In working through topics pertaining to fractions, I find that questions about improper fractions are common. Today we’ll look at questions about the definition of the term, and next time we’ll move on to mixed numbers and how to convert between the two forms.

## How can you have more than 9 ninths?

First, here is a question from 2002, about how improper fractions should be read (and how they can exist at all):

Improper Fractions My math teacher says that we read 14/9 as "fourteen ninths", but I think we should read it as "fourteen over nine" because the 14 can't be a part of nine parts. And do we have a fraction called "six halves"? If yes, how do we write it? Please help. I am really confused. Best regards.

Dana seems specifically concerned about the use of words, feeling that an improper fraction shouldn’t be *read* as a fraction; but probably also is unsure about the concept itself. I replied:

Hi, Dana. In everyday speech, we think of a fraction as something less than one. If I said "a fraction of my coworkers were laid off today", I would mean a small part; if I said "a fraction of a percent of students understand the meaning of this word", I would mean less than one percent. But the root meaning of the word "fraction" is just "broken"; mathematically, a fraction can be made up of any number of pieces, even if, together, they make up more than a whole.

This is yet another example of a mathematical word starting with an everyday concept (a piece of something), and extending it so that some of the original implications (smallness) are no longer present. Mathematicians love to generalize.

Just imagine taking a bunch of pizzas and cutting each of them into eight parts. Now I might eat three of them myself, so I've eaten three eighths: 3/8 of a pizza. But someone else might take 21 pieces for his family of seven; he's taken 21 eighths, which is the fraction 21/8. Do you see that it really makes just as much sense to read this fraction as "twenty-one eighths", even though it is not part of just one pizza? Similarly, if I cut three apples in half, I would have six halves, which we can write as 6/2.

All of these – the proper fraction \(\frac{3}{8}\),

the improper fraction \(\frac{21}{8}\),

and the whole number in disguise \(\frac{6}{2} = 3\),

are fractions, in the broad sense.

Certainly there is a good reason for preferring, in many situations, to rewrite such an improper fraction as a mixed number, so that the fractional part is less than one as we expect;we call these fractions "improper" just because they do feel "wrong" to us. But since improper fractions still make sense, and are very useful in many situations, it is correct to read them the same way we read proper fractions.

As we’ll see below, this sense of “wrongness” is the core of the idea of “improper”.

## Is 4/4 an improper fraction, or not?

This 2003 question deals with a commonly misstated aspect of the definition:

Is the Fraction N/N Considered Proper or Improper? I am developing a technical math review program for adults and have a question regarding the correct term to use when attempting to identify the fraction n/n when n is greater than 0. It does not seem to fit into any of the common fraction categories, (ie, proper, improper, mixed). I understand that it can be identified as "1", but was wondering if n/n is just a unique situation that does not fit into any of the fraction groups. It would seem that it could be identified as a mixed number, (n/n = 1 0/n but that seems too obvious). Any comments would be appreciated.

Richard had found definitions that say an improper fraction is one whose numerator is **larger** than its denominator. Does this make 4/4 proper, because 4 is not greater than 4?

I answered:

Hi, Richard. It certainly is not "mixed", since it has only a fractional part, and no whole part. The numerator of a "proper" fraction is less than the denominator, so I would call this an improper fraction. You can find definitions of "improper" that say the numerator has to be greater, not just equal; but I think that is wrong. See this page, which is a little ambivalent: Improper Fraction http://mathworld.wolfram.com/ImproperFraction.html A fraction p/q > 1. A fraction with p/q < 1 is called a proper fraction. Therefore, the special cases 1/1, 2/2, 3/3, etc. are generally considered to be improper. According to Merriam-Webster (m-w.com), an improper fraction is a fraction whose numerator is equal to, larger than, or of equal or higher degree than the denominator.

I find that today, the Mathworld page is the same except that they have changed the first sentence to “A fraction p/q ≥ 1.” That makes everything consistent, while still using the word “generally” to suggest that not all authors say this. The dictionary still gets it right. (They don’t always do so for math terms.)

## Is 5 a mixed number?

To elaborate on what I mentioned there, here is a similar question from 2004 about mixed numbers:

Definition of Mixed Number Is 5 a mixed number? Why or why not? I think it is, because you can write 5 0/3, can't you?

Doctor Rick answered:

Hi, Brian. I think it's important, when considering this question, to keep in mind the difference between anumberand anumeral. A number is more of an idea, while a numeral is a name for a number--a way of telling others what number you're thinking of. For instance, 1.5, 3/2, and 1 1/2 are three different numerals that all represent the same number. A mixed number would be better called amixed numeral, because it is a way ofwritinga number. When you ask if the number 5 is a mixed number, you're really asking about this particular way of writing it: is the *numeral* 5 a mixed number?

There are many ways to write 5, some of which can be called mixed numerals, though “5” is not:

You *can* represent the number 5 by a mixed number 5 0/3, though I wouldn't do so except while working a problem with mixed numbers (even then I'll probably only "write" it this way in my head), or because I was asked to do it. Butthe numeral 5 is not a mixed numberbecause it doesn't have a fraction part. If a mixed number meant a number that *could* be written with a whole number and a fraction, then *every* number would be a mixed number. For instance, 0.5 would be a mixed number because it can be written 0 1/2. There would be no reason to call anything "mixed number", because it wouldn't tell you anything about the number! That's my opinion. I haven't checked to see whether everyone agrees with me. If you find out that someone disagrees, I'd like to see their reasons.

An additional comment I might make, from my more recent experience teaching, is that sometimes a problem will ask for a mixed number as an answer, but you might find the answer is 5. In that case, the instructions tell you what to do in the “worst case”: If you can’t write it as a whole number or proper fraction, use a mixed number rather than an improper fraction. In that sense, 5 would fall under the umbrella of “mixed number”.

## Is -3/4 a proper fraction?

In 2004, we got a question at another edge of the definition:

Can Negative Fractions Also Be Proper or Improper? How do you classify negative fractions? Is a fraction like -3/4 considered a proper fraction? Is -8/3 considered an improper fraction? I've generally thought thatall proper fractions lie between 0 and 1 on a number line. However, what about the negatives?

That’s a new one! I answered:

Hi, Laurel. The usual definitions are aimed at kids who don't know about negative numbers; it's rare to see a definition that takes negatives into account. When you consider negative fractions, it is theabsolute valuethat has to be less than 1 for a proper fraction; a fraction between -1 and 1 therefore is proper.

Quite a few terms are defined for kids (or for mathematicians living before modern times!) and therefore apply only to simple cases with positive numbers. Another example is direct proportion, which is commonly described by saying that both quantities involved increase together – but that isn’t true if one can be negative. Formal mathematical definitions are stated carefully to cover all cases, but common definitions are not so careful.

An alternative way to deal with this might be to restrict the word "fraction" to positive numbers, and use "negative improper fraction" to refer to "the negative of an improper fraction". That avoids the question, but might sound too tricky to kids. I can't be sure which way people generally think of it when they do talk about negative fractions; I just know that -8/3 is improper and -2/3 is proper.

This would allow us to say a proper fraction is positive, ignoring the sign issue; but then we would seem to be saying that any fraction must be positive, which is definitely not true.

But “I just know” always feels dangerous to me:

I like to confirm my impression in cases like this, so I searched on Google for "negative improper fraction", but I found nothing but this, which doesn't clarify the exact meaning: Positive and negative numbers http://www.mathleague.com/help/posandneg/posandneg.htm What is the reciprocal of -5 1/8? First, we convert to a negative improper fraction: -5 1/8 = -41/8, then we switch the numerator and denominator, and keep the same sign: -8/41.

They could be taking “negative improper fraction” either way.

Unfortunately, the Math League'sdefinitionof improper fractions ignores negatives: Fractions http://www.mathleague.com/help/fractions/fractions.htm Improper fractions have numerators that are larger than or equal to their denominators. Examples: 11/4, 5/5, and 13/2 are improper fractions. I'm inclined to interpret "larger than" here as "greater in absolute value" in order to make the definition work with negatives. That is, p/q is improper if |p| >= |q|.

That’s another language workaround; “larger” does sometimes have this meaning, as opposed to “greater than”, which is more formally mathematical.

Laurel replied, explaining the reason for the question:

Thank you, Dr. Peterson! I am a 6th grade math teacher of a top level class. When I asked the students to write where all proper fractions lie on a number line, the answer I expected was between 0 and 1. But one student claimed that they were between -1 and 1. I'd never thought of that, and I wanted to be sure before I saw her again that it was, in fact, correct. I appreciate your very prompt reply.

Students thinking outside the box (because they aren’t quite sure where the box is!) can be the source of some of our best questions.

## Why is it called improper?

Those questions lead up to a deeper question: What is the basic meaning of “improper” that determines how we think about any definition issues? Here is a question from 2006 that takes us to that level:

Origin of the Term "Improper" Fraction I love your website and use it often. Do you know why fractions with larger numerators than denominators are called "improper"? Where did the terms "proper" and "improper" originate in relation to fractions? Just curious! Thank you. AnneMarie

Doctor Ian answered:

Hi AnneMarie, It might help to consider some other uses of "proper" in mathematics. For example, what are the divisors of 12? They are 1, 2, 3, 4, 6, 12 But a number is _always_ a divisor of itself, so we use the term "proper divisor" to refer to the _other_ divisors. Similarly, a set is always a subset of itself, so we use "proper subset" to refer to subsets missing at least one element.

These examples of other uses of “proper” within mathematics suggest that it means “other than the whole”.

You can find other examples by searching for the keyword "proper" at http://mathworld.wolfram.com/ So by extension, we might consider "proper" to be used, when talking about fractions, to exclude the analogous case, i.e., where the numerator isn't at least one less than the denominator. Does that make sense?

I jumped in with an addendum, looking outside of math for more examples and history. I noted that the question, literally, was “*where* did the terms originate”:

Hi, AnneMarie. To add a bit to what Dr. Ian said, I have two slightly different ways to approach the question. First, historically, if I want to strictly answer your question, I should tell you WHERE the usage first occurred. For that, we look here: Earliest uses of mathematical words http://jeff560.tripod.com/mathword.html That site says PROPER FRACTION appears in English in 1674 in Samuel Jeake Arithmetic (1701): "Proper Fractions always have the Numerator less than the Denominator, for then the parts signified are less than a Unit or Integer" (OED2).

This gives a definition (which applies to positive fractions), and a reason: A proper fraction is less than 1. This ties in with what we saw before, “proper” meaning “less than the whole”.

Then, where did the word itself come from?

Second, to find WHERE a word comes from, and WHY, we can look in a dictionary. I looked up "proper" and found that it originally meant "belonging to one" or "peculiar to something"; then came to mean "appropriate or suitable" and "strictly correct". That last meaning, I think, is where the common mathematical use originated. A "proper fraction" is one thatreally is a fraction(a broken part, as it is etymologically)in a strict sense, rather than following the extended meaning mathematicians have given it (anything expressed as a/b). In the same way, a "proper subset" isREALLY a subset, strictly speaking (a part, not the whole), and a "proper divisor" isSTRICTLY a divisor, not just the same number, and so on.

In Latin, “proprius” means the adjective “own”; from it we also get “property” (one’s own possessions, or a characteristic), “propriety” (properness), and “appropriate” (suitable).

Another related usage is "the city proper", meaning "the city itself, taken in a narrow sense rather than a broad sense". In m-w.com, where I looked, this is defined as "strictly limited to a specified thing, place, or idea". Of all the definitions they give, I think this is the closest to the general mathematical usage.

So “proper” means “in the narrowest sense”.

Then I looked farther into the MathWorld site, as Doctor Ian suggested, for more definitions.

Finally, taking Dr. Ian's advice, I found this page: Proper http://mathworld.wolfram.com/Proper.html In general, the opposite of trivial. SEE ALSO: Proper Divisor, Proper Ideal, Proper Subset, Proper Class, Proper Submodule, Proper Fraction, Proper Subfield, Proper Extension, Proper Superset, Trivial.

That gives you a sense of other uses of the word in math, which are all similar to those Doctor Ian and I raised. “Trivial” means “related to or being the mathematically most simple case”.

Taking that general sense of the word, one might wonder whether it would also exclude "trivial" cases at the other end of the spectrum: are the fraction 0/1, the subset {}, and the divisor 1 "improper"? Usually, the answer is no; but sometimes one might want to take it that way: Proper Divisor http://mathworld.wolfram.com/ProperDivisor.html To make matters even more confusing, the proper divisor is often defined so that -1 and 1 are also excluded. Using this alternative definition, the proper divisors of 6 would then be -3, -2, 2, and 3, and the improper divisors would be -6, -1, 1, and 6. We've more than answered your question, but I think it's interesting!

## How about improper fractions in algebra?

The last question we’ll look at takes us beyond arithmetic, but is useful in thinking about how a concept changes as it is applied to new things. This is from 2017:

Inconsistently Improper? Why do we consider somealgebraic fractionswith thesame degreeof x to be improper? For example, why is (x - 1)/(x + 1) improper? I understand that a fraction is considered improper if its numerator is greater than its denominator. However, for (x - 1)/(x + 1), if x was 2, then (x - 1)/(x + 1) would be 1/3. I think for any real x, the numerator would still be smaller than the denominator. So why is (x - 1)/(x + 1) improper? Or, more generally, why is a fraction such as (x - 1)/(x + 1) -- which would be a proper fraction if x was replaced by a real number -- still considered to be improper?

I answered:

Hi, Garand. When we talk about algebraic fractions, we can't compare polynomials by saying which one is greater, as we do with numbers, since typically one will be greater for some values, but less for others. Instead, we compare their degrees. So by definition, if the degree of the numerator is not less than the degree of the denominator, we call it improper. This is entirely unrelated to the values of the numerator and denominator for any particular value of the variable.

In moving from arithmetic to algebra, and specifically polynomials, **values** are not as significant as **degrees** (in terms of their overall behavior), so we transfer the role of size of a numerator to its degree. Interestingly, this means we have abandoned any explicit idea of being “less than the whole”, but simply taken an *analogy* to fractional numbers.

In particular, when the degree of the polynomial in the numerator is greater than or equal to the degree of the one in the denominator, we can divide the top by the bottom to obtain a polynomial plus a proper fraction (in the algebraic sense), in just the same way we do with improper fractions (in the numerical sense) to obtain mixed numbers. The parallel is ultimately the reason for using the same term.

In the example, the improper fraction \(\displaystyle\frac{x-1}{x+1}\) reduces to the “mixed expression” \(\displaystyle 1 – \frac{2}{x+1}\), where the fraction part is proper.

For those of you paying attention to the algebra, I added a correction to the question:

By the way, in your example of (x - 1)/(x + 1), you are wrong in your assumption that the value will be a proper fraction for ANY real value of x. Consider x = -5: (x - 1)/(x + 1) = (-5 - 1)/(-5 + 1) = (-6)/(-4) = 3/2, which is an improper fraction! It may be true that there are SOME improper algebraic fractions that always evaluate to proper fractions, but that would be a very special circumstance.

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