# Is it poor grammar to replace normal phrases with mathematical symbols in sentences in a mathematical paper?

English is not my native language and when I read mathematical papers, sometimes I saw sentences such as

The matrix A has rank ≥ n.

I am wondering if this sentence should be considered as grammatically wrong. I think the correct expression should be

The rank of the matrix A is greater than or equal to n.

Are expressions such as "The matrix A has rank ≥ n" considered as acceptable in mathematical papers/theses/textbooks?

• I've never seen an argument against it, @Zuriel. On the other hand, I don't read mathematical papers. My main disagreement with your both of your examples is the phrase the matrix A. I would say either the matrix or matrix A. (The A is used to specify the matrix in question; the the is used to indicate that you are talking about a previously specified matrix. Combining the two seems clumsy.) – TRiG Dec 22 '14 at 14:54
• @TRiG I disagree. Your example without "the" would be grammatically incorrect. In the example at hand I would write "at least" rather than either suggestion. – Tobias Kildetoft Dec 22 '14 at 15:00
• I don't think this is a good fit for English Language & Usage (it's more about academic writing conventions than the English language). One could debate whether it fits better here or on math.stackexchange.com, but I think it's fine here (and indeed this issue comes up in more fields than just mathematics). – Anonymous Mathematician Dec 22 '14 at 15:12
• @TRiG I would say 'the matrix A' is correct (and frequently used). I wouldn't say 'Matrix A' is incorrect, but I think it's less common. On the otherhand, there definitely is an argument against using the ≥ sign, namely that it is not formal English, rather a form of short-hand. – Jessica B Dec 22 '14 at 16:03
• @D.W.: This question is about mathematical equations in academic writing: the scope of that is beyond one academic field. "We expect people to do a significant amount of research before asking, and to show us in the question where they've looked." I don't think that the majority of questions asked here meet that standard. You are right though that the issue is covered in math style guides; an answer which includes a reference to one would be helpful. – Pete L. Clark Dec 23 '14 at 2:44

Are expressions such as "The matrix A has rank ≥ n" considered as acceptable in mathematical papers/theses/textbooks?

No, it's often considered poor style to incorporate fragments of equations like this into text. I wouldn't go so far as to say it's ungrammatical, but many people consider it bad writing. Some others don't care about this issue, which is why you sometimes see it done, but this is more common in informal or unedited writing.

The issue is that "rank ≥ n" is mixing together English and mathematics within the same construction. If this doesn't bother you, imagine a more dramatic case like "n + five". (By contrast, when someone writes "if x ≥ y", the inequality "x ≥ y" is a self-contained unit within the sentence.) There's no logical reason why mathematical writing conventions couldn't allow this sort of mixing, but they don't.

Saying "The matrix A has rank at least n" is shorter and cleaner than "The rank of the matrix A is greater than or equal to n", but they are both acceptable. I'd recommend avoiding "The matrix A has rank ≥ n" (I can't think of a good reason to prefer it, and avoiding looking bad is a reason not to use it).

• At least here, you could write "For matrix A, rank(A) > n," and avoid the issue altogether. – aeismail Dec 22 '14 at 16:31
• @aeismail Eeeew. That's just awful. I vote for "Matrix A has rank at least n." – JeffE Dec 22 '14 at 20:44
• @aeismail Having only a comma separating the two mathematical expressions $A$ and $rank(A)>n$ is at issue. I agree with mbork. – Steven Gubkin Dec 22 '14 at 22:19
• @yo': To my ear, "Matrix $A$ has rank..." sounds like it was written by a non-native English speaker (despite the fact that at least one native speaker supports it). It's a subtle issue: the name of the matrix is not "Matrix $A$", it's "$A$". The word "matrix" has been added so as not to begin a sentence, clause or phrase with a mathematical symbol. Saying "Matrix $A$" instead of "The matrix $A$" carries a whiff of anthropomorphism. I guess I am also a little annoyed at the amount of "article dropping" that takes place in math papers, so I would not want to sound like an article-dropper. – Pete L. Clark Dec 23 '14 at 18:02
• If I read a sentence starting "For matrix A..." instead of "For the matrix A" I would definitely assume the speaker was not a native American English speaker. Sesame Street doesn't say "This episode was brought to you by letter A and number 5." – Noah Snyder Dec 24 '14 at 3:56

(Copy Editor and mathematician speaking)

I, for one, allow these mixed constructions when editing the articles. I do know that it is not the best grammatical style, but not everything in math is easy to put down in proper English grammar. The two rules of thumb I use for these boundary cases is: Is the text clear to the reader? Can you easily make it grammatically correct?

For instance, the sentence For matrix $A$, the equation $\rank(A)\geq5$ holds. is cryptic and long. A better option might be For matrix $A$, we have $\rank(A)\geq5$. or Matrix $A$ satisfies $\rank(A)\geq5$. I would be fine with Matrix $A$ has rank at least 5. However, this gets complicated if you have more such expressions in a row, like in:

... which is defined as a non-real algebraic integer in modulus $>1$ whose
Galois conjugates except its complex conjugate are in modulus $<1$.


vs.

... which is defined as a non-real algebraic integer in modulus greater than $1$ whose
Galois conjugates except its complex conjugate are in modulus less than $1$.


I prefer the first option. This went through the AMS language editoral, as far as I remember, without any problem.

I let it pass if the sentence is unambiguous and can be pronounced normally when reading without any special effort like in "If $A$ is $\ge B+C$ and $f:[0,A]\to\mathbb Z$, then... (If the quantity/parameter/number $A$ is larger than the sum $B+C$ and the function $f$ maps the interval $[0,A]$ to the set $\mathbb Z$, then...) because in this case the extra words just slow the reader down. However, when seeing any ambiguity like "If A, B, C." (which comma is "and", and which is "then" here?) or something that, if attempted to be read as a sentence, violates not only the rules of the grammar, but also those of common sense as far as structuring sentences is concerned and which, if one needs it to be said at the board in a classroom, will have to be split into separate sentences and totally restructured to be comprehended by ear, I usually object.

Side note: what's the point of not enabling mathjax on Academia?

• I love your side note! – Zuriel Dec 23 '14 at 14:44
• upvoting the side note, but I can guess the limited audience (within academia.SX) needing it works against it. – Matthew Leingang Dec 23 '14 at 17:10
• The nice thing about mathjax, however, is that if there is no LaTeX code in the post, it is not even invoked, so what is there to lose? Anyway, that should rather be discussed on Meta from this point on :-) – fedja Dec 23 '14 at 17:30
• If you support adding mathjax to the site, please weigh in at meta.academia.stackexchange.com/questions/740/…. Apparently the point of not having it is that it one has to petition the SE administration to get it and there is concern that it will happen more often that people who don't know about mathjax will want to include two dollar signs on the same line more often than people will want to include mathematical equations. This is basically an argument from ignorance... – Pete L. Clark Dec 23 '14 at 17:49

In principle, it is generally acceptable to mix together mathematical and prose statements, as in your example: either construction would be technically be grammatically correct.

In practice, which to choose depends on how you want your reader to think about the statement that you have written. Prose emphasizes the relationship, in your example focusing the reader on "greater than." A mathematical statement tends to instead be thought of as a unit, in your example focusing the reader on "rank." You should thus choose accordingly.

One exception: small integers referring to counting within a small range should always be written as prose. Some examples of this distinction:

We selected eight conditions to test.

We found that 8 of the 73 samples were positive.

The boundary of "small" is a bit hazy: certainly less than 10, usually less than 20.

• it is generally acceptable — [citation needed] – JeffE Dec 22 '14 at 20:44
• @JeffE My reviewers don't seem to complain when I do it in good taste. – jakebeal Dec 22 '14 at 20:48
• @jakebeal Reviewers rarely, in my experience, comment on writing style, except for egregious errors. All the advice I've seen on mathematican writing recommends not mixing words and symbols as in "the rank is >2". – David Richerby Dec 23 '14 at 1:51
• @DavidRicherby Perhaps we have different experiences in our different communities - with some conference and journals, I often receive detailed remarks on wording and grammar. – jakebeal Dec 23 '14 at 2:28
• small integers referring to counting within a small range should always be written as prose. J.E. Littlewood (A mathematician's Miscellany, p. 39) humbly disagrees. I don't know the official position of his long time co-author who left us one of the most exquisite samples of English prose among other works, but I suspect it would not be very different either. The only real rule is that you try to do what does make sense, try not to do what does not, and don't insist that your way to distinguish between the two is the only correct one. Everything else is allowed :). – fedja Dec 23 '14 at 17:21