner if "improper" was the complement of "proper". But those are aesthetic considerations that don't trump verifiability requirements. Further, I'm aware of two other sources which contradict this article's definition:

• The Hutchinson Pocket Dictionary of Maths defines an improper fraction as "fraction whose numerator is larger than its denominator".
• The Cambridge Dictionary defines it as "a fraction in which the number below the line is smaller than the number above it".

I wouldn't consider either of those incontrovertible sources. But until somebody can provide something more authoritative, the article needs to be edited. It should be obvious that it's not appropriate for Wikipedia to provide a definition which is contradicted by the cite offered for that definition. If you're not willing to edit the text to match the source it cites, then I request that the cite be removed and replaced with a citeneeded tag. 110.23.152.248 (talk) 16:11, 27 June 2023 (UTC)

Noting that the claim is repeated in the following paragraph, with a second source cited, which *also* contradicts the definition here.

The Wiki text: "It is said to be an improper fraction, or sometimes top-heavy fraction,[16] if the absolute value of the fraction is greater than or equal to 1", listing "3/3" as an example of an improper fraction. But the cited source, Greer's "New Comprehensive Mathematics for O Level"[1] actually says "if the top number of a fraction is **greater than** its bottom number then the fraction is called an improper or top heavy fraction... **note that all top heavy fractions have a value which is greater than 1**." 110.23.152.248 (talk) 23:47, 27 June 2023 (UTC)

I think the sources do not consider ±n/n to be a fraction, as they reduce to 1 or -1, or for 0/0 they are not a valid fraction at all. So they assume the two values are unequal and thus never state whether equal ones are improper. This does not conflict with the text. Spitzak (talk) 16:29, 29 June 2023 (UTC)

Even if the sources genuinely were ambiguous on the "or equal to" question the article would still be giving a definition that goes beyond what the cited sources support, and that would still be open to challenge and removal in the absence of a cite.

But they are *not* ambiguous. The Greer quote states that "all top heavy fractions [which it's just established as a synonym for 'improper fractions'] have a value which is greater than 1". A fraction equivalent to 1/1 clearly does not have a value "greater than one", therefore by Greer's definition it cannot be an improper fraction. The other sources discussed above all state that an improper fraction has numerator **larger than** denominator - not "larger than or equal to" - which, again, is incompatible with the definition given here.

At this point, given the number of sources that impose a strict "greater than" definition, the article should recognise that definition. If somebody can provide other, reliable sources which offer a "greater than or equal to" definition, then the article can cite those too, acknowledge both definitions, and observe that terminology is inconsistent. Until then, it should match what is given in the sources discussed. 110.23.152.248 (talk) 03:06, 1 July 2023 (UTC)

@Spitzak: The text currently says "It is said to be an improper fraction… if the absolute value of the fraction is greater than or equal to 1." As I read the source cited, from Greer, it says that a fraction is improper only if its value is greater than 1 (as 110.23.152.248 has pointed out). Therefore this part of the text contradicts the source: "or equal to"; and these three words should be removed. What do you think? (Whether Greer considers 8/8 to be a fraction at all would seem to be a separate topic.) —Ben Kovitz (talk) 05:21, 1 July 2023 (UTC)

I posted at NOR/Noticeboard to seek further opinions on this. Another commenter there has located some sources which *do* support the "or equal to" definition. Given that we now have multiple sources for both definitions, I now think the best thing is for the article to acknowledge that conventions vary on this question, and to cite sources for both versions. 110.23.152.248 (talk) 06:06, 1 July 2023 (UTC)

I think the question here is not when a fraction is improper, but whether an expression such as 8/8 is a fraction. Essentially, calling fractions proper and improper is language most common in elementary school, and seldom used after elementary school.

The essential point is that the rational numbers include the integers, which in turn include the natural numbers. And every rational number can be written as a fraction. So the rational number "one" can be written as a fraction, and 8/8 is as good a way to do that as any. The ratio of 8 to 8 is 1.

Any part of a whole can be called "a fraction", as in "You have only completed a fraction of the work you've been assigned." That usage is not the subject of this Wikipedia article. The subject of this Wikipedia article is notation: the use of a horizontal or slanting line to write a number which can always be written in other ways: decimals, percents, etc. So it is a notation. Numbers that are not rational, such as pi/2, are still called "fractions" because of the use of the slanting line. If we talked about "proper" and "improper" fractions, then pi/2 would be an improper fraction, because it is bigger than one. But in all my years of teaching math, I've never heard it called that.

So, maybe we should say something like this:

In elementary school, students are taught to call a fraction larger than one, or sometimes a fraction greater than or equal to one, an "improper fraction", while a fraction between zero and one is called a "proper fraction". When students learn about negative numbers, they may be told that a fraction is proper or improper depending on whether its absolute value is proper or improper, and they may be taught that 0/1 and 1/1 are one or the other, but there is no general rule about this. Different elementary textbooks say different things, and the words "proper" and "improper" are seldom used to describe fractions after elementary school. Rick Norwood (talk) 10:55, 1 July 2023 (UTC)

recent edit saying "citation needed"

The paragraph in question is common knowledge to math historians, and has two citations. I don't see the problem. Rick Norwood (talk) 10:36, 29 June 2023 (UTC)

Geometric line for constructing fractions

After example in page 36 of Growing ideas of number (by John N Crossley) JMGN (talk) 10:14, 26 August 2023 (UTC)

Perhaps, you are trying to tell us that there is a geometric way to represent or estimate fractions by drawing parallel lines on a graph. This is something that the article is currently not addressing. Should it be? I think this is an interesting idea. What do other editors think? Dhrm77 (talk) 11:12, 28 August 2023 (UTC)

https://matheducators.stackexchange.com/a/26695/12046 JMGN (talk) 16:11, 28 August 2023 (UTC)

This was well known to Euclid, and is addressed in Straightedge and compass construction#Constructible points and Intercept theorem#Algebraic formulation of compass and ruler constructions. This is also one of the key points in Emil Artin's proof of the equivalence of the Euclidean spaces of synthetic geometry and analytic geometry, in his book Geometric algebra. D.Lazard (talk) 16:43, 28 August 2023 (UTC)

BTW, the "length 1" on the abscissa axis is four times as long as that on the y-coordinate. I don't know why tho'... JMGN (talk) 11:23, 5 September 2023 (UTC)

@JMGN: Please see WP:TPG and WP:NOTFORUM. Eyesnore 22:36, 6 September 2023 (UTC)

Weird speculation in "mixed fraction" section

I find this text a bit odd:

This tradition is, formally, in conflict with the notation in algebra where adjacent symbols, without an explicit infix operator, denote a product. In the expression $2x$, the "understood" operation is multiplication. If x is replaced by, for example, the fraction ${\tfrac {3}{4}}$, the "understood" multiplication needs to be replaced by explicit multiplication, to avoid the appearance of a mixed number.

When multiplication is intended, $2{\tfrac {b}{c}}$ may be written as

$2\cdot {\frac {b}{c}},\quad$ or $\quad 2\times {\frac {b}{c}},\quad$ or $\quad 2\left({\frac {b}{c}}\right),\;\ldots$

• First, I'm not convinced that there is any "formal conflict" in the first place. You might as well say that the expression 23 is in conflict with the "product" convention, because 2 times 3 is 6 rather than 23.
• Second, it's ludicrous to suggest that $2{\tfrac {b}{c}}$, with literal letters $b$ and $c$, would ever be read as a mixed fraction $2+{\tfrac {b}{c}}$. Mixed-fraction notation always has literal numerals in the fraction part, never ever ever variables or even expressions (and in fact I think it pretty much has to be a proper fraction, with the numerator less than the denominator). What's true is that if you want to write 2 times ${\tfrac {3}{4}}$, then you have to do something to avoid the reading as a mixed fraction.

I'm torn as to whether to simply remove this whole bit of text, as I'm not sure there's enough worth saving. The second point about $2{\tfrac {3}{4}}$ versus $2\left({\tfrac {3}{4}}\right)$ might possibly be worth saying something about. --Trovatore (talk) 21:01, 14 January 2024 (UTC)

I agree this is nonsense. Nobody thinks "123" means 1×2×3 even if they say 2a means 2×a. There is no "conflict", this is just a way of writing the number. Spitzak (talk) 02:00, 15 January 2024 (UTC)

@Jacobolus: OK, time we hashed this out by discussion rather than by duelling edit summaries. I think it's simply incorrect to claim that the reason for not using mixed fractions is some sort of "ambiguity". Please defend that proposition or allow it to be removed. --Trovatore (talk) 03:50, 15 January 2024 (UTC)

The previous text was awkward and confusing and is fine to remove. However, it is worth mentioning in this section that juxtaposition in mathematics conventionally is reserved for multiplication, and addition is written explicitly. If you write something like

$a{\frac {x}{y}}$

that always means $a\cdot (x/y)$ and never means $a+(x/y).$ Even when the constituent parts are numbers, in the context of mathematics at the high school level or beyond, the explicit notation with a + sign is used instead of juxtaposition for a 'mixed number' (or more commonly, an improper fraction is left). –jacobolus (t) 04:00, 15 January 2024 (UTC)

I'm not convinced that this is "worth mentioning". How is it more relevant than the fact that 23 doesn't mean 2 times 3?

Echoing your always and never, $2{\tfrac {3}{4}}$ always means $2+{\tfrac {3}{4}}$ and never means two times ${\tfrac {3}{4}}$. Anyone who uses it the latter way has simply made an error. Therefore there is no ambiguity. The fact that juxtaposition "in algebra" means multiplication is irrelevant here, because this isn't algebra; this is just the meaning of numerals. --Trovatore (talk) 04:03, 15 January 2024 (UTC)

The expression $2{\tfrac {3}{4}}$ is never used in mathematics at or beyond the high school level, because it might otherwise cause confusion. People just write $2+{\tfrac {3}{4}}$ or ${\tfrac {11}{4}}$ instead. –jacobolus (t) 04:10, 15 January 2024 (UTC)

I agree it's at least rare in mathematics. I'm not convinced it's because it "might cause confusion". The expression ${\tfrac {11}{4}}$ is used because it's more compact and easier to read and manipulate, not because $2{\tfrac {3}{4}}$ is actually unclear. --Trovatore (talk) 04:15, 15 January 2024 (UTC)

If you start mixing $2{\tfrac {3}{4}}$ meaning $2+{\tfrac {3}{4}}$ with $2{\tfrac {x}{4}}$ meaning $2\cdot {\tfrac {x}{4}}$ in the same context, you have a recipe for serious confusion. –jacobolus (t) 04:51, 15 January 2024 (UTC)

This paper suggests that adults with a college education routinely have difficulty interpreting mixed numbers because of confusion about the meaning of concatenation, while introductory algebra students have trouble with concatenation meaning multiplication because of past experience with multi-digit numbers and mixed numbers. –jacobolus (t) 05:00, 15 January 2024 (UTC)

I agree this does not seem "worth mentioning". We all have the example that everybody agrees "123" does not mean 1×2×3, even if in the same text abc means a×b×c. The fractions where everything is a numeral are the same, they are not multiplied. Spitzak (talk) 04:11, 15 January 2024 (UTC)

The notation $123$ always means $1\cdot 10^{2}+2\cdot 10+3$ in every context. The several digits are treated as a lexical unit representing a single number. In algebra, the notation $abc$ using letters always means $a\cdot b\cdot c,$ and the letters are variables of arbitrary numerical value, not digits. However, when someone wants to multiply numerical constants, they are forced to adopt the more explicit notation of e.g. $30=2\cdot 3\cdot 5,$ because $235$ already means something else. Conversely, if someone wants to use a letter to represent a digit, they need to jump through hoops to make the meaning explicit.

Fractions are a higher-level aggregated structure. Deciding how to treat fractions combined with other operations depends on the context.

In modern use, the notation $2{\tfrac {1}{2}}$ is a more concise shorthand for $2+{\tfrac {1}{2}}.$ This notation descends from an older and more complicated mixed-unit notation used by medieval traders of the Islamic world, as recounted by Fibonacci. Unit systems have been somewhat simplified since then, and this notation is now an unused historical relic; "mixed numbers" are a lingering remnant that persists in daily life and trades (especially carpentry) in non-metricized regions, but is not used in more technical contexts.

In a somewhat similar way, percentages were a way of writing some kinds of decimal fractions before the adoption of general decimal fractions per se, and today are something of an anachronism. Just as you won't find mixed numbers, you also won't find percentages used in mathematics beyond the elementary level, because decimal fractions are a better and better-integrated notation. –jacobolus (t) 04:49, 15 January 2024 (UTC)

As far as I know, there is no context whatsoever in which $2{\tfrac {1}{2}}$ means anything other than ${\tfrac {5}{2}}$. I don't disagree that it's better to avoid this format in serious mathematics, but it's not because the meaning is in any way unclear. --Trovatore (talk) 06:33, 15 January 2024 (UTC)

Here's one more source commenting in the general vicinity of this discussion:

We next introduce a staple in the school curriculum, the concept of a mixed number.[...] By tradition, whenever we have the sum of a whole number and a proper fraction, the addition sign is omitted so that $4+{\tfrac {2}{5}}\mathrel {\stackrel {\text{def}}{=}} 4{\tfrac {2}{5}}.$ In general, for a whole number $q$ and a proper fraction ${\tfrac {r}{\ell }}$ the notation $q{\tfrac {r}{\ell }}$ stands for $q+{\tfrac {r}{\ell }},$ and it is called a mixed number. Please be warned that the symbolic notation for a mixed number is a confusing one, because $q{\tfrac {r}{\ell }}$ suggests the product of $q$ and ${\tfrac {r}{\ell }}.$ Our advice is to avoid using it whenever possible. [...]

Although the tradition in school mathematics is to insist that every improper fraction be automatically converted to a mixed number, there is no mathematical reason for this practice. This tradition seems to be closely related to the one which insists that every answer in fractions must be in reduced form. ¶ Given the general confusion about mixed numbers in school mathematics, there is a strong case for the suggestion that this concept be used sparingly in the school curriculum at present.

This source doesn't explicitly point out that mixed numbers are rare after school mathematics. It's often difficult to find sources explicitly making negative claims of this general type. –jacobolus (t) 07:01, 15 January 2024 (UTC)

The italicized portion (your emphasis? doesn't really matter; just curious) warns about the "symbolic notation for a mixed number", which is indeed confusing. Actually I'd go further than confusing; it's incorrect. You can't substitute symbols. $2{\tfrac {1}{2}}$ unambiguously means $2+{\tfrac {1}{2}}$, whereas $q{\tfrac {r}{\ell }}$ unambiguously means $q\cdot {\tfrac {r}{\ell }}$. Again, this is not fundamentally different from $r\ell$ meaning $r\cdot \ell$ whereas 23 does not mean $2\cdot 3$. --Trovatore (talk) 07:15, 15 January 2024 (UTC)

No, the source had the italics. I substituted a different claim about mixed numbers being phased out from secondary school, with a source. Is that better? –jacobolus (t) 08:09, 15 January 2024 (UTC)

I think it reads pretty well at the current revision. Thanks, jacobolus. --Trovatore (talk) 23:02, 15 January 2024 (UTC)

Mixed numbers have their place

There have been some dismissive comments about mixed numbers. They have their uses, especially in recipes (two and a half cups) and common measurements (four and a half inches). I would hope all grade schools would teach mixed numbers, but only in examples where the fraction part is 1/2 or 1/4 or some other frequently used fraction. The important point is that usually an understood operation is multiplication and that mixed numbers are an exception. I don't think they are ever useful in algebra. For that matter, the only useful fractions are the "small" fractions.Rick Norwood (talk) 11:54, 16 January 2024 (UTC)

How do you feel about the current text of the section?

In primary school, teachers often insist that every fractional result should be expressed as a mixed number. Outside school, mixed numbers are commonly used for describing measurements, for instance $2{\tfrac {1}{2}}$ hours or $5\ 3/16$ inches, and remain widespread in daily life and in trades, especially in regions that do not use the decimalized metric system. However, scientific measurements typically use the metric system, which is based on decimal fractions, and starting from the secondary school level, mathematics pedagogy treats every fraction uniformly as a rational number, the quotient ${\tfrac {p}{q}}$ of integers, leaving behind the concepts of "improper fraction" and "mixed number". College students with years of mathematical training are sometimes confused when re-encountering mixed numbers because they are used to the convention that juxtaposition in algebraic expressions means multiplication.

I tried to preserve some of the criticisms / concerns about 'mixed number' notation while not trying to make direct value judgments or claims unsupported by reliable sources. –jacobolus (t) 09:44, 17 January 2024 (UTC)

You have done very good work on this article. Rick Norwood (talk) 11:36, 17 January 2024 (UTC)

References