I don't have any old checks, but I can log into my bank's website. What can I do to learn the number?
Every banking system I have seen has the account info somewhere in the area you see after you login.
I was going to say that you should also be able to find it on the electronic version of the monthly statement, but my bank X's out most of the account number on the electronic statement.
If there are checks that you wrote, that you can still see the transaction on the website, many banks attach to the transaction a copy of the check. You should be able to see it there.
Depending on your country the the bank may have sent you your copy of a tax form showing the interest earned in a year. In my case the account number is listed on the tax form.
Otherwise call the bank...