Broadband connections don't work on a dial-up system, there's no number to dial to.
If you have a BT landline broadband connection the line is split into separate channels. A good analogy is that of a pipe with a narrower pipe inside it. The wide pipe takes digital ADSL traffic - what we call broadband - and the narrow pipe is for analogue traffic (dial-up). The two data streams never mix, they run separately to the exchange, where the analogue signal is taken straight into the normal BT telecoms network. The ADSL connection goes into a gateway, and thence via some more complex hardware into the Internet. If your ISP is not BT then your connection to the internet may be at the ISP's premises (if the company is big enough), where there will be similar hardware.
So, as you can see, there's no dialling involved in a broadband connection - your modem automatically handshakes with a device in your local BT exchange.