Talk:Fraction

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

Semi-protected edit request on 26 April 2024

Please change 'number' to 'real number' in the fourth paragraph of the introduction section Fraction. Details: It says "In mathematics the set of all numbers that can be expressed in the form a/b, where a and b are integers and b is not zero, is called the set of rational numbers " Change 'numbers' to 'real numbers' possibly with a blue hyperlink. I mean what kind of number is being referred to by the word 'number'? Complex numbers? integers? rational numbers? Hyper-real numbers? Of course it would be circular, or a tautology, to say the rational numbers are the set of all rational numbers. But we can define rational numbers using real numbers as the superset. 207.244.169.9 (talk) 19:57, 26 April 2024 (UTC)

The rational numbers can certainly not be defined using the real numbers, since the real numbers are defined using the rational numbers, and this would be a circular definition. The phrase "all numbers that can be expressed in the form a/b, where a and b are integers and b is not zero" does not imply thaat there are other numbers. It simply means that an expression a/b represents a number that is called a rational number. However, the sentence can be simplify and clarified, and I'll do it. D.Lazard (talk) 20:22, 26 April 2024 (UTC)

Understanding fraction operations visually

Visual representations play a pivotal role in elucidating the concept of fractions, providing learners with intuitive tools to grasp abstract mathematical ideas.

Visual Representations of Fractions:

Pie Charts: Fractional parts of a whole are often depicted using pie charts, where each sector represents a fraction of the entire circle. This visualization allows learners to visualize fractions as proportions of a whole and understand relationships between different fractions.

Bar Models: Bar models represent fractions using segmented bars, with each segment corresponding to a fraction of the whole bar. This visual tool aids in comparing fractions, understanding equivalent fractions, and performing arithmetic operations.

Number Lines: Fractions can be represented on number lines, where each point corresponds to a fraction between 0 and 1. Number lines provide a linear representation of fractions, facilitating comparisons, addition, subtraction, and identifying fractional positions.

Area Models: Area models partition geometric shapes, such as rectangles or circles, into equal parts to represent fractions visually. This method helps learners visualize fractions as areas or regions, fostering a deeper understanding of fraction values and operations.

A tool like https://www.visualfractioncalculator.com/ can help understand fraction addition, subtraction, multiplication and division. Swapsshah (talk) 09:26, 14 May 2024 (UTC)

Compound and complex fractions

Hi, the present section on compound and complex fractions seems outdated and/or incorrect:

• It explains a distinction between the two terms in > 200 words, only to then state that both terms can be considered outdated. The two works cited for the distinction are from 1814 and 1853, respectively! And the one from 1814, by Peter Barlow, even says that these distinctions "are certainly improper".
• The source used to back up the claim that both terms can be considered outdated, an archived version of the online Collins Dictionary, does in fact not make any such claim. It says just what the current version of this dictionary says: that complex fractions are fractions where the numerator and/or the denominator are fractions themselves, and that they are also called compound fractions.
• The source used to back up the claim that the terms are also used to denote mixed numerals is an archived version of an online mathematics course called "S.O.S. Math", which is not active any more.

I suggest a complete rewrite by a native speaker, just stating that both terms are used synonymously for what is presently described as complex fractions here, using the Collins source, and getting rid of everything else. Biologos (talk) 17:44, 8 November 2024 (UTC)

Note that this is explicitly in a section about "historical notions".

The term compound fraction meaning several fractions composed by multiplication, often written as "a/b of c/d of e/f" or the like, was certainly commonly taught in the 18th–19th century (here's another example from 1870, JSTOR 44859615), and the concept is still in use today though it is no longer necessarily named or given prominence in educational materials (I think today it would more likely just be called a "product of fractions" or similar). From what I can tell the term compound fraction is not too widely used today, and a few examples I found in a quick literature search were inconsistent and somewhat unclear. I did find somewhat more recent examples of compound ratio and compound proportion used to mean the same thing. The term complex fraction is still in currency, used in elementary education, with the meaning described here; we could definitely include better sources though. The distinction made between these two concepts seems accurate (here's another source comparing the two, from 1856, JSTOR 44364747), though as with any subject so common there is always some inconsistency between regions and individual authors.

Dictionaries are unfortunately usually poor sources for this kind of thing, typically unreflective of real-world use. However, Barlow accurately describes the meaning of those two terms as found in early 19th century arithmetic instruction. His comment that this is "improper" is about his own interpretation of what "compound" and "complex" mean: he is complaining that whoever turned these names into technical jargon made a poor choice conflicting with the plain-language meaning of the individual words. I'm not sure I agree; "complex" comes from Latin for "braid together" whereas "compound" comes from Latin for "put together". These words are both used in a wide range of technical jargon across several fields, without tremendously strong consistency. The use of "compound" to mean multiplication by some scaling factor is also found e.g. in compound interest.

I think this section is more or less fine in general content, though the text could be clearer and the sourcing could be improved. The main purpose is to give a target for the wikilinks compound fraction and complex fraction so someone encountering these terms and looking them up in Wikipedia can figure out what they mean. Giving examples seems helpful, so I wouldn't significantly reduce the length of this section. –jacobolus (t) 21:04, 8 November 2024 (UTC)

Thank you for your prompt and comprehensive reply! Indeed, I had not noticed the Historical notions section heading, because I found the subsection by searching for the term complex fraction in the article and jumping directly to its location. Since you are saying that the main purpose is to give a target for the wikilinks naming the two terms, the Historical notions heading might be overlooked frequently. To guide the reader, could the first sentences in the subsections for the terms again state that the terms were used to describe what is described then? And then, the last paragraph could be shortened and updated somewhere along the following lines: "The terms "complex fraction" and "compound fraction" are now used in no well-defined manner, partly even taken synonymously for each other[25]."? Biologos (talk) 07:50, 11 November 2024 (UTC)