A number of vs the number of

Why is a number of plural, but the number of singular?

We get asked this question frequently by our students.

The confusion arises because, overtly, there seems to be little difference between the two except for the indefinite article in one and the definite article in the other.

Yet, small though the difference is, the two differ in subject-verb agreement. The expression a number of takes a plural verb, whereas the number of takes a singular verb:

(1) A number of students have raised this question.
(2) The number of students has increased over the years.

Why?

Well, a number of means several:

(3) Several students have raised this question

 On the other hand, the number of refers to a specific number of something:

(4) The number of participants (at over 12,000) is high.
(5) The number of stray cats has grown in our neighbourhood. (It was 20 last year, but it is now 35.)