In my exercises I’m reading through it says:
Wir haben den dritten Oktober — It is the third of October
Is this the common way to express the date in German?
would no one say:
Es ist der dritte Oktober
as a more direct translation?
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For me as a native German speaker, I would say
"Heute ist der dritte."
as my first option. (See that I left the month part out since it is obvious most of the time that we know which month we have). This would probably the most common answer in spoken German language since pretty much every spoken language tries to abbreviate as much as possible.
If you wanna express that today is the "third of October" you would probably go with
"Wir haben den dritten Oktober"
when you are speaking, while written language prefers to use
"Es ist der dritte Oktober"
Note that this is only a fine, slight difference, so you should be fine either way, but if you wanna show that you have a deeper understanding of the language you could respect and use it the way I showed you.