die Rufnummer
phone number
A sequence of digits at which a phone line or mobile device can be reached.
Können Sie mir bitte Ihre Rufnummer geben?
Could you please give me your phone number?

phone number
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phone number
A sequence of digits at which a phone line or mobile device can be reached.
Können Sie mir bitte Ihre Rufnummer geben?
Could you please give me your phone number?

The word 'Rufnummer' (phone number) is made up of 'Ruf' (call, from 'rufen' — to call) and 'Nummer' (number). So it is literally the number you can use to call someone.
Bitte notieren Sie sich meine neue Rufnummer.
Please write down my new phone number.
Typical verbs used with 'Rufnummer' (phone number) are 'wählen' (to dial, in order to call), 'speichern' (to save, in the phone book), and 'hinterlassen' (to leave, on an answering machine).
Ich habe eine unbekannte Rufnummer auf meinem Display, die ich nicht gespeichert habe.
I have an unknown phone number on my display that I haven't saved.
A common and useful phrase is 'die Rufnummer unterdrücken' (to withhold/hide the phone number). This means that your own number is not shown to the person being called.
Wenn du anonym anrufen möchtest, kannst du deine Rufnummer unterdrücken.
If you want to call anonymously, you can withhold your phone number.
'Rufnummer' (phone number) is a rather formal and technical term. You often find it in contracts, on official forms, or in device settings.
Für die Vertragsänderung müssen Sie Ihre Kundennummer und Rufnummer angeben.
For the contract change, you must provide your customer number and phone number.
number
'die Nummer' and 'die Rufnummer' can both mean a sequence of digits used to reach someone. 'die Nummer' is very general and can also mean many other kinds of numbers. In conversation, 'Nummer' can be short for phone number when the context is clear. 'die Rufnummer' is more precise and sounds somewhat more formal, for example with authorities, contracts, or customer service. The confusion arises because both can fit when talking about phones. In everyday life, people often say 'Nummer'; in official texts, 'Rufnummer' is more common.

Bitte geben Sie auf dem Antrag Ihre aktuelle Rufnummer an.
Please enter your current phone number on the application form.
Ich habe die Nummer der Ärztin in meinem Handy gespeichert.
I saved the doctor's number in my phone.