I'm writing an article and I have used this sentence.
[…] although there are a significant number of computers […].
After I checked it with an online spellchecker, it said that I should use is.
Should I change the phrase as suggested?
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I'm not sure what your grammar checker is on about. No matter how you break down the sentence, the verb must be plural.
If you figure that "computers" is the noun and "a significant number of" is a modifying phrase, then it is dead obvious that the verb should be "are".
If you figure that "a ... number" is the noun and both "significant" and "of computers" are modifying phrases, then you as a learner could be excused for thinking that the verb should be singular; but anyone with some grammatical knowledge, and certainly a grammar-checking algorithm, should know that "a number" is a plural noun phrase. Perhaps the algorithm wasn't very well-written and got confused by that intervening "significant"?
Anyway, bottom line is, you had it correct in the first place; please ignore the grammar checker.