formula to calculate the number of days a case was open based on the date it was created

Try this as a formula numeric field

ToDays([Date Closed] - ToDate([Date Created]))

But you probably want this better one to also show the running days open for unclosed cases.

var date EndDate = IF(IsNull([Date Closed]), Today(), [Date closed]);

ToDays($EndDate - ToDate([Date Created]))

You also have to think if you need to add 1 to the result.  For example if a case was opened on Monday and it's now Tuesday has the case been open 1 day or 2. That's your call based on how your organization thinks.