If the numerator is greater than the denominator, it's improper. The other way around, it's proper. What if the numerator is the same as the denominator?
2026-03-28 13:59:37.1774706377
On
On
is x/x a proper or improper fraction?
1.2k Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail At
3
There are 3 best solutions below
0
On
A quick google search yields this site that clearly says (emphasis added):
An Improper Fraction has a top number larger than (or equal to) the bottom number.
0
On
The term "fraction" (from the Latin frangere, "to break") essentially means "fragment". A proper fraction, then, is one that that respects this intention, representing a fragment of the unit.
The unit itself is clearly a special case; I suspect textbook definitions lean toward calling $x/x$ an improper fraction, perhaps because the unbroken whole isn't usually what comes to mind when one thinks of a "fragment". Reasonable people may disagree, however, so keep an open mind.
Consider, for instance, these entries from the "Earliest Known Uses of Some of the Words of Mathematics" (EKU) pages ...
IMPROPER FRACTION was used in English in 1542 by Robert Recorde in The ground of artes, teachyng the worke and practise of arithmetike: "An Improper Fraction...that is to saye, a fraction in forme, which in dede is greater than a Unit." [Under FRACTION, EKU notes that Recorde wrote "A Fraction in deede is a broken number", which recalls the notion of a fragment.]
... but also ...
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).
Interestingly, EKU doesn't cite either of these sources as explicitly defining the opposite terms; perhaps they did. (EKU isn't an absolute authority.) In any case, if we interpret the given citations such that anything that isn't "improper" is "proper", and vice-versa, then these authors differ in their classification of the unit.
In a comment to the question, @egreg notes that "apparent fraction" applies to $x/x$. I've never heard of this term. EKU doesn't have it, nor does (English) Wikipedia. I finally tracked-down this definition via Answers.com and thought I'd share it ...
An apparent fraction is a fraction with one of the following properties:
- The numerator and denominator are the same (eg 2/2 or 10/10)
- The denominator is one (eg 4/1 or 20/1)
- The numerator is a multiple of the denominator (eg 4/2 or 10/5)
So, it seems that an apparent fraction is simply an integer that appears in fractional form. I guess it's nice that there's a word for such a thing, but I'm with @egreg: I don't know that I'd have much use for it.
At the end of the day, it doesn't matter. It comes down to a matter of convention. In fact, I've never heard of the terms, proper, improper (or, especially, apparent) fractions anywhere beyond primary school. No theorems apply exclusively, for example, to proper fractions. Hence it's best to just use whatever definition you like; if you feel that $10/10$ is improper, then say it so, otherwise, say it's proper. You do you, it doesn't really matter.