What is the rationale for distinguishing between proper and improper fractions?

I cannot recall ever hearing the terms "improper fraction" and "proper fraction" outside of an elementary and middle school setting. At some point in my mathematics education people began to simply say "fraction".

Has there been any research into the benefits of differentiating between these two concepts?

What is the rationale for this differentiation?

• See also partial fraction decomposition. Step 1: Use polynomial division to reduce the potentially improper fraction to a proper fraction. Of course, by this point, you're calling the various objects "rational functions". Oct 2, 2021 at 22:05
• @EricTowers Interesting. So your claim is that writing$\frac{x}{1+x} = 1-\frac{1}{1+x}$ is analogous to going from an improper fraction to a mixed number? Oct 2, 2021 at 22:07
• It's not just my claim. dummies.com/education/math/calculus/… (See also en.wikipedia.org/wiki/Rational_function , where the change to "rational function" has already happened.) Oct 2, 2021 at 22:09
• I am not going to be at all surprised when the WIkipedia has errors of detail. Fixed. Oct 2, 2021 at 22:16
• I don't think I've seen numbers in the form 1¼ outside of the US (in grade school in Italy I was always taught to write 1.25 or similar when necessary). Indeed when I lived in the US, writing numbers this way always struck me as confusing and needlessly complicated. Of course we say something like "one hour and a half" etc, but only in words. I suspect this is very cultural dependent Oct 4, 2021 at 8:42

The reason mixed numbers are found in US education is that mixed numbers are found outside of school in the US, so the children need to learn to understand them. Mixed numbers are found in road signs, cooking recipes, length measurements, and so on. (Denis Nardin commented that mixed numbers are never seen in Italy. Meters, centimeters, and millimeters easily become decimals; miles, yards, feet, and inches do not.) (But see the Hong Kong roadsign below, with ½ km.)

Also no research, just an explanation.

In order for the general public to understand you, you cannot say $$\text{Add }\frac{4}{3}\text{ liters of water.} \tag{1}$$ instead, you must say $$\text{Add }\; 1\,\frac{1}{3}\;\text{ liters of water.} \tag2$$ Thus, gradeschool kids need to learn how to get their answer in the form $$(2)$$. To teach this, there must be terminology for these two forms of the answer, such as "improper fraction" and "mixed number".

Can you imagine a road sign "Fenstanton $$\frac{9}{4}$$" ?

• I don't find this argument particularly persuasive. I think it feels a bit circular: people are taught they are somehow different because people are not used to using them in everyday life; people are not used to using them in everyday life because they are taught they are somehow different. Oct 2, 2021 at 18:30
• @RichardWard, I'd argue that human intuition for measurement tends to operate using mixed numbers, where we imagine N of something, then add some fraction of that something to the end. If we're given a fraction or even a decimal representation of a number with a physical unit attached and asked to intuit it, we'll likely break it down into countable multiples of some reference we're familiar with. So it's not a matter of the general public being uneducated and thus they can't tell you what 5/4 means, but it being a lot more intuitive to think about 1 1/4 of something. Oct 2, 2021 at 19:23
• For what it's worth, I find those road signs needlessly complex, and hard to parse. "1³₄" looks too much like "134" to me. Would it be really bad to write "1.75", "1.8" or even "2" instead? "6 7/8" looks completely ridiculous too. Speedometers are not very accurate anyway. They should simply write "6" or "7" instead. Oct 3, 2021 at 14:14
• These are horrible signs. Bad typography, hard to parse, and yes, decimal would be easier. The last one is a cringeworthy joke. The point of a "mixed number" is to say "about 3 miles", but decimal is no worse for this purpose and better for others. Oct 3, 2021 at 15:32
• @RustyCore bad typography?! The typeface was designed by Margaret Calvert specifically to improve legibility over other countries' road signs. In particular the use of mixed case makes it much easier to read than American equivalents. I'd be curious to know what you specifically find to be "bad" about it unless you're just referring to the fractions. Personally I've never had an issue parsing them; I've certainly never heard anyone complain about reading 2³ as 23 for instance. Oct 4, 2021 at 13:40

I do not know of any relevant research.

Here are my own not-research-informed ideas.

Most people refer to fractions as parts of a whole. If someone says "I lost a fraction of a pound on my diet", you can be fairly certain that they didn't lose $$\frac{23}{1}$$ pounds.

Since the common usage of the word and the mathematical usage differ, it is useful to draw attention to the fact that fractions can be larger than one by giving them a special name. I think the word "improper fraction" is a bit unfortunate because it carries the implication that such fractions are undesirable.

This is a little bit similar to the word "or". In English the word can be used in at least two ways. In mathematics, when we need to be precise, we invent the terms "inclusive or" and "exclusive or" to make the distinction.

• Interesting. I've never seen "inclusive or" anywhere. In mathematics, this is just "or", isn't it? Oct 3, 2021 at 14:16
• @EricDuminil It is only used to make the distinction. Oct 3, 2021 at 14:59
• @EricDuminil: E.g., Rosen Discrete Mathematics Sec. 1.1: "The use of the connective or in a disjunction corresponds to one of the two ways the word or is used in English, namely, as an inclusive or" (and many other math books). Oct 7, 2021 at 18:00

Student are introduced to fractions as part of a whole. They are then taught that improper fractions can be more than a whole - this is not ideal terminology or helpful for understanding. Improper fraction is a terrible name since it implies that there is something wrong with the fraction.

Once student start to do calculations with fractions greater than one, the distinction helps make calculations easier.

1. When multiplying and dividing you would convert mixed numbers to "improper fractions" to make the calculation easier.
2. When adding and subtracting you would work with mixed numbers and not improper fractions.
3. Converting between mixed numbers and improper fractions (which is necessary in many different problems) can strengthen understanding of fractions if taught properly.

The distinction is important at the elementary school level. From the comment below I see it is important at the high school level too.

• It is definitely needed later, as many students still don't quite "get" fractions. Also, the distinction is important when describing the best form for an answer. For example, in beginning algebra we want students to understand that slope is best left as a "improper" fraction, so that it's saying rise over run, unless a decimal version works better in the situation. y-intercept is best written as a mixed number, so you can see where it is on a number line. Oct 2, 2021 at 17:38
• @SueVanHattum You are so right - I modified my answer. Oct 2, 2021 at 17:58

Interestingly enough, we do not have such a distinction in France. From the French Wikipedia article about fractions (emphasis mine):

Dans l'enseignement français depuis la fin du xixe siècle, la fraction est définie comme le quotient de deux nombres entiers sans contrainte sur la taille du numérateur et du dénominateur (...)

In French education, since the end of the 19th century, a fraction is defined as a division of two whole numbers, without constraints on the size of the numerator and denominator (...)

I think I recall that at the very, very first introduction of fractions my children saw the version $$1\frac{3}{4}$$ but it was quickly replaced by $$\frac{7}{4}$$ and never came back. I do not remember how it was when I was learning fractions, but I do not remember ever having used the $$1\frac{3}{4}$$ version.

On a related note, I find the $$1\frac{3}{4}$$ version particularly unintuitive, it suspiciously looks like $$1\times \frac{3}{4}$$.

EDIT: Now that I think of it, it may be that the US use of proper vs improper fractions come from their very heavy use in measurements.

In Europe, we would never say that something is $$1\frac{2}{11}$$ meters. We would say it is 1,18 m (with a comma :)) - because of the decimal nature of the metric system.

We basically never have a need to use fractions in everyday life, only in calculations (where the "improper" form is easier to use)

• "Fraction" simply defines a rational number (that is, one number over another), even in English. In math theory, there are different types of fractions, including simple, complex, proper, improper, reducible, irreducible, etc. These are adjectives that describe a particular property of the fraction. Note that the page you linked mentions the section "Nombre mixte et fraction impropre", which goes in to more detail. Oct 5, 2021 at 1:03
• @phyrfox: OP's question is about proper and improper fractions. In France, we do not have that distinction - contemporarily and in the mainstream education system. The quote I gave is the first paragraph of the part you refer to, which explains how it was in the past (and is not anymore in the present).
– WoJ
Oct 5, 2021 at 6:59

The use of rulers with fractional inches is the first thing that springs to mind.

Like this:

The four keys have a width of $$2 \frac{11}{16}$$ inches at the tops of the key caps.

If I calculated a length, and got $$\frac{43}{16}$$ inches, I'd have to convert it to $$2 \frac{11}{16}$$ to actually measure it.

• The "improper fraction" $$\frac{43}{16}$$ is easier to use for further calculations.
• The "proper fraction" $$2 \frac{11}{16}$$ is easier to use in the real, physical world.
• I wouldn't call "2 11/16" a "proper fraction". I think 11/16 is a "proper fraction", but "2 11/16" is a "mixed fraction" or "mixed number". Wikipedia appears to agree with me: Proper and improper fractions and Mixed numbers
– Stef
Oct 4, 2021 at 13:30
• "The improper fraction $43/16$ is easier to use for further calculations." It depends. But the advice should be: leave it as improper fraction until you need it in some other form (such as additional computation or final answer). Oct 5, 2021 at 15:44
• From the point of view of a person used to SI units and decimal systems : This just seems to explain a weird convention with a weird measurement tool. Oct 8, 2021 at 10:02

Here's another use-case that came up in my college remedial algebra class tonight (and again, this boils down to translations to mixed numbers): Finding fractions in a graph.

So the specific example that presented itself tonight was a book exercise: "Graph the equation: $$y = - \frac 5 3$$". At that point, my students could tell me that this would be a horizontal line in the Cartesian plane, and that it should go through $$-\frac 5 3$$ on the y-axis. But where is that? Nobody could find it.

ME: Can anyone tell me what two integers $$-\frac 5 3$$ is between?

STUDENT: Between 3 and 5.

ME: No.

So given that none of the students could answer it, I suggested: it's probably helpful if we convert to a mixed number. What is that? Well, now I'm talking about taking this improper fraction and writing as a whole number plus a proper fraction (with little underlined blanks with those words underneath). But no one can accomplish that, so I reviewed the long division algorithm, and came up with $$-\frac 5 3 = -1 \frac 1 3$$. This at least lets us answer the question above: "This value is between $$-1$$ and $$-2$$" (and then finish the exercise by drawing a horizontal line at that height).

Then my students asked for another one like that (presumably because it seemed opaque to them). The next exercise in the book was, "Graph the equation: $$y = - \frac {15} 4$$". Again, none of my students could find that location, i.e., say what two integers straddle it, because none of them could convert to a mixed number. I went through a long division again, etc. One student said he had no idea how I was coming up with "the fraction part". So for the purposes of this class I had to refer him to tutoring -- where I assume they'll be using the terms "proper fraction" and "improper fraction" to talk about those parts of the translation-to-mixed-number process, and so actually find where the value is on a number line.

You can see an identical problem in finding other points, intercepts, slopes, etc. that have fractional components, if people are unable to recognize the need and convert to the mixed number format.

• I find this the most persuasive response so far. It is useful in practice to recognize that $32/11$ lies between $2$ and $3$, and being forced to express $32/11$ as an integer plus some fraction less than $1$ teaches a student a way of making that recognition. Oct 15, 2021 at 16:48

I did some research, which you can follow here, that explains the why of the nomenclature we use. Basically, something is proper if it is contained in something else, and improper otherwise. For example, the proper divisors of 12 are 1, 2, 3, 4, and 6. This set excludes 12, because that represents the entire portion, e.g. no fraction occurs. Similarly, in set theory, a proper subset is a set that contains no elements not in the parent set, and is missing at least one value from the parent set. Similarly, an improper fraction contains at least the whole.

The reason why we learn improper fractions briefly is to (a) introduce a useful form for operating with rational numbers (e.g. $$\frac32\times\frac34$$ is slightly easier to math than $$1\frac12\times\frac34$$), and (b) also to introduce the concept of proper and improper sets. In addition, in order to have a proper solution, one cannot use solutions which are not irreducible, as those are improper solutions. For example, $$1+1=\frac42$$ would be an improper solution, for hopefully obvious reasons.

This is obviously not really discussed in higher maths, by the time this topic is revisited, you're now talking about rational numbers and rational expressions, rather than simply fractions. This could be the reason why it's not talked about using that exact nomenclature in higher maths.

The appearance of "mixed numbers" is inherent to the Euclidean algorithm. For example,

\eqalign{\scriptsize{\frac{355}{113}}\rightarrow& 335=\color{red}{3}\cdot113+16,\ \scriptsize{\frac{355}{113}}=3\scriptsize{\frac{16}{113}}\\ \scriptsize{\frac{113}{16}}\rightarrow&113=\color{red}{7}\cdot16+1,\phantom{xx}\ \scriptsize{\frac{113}{16}}=7\scriptsize{\frac{1}{16}}\\ \scriptsize{\frac{16}{1}}\rightarrow&\ 16=\color{red}{16}\cdot1+0,\phantom{xxx} \scriptsize\frac{16}{1}=16\frac{0}{1}&}

and then we encode this as the continued fraction

$$\frac{355}{113}=\color{red}{3}+\cfrac{1}{\color{red}{7}+\cfrac{1}{\color{red}{16}}}$$

Notice the natural appearance of mixed numbers $$7\frac{1}{16}$$ and $$3\frac{16}{113}$$.

The Euclidean algorithm, which is one of the oldest known algorithms, involves repeated conversions of positive rational numbers $$p/q$$ where $$p>q$$ to mixed numbers:

$$\frac{a}{b}=n_0+\frac{r_0}{b}\rightarrow\frac{b}{r_0}=n_1+\frac{r_1}{r_0}\rightarrow\cdots$$

• 355/113 is not equal to 3*113+16. Oct 14, 2021 at 21:35
• @StevenGubkin Thanks, I had a typo that is corrected. The second appearance of 355 was incorrectly typed at 335. Oct 14, 2021 at 23:58
• Sorry, I meant to point out that the LHS is a number which is a little more than 3, while the RHS is a number which is exactly 355. For there to be equality, they should either both be equal to 355 or both be equal to 355/113. I call this "abusing the equals sign" when I talk with students. Oct 15, 2021 at 10:42
• @StevenGubkin Yes you are correct. I was being sloppy with equality. How embarrassing! Now it's clarified. Oct 15, 2021 at 14:44

For me it is simply a difference between diving 1 unit vs. dividing a group(more than 1 unit). I teach very small kids and sometimes I explain them a difficult topic for example fractions and try to find out WHAT EXACTLY are they not understanding, why they don't understand. So I have found that difference of diving/splitting 1 vs dividing/splitting group is important...