Q:

How to know the number of database connections

Please note that even though it looks very similar, it's not a duplicate question with this link:
How to list active / open connections in Oracle?
I'm not asking about the number of sessions, but connections. I'm aware that I can query the v$session view, but I don't know how many connections are being used there. If there is a way to derive from it, please enlighten me.
EDIT:
I'm asking about the physical database connection to the database.

A:

Bit confused with your statement I'm not asking about the number of sessions, but connections.
Conceptually both are same. Every active session will correspond to a underlying active connection to the database.
Now, if you meant to know the max allowed connection limit then Documentation says

Maximum number of connections (system and application) across all
  databases in an instance  = 2048

To know the allowed session configured to your database, you can query v$parameter view like
SELECT name, value 
  FROM v$parameter
 WHERE name = 'sessions'

If you want to know the Active session at any instance Out of total configured to allow then you can query v$session view using the Status column like
SELECT COUNT(*)
  FROM v$session
WHERE STATUS = 'ACTIVE'

You may want to refer this post How to check the maximum number of allowed connections to an Oracle database?

