when a phone call is dialled in Shadowrun, how is represented? By numbers? Do area codes exist? Do telecom systems even work the same way telephone systems do?

I would think they would be more like ipv6 internet addresses.

Phones would be just another integrated application or device.

Your telecom would have a physical presence in the matrix, that
people can visit or patch through to talk with you.

At least that is how I would imagine it ... data is data.

I suppose technically they look like this: NA/UCAS-SEA 5206 (15-1667)

So you can see some mockery of the original phone system simply tacked onto an international one, which is believable in a way, but not as a "people interface". I hypothesize that phone numbers are a bit like webpages in usage. In other words, to call a known contact you pick up the phone and say, "Give me Velocity" and the phone calls you however you've set it up to be called (ie, sent to the mobile first, then telecom, then voice mail or voice mail everything, etc). An unknown would be like a google search, "I'm looking for a Velocity in Montréal," and the phone would list whatever public profiles for "Velocity" located in Montréal.

And, for those with bad memories, they can search by pictures.

Just imagine what happens when you do that and get multiple results...

According to SR2, they look like:
NA/UCAS-SEA-2206

So that's:
(continent)/(nation)-(optional region)-(four digit LTG #)

... which provides a maximum of 10,000 numbers for areas frequently containing multiple cities with millions of residents, and requires remembering numbers, which the real 'net hasn't required for decades, even since before Shadowrun was created.

Personally, I figure they should look something like:
telecom://nakamura.susan@hr.sea.renraku.com
or
voice://elfch1xx0r69@ucasonline.com

URLs, email addresses, not (-shudder-) phone numbers.

You guys are great, thanks very much for the info.

Keep in mind, most telecoms will probably have an automatically updating phone book. So the first time you can put in the number (or likely, if the person has publicly available information, a name and address), then the next time you can simply say 'Dave' or 'Jimmy the Squid' and the telecom will take care of the nitty gritty on your behalf.

Mine tend to look like:

"There's a tridphone number scrawled on the wall above the wreckage. Very shaky handwriting. Underneath it the maniac wrote 'Call me if you want to see your cat again! I'm getting hungry! MWAAHAHAHAAA!' Yes, he spelled that out."

It's fairly rare that the actual format of the number matters in game. Every once in a while I'll make some sort of in-game artifact (a newssheet, a corporate memo, etc.) and then this sort of thing matters, but if you start getting too deep it gets hard to be consistent, and you realize that a lot of what FASA defined really wouldn't make sense if implemented in practice.