"I do not know his number."
Translation:Nie znam jego numeru.
You can't say "Nie wiem jego numeru". You should say "Nie wiem, jaki jest jego numer" or similarly.
As the English wiktionary says:
znać - to know somebody/something (be acquainted or familiar with)
wiedzieć - to know (be certain or sure about something)
While this is true , when you look closely you will find holes in this theory.
the difference between znać i wiedzieć is also in what follows them .
Znać + kogoś/coś ( a noun in accusative)
wiedzieć+ a phrase or about sth
Sometimes liczba is listed as sinonym with numer. Is this not true when talking about a phone number?
Both are translated as "number", but:
liczba is an abstract entity in mathematics, it's the grammatical number, and the amount of something countable.
numer is "used to show the rank of something in a list or sequence", so it's a house/flat/room number, a number on your locker, on a footballer's shirt, also a telephone number.
Technically, without a clear context, "liczby" could be okay here... but maybe let's stick to "numeru".