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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?
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).
(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) ≥ 5 holds." is cryptic and long. A better option might be "For matrix A, we have rank(A) ≥ 5." or "Matrix A satisfies rank(A) ≥ 5." 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 editorial, 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?
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.
