The simple answer is NO. WhatsApp doesn’t give you a new phone number at all, nor does it hide yours from your contacts the way Kik can (if you use it with an email address rather than a number).
Instead, the app is linked to your existing phone number. Each copy of WhatsApp can be bound only to one number, though it is possible on some phones to install multiple copies of the same app, thus linking to multiple numbers, but always in separate instances of the app.
In order to start new chats, you need to add contacts – doing so means that you need to save their number on your WhatsApp or phone contact list. When that’s done, you can send them a message. If they don’t have it already, this will show them your phone number along with the message you sent.
The main difference between WhatsApp and your regular texts is that instead of using your text allowance (which many providers restrict) you can send as many messages as you like via your Internet connection instead. You can also make voice and video calls without having to use your minutes. This will consume your data allowance at a fairly slow rate unless you are connected to a Wi-Fi hotspot.
A quick summary: No, WhatsApp does not give you a new phone number. It’s tied to an existing one and uses data services in order to send messages and make calls.