I suppose 'the number of' is followed by a singular, but what about the following sentence where they are two subjects:
Both the number of entries and exits is/are high.
I suppose 'the number of' is followed by a singular, but what about the following sentence where they are two subjects:
Both the number of entries and exits is/are high.
I agree with the comments by tchrist and RegDwigнt that the underlying problem here is how to manage the parallelism implied by the word both in the example sentence. The wording
Both the number of entries and exits [is/are] high.
attempts to establish "the number of entries" and "exits" as parallel branches. But I think that tchrist is right in asserting that opting for "the number of entries" and "the number of exits" is a better way to convey the equality of the two branches. That would yield this sentence:
Both the number of entries and the number of exits [is/are] high.
and it should be fairly clear from the repetition of the word number that are is the better verb choice for that sentence.
On the other hand, you could slide the node-word both farther up the trunk of the sentence and make the parallel branches "entries" and "exits":
The number of both entries and exits [is/are] high.
In this case I would be inclined to go with is as the verb in the sentence, on the theory that "number of both" in that sentence means "total number of items in both categories of things, taken together." But given the choice, I would also be inclined to change number to numbers:
The numbers of both entries and exits [is/are] high.
and again revert to are as my verb choice.
The question of how to properly frame the sentence hinges on whether the author wants to emphasize that the total of x + y, where x = entries and y = exits, is a large number or whether the author wants to emphasize that both x taken separately and y taken separately are large numbers. Referring to a design that has 122 entries and 3 exits, you could assert that "The number of both entries and exits (taken together) is large"; but it would not be true in that case that "Both the number of entries and the number of exits are large."