Non-integral number vs non-integer number

Question

Perhaps a rather silly question but what exactly is the difference between a non-integral number and a non-integer number?

I have heard both being used but I can't really make out the difference between the two. Is a non-integral number something concerning integrals?

An integral number is synonymous for an integer. A non-integral number is a number that is not an integer. — MeltedStatementRecognizing, Jul 31 '22 at 13:20
So "integral" can both mean "integral" as in the area between under a graph and "integer"? — tempdev nova, Jul 31 '22 at 13:23
Yes, $\int f(x)dx$ is also called "integral". So the name is the same as for an "integral" solution to $x^2-2y^2=1$, but is has nothing to do with it. So it always depends on the context. — Dietrich Burde, Jul 31 '22 at 13:29
True story. There was a lecture announced, the title was something about "integral equations". So I went, expecting equations involving integrals. But no, it was about equations with coefficients in $\mathbb Z$. — GEdgar, Jul 31 '22 at 14:12

score 3 · Accepted Answer · answered Jul 31 '22 at 13:30

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Some people call integers integral numbers, but it is not very common. Usually you call "integer numbers" just integers, so you say 2 is an integer, 1/2 is not an integer.

answered Jul 31 '22 at 13:30

trula

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It is very common to ask for "integral" solutions of a Diophnatine equation, i.e., like $x^2-2y^2=1$, or $x^3+y^3=z^3$. – Dietrich Burde Jul 31 '22 at 13:31
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Yep. Someone who knows how to use Google Ngram should probably check, but in my experience "integral solutions" is very commonly used, but describing integers as "integral numbers" is rare, and even sounds a bit off. (I'm a native American English speaker, FWIW.) – JonathanZ Jul 31 '22 at 16:11

ryang · Answer 2 · 2022-08-01T03:28:33.103

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To echo Dietrich Burde's comment, the phrase “integral solutions” (i.e., solutions that are integers) is rather common in number theory.

I find myself going out of my way to avoid writing the word “integral” to refer to integers—opting for “integer solutions”, for example, instead—because I irrationally worry that the word gets crucially misunderstood, not as being related to integration, but as meaning essential/fundamental!

I'd be happier to use the word if it has a more distinctive spelling, like “integeral” or “integereal”, instead.

edited Aug 01 '22 at 03:28

answered Jul 31 '22 at 16:01

ryang

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Is "integer solutions" grammatically dubious? It seems perfectly understandable to me. – qwr Jul 31 '22 at 21:52
"Grammatically dubious" is probably overly forceful; my point is that "[fill this in] solution" seems to require an adjective, which 'integer' isn't. – ryang Aug 01 '22 at 02:55
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nouns very well can act as adjectives too – qwr Aug 01 '22 at 03:02
Oh yes, nouns too can function as adjectives. So, "seems to require an adjective" ought to be "requires a modifier". Will edit away "grammatically dubious". – ryang Aug 01 '22 at 03:07
Grammar aside, understandability is the most important – qwr Aug 01 '22 at 03:08
For sure, clarity is the most important, which is precisely why I said I prefer the perfectly-understandable "integer solutions”. $\quad$ Anyhow, now that we've established that "integer solutions" is grammatically correct, I'm led to wondering: how (in what contexts) might "integral solutions" be advantageous?? – ryang Aug 01 '22 at 03:26

Non-integral number vs non-integer number

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