
aufschreiben
to write down
To record something in writing so as not to forget it; to make a note.
Kannst du mir bitte deine Adresse aufschreiben?
Can you please write down your address for me?
to write down
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to write down
To record something in writing so as not to forget it; to make a note.
Kannst du mir bitte deine Adresse aufschreiben?
Can you please write down your address for me?

to prescribe
For a doctor to prescribe a medication or treatment to a patient; to issue a prescription.
Der Arzt hat mir ein starkes Schmerzmittel aufgeschrieben.
The doctor prescribed me a strong painkiller.

to put on a tab
To put goods or services on a bill that will be paid later.
Schreiben Sie es bitte auf, ich bezahle dann am Ende des Monats alles zusammen.
Please put it on my tab; I'll pay for everything together at the end of the month.
'Aufschreiben' (to write down) is a separable verb. The prefix 'auf-' goes at the end in a main clause. In subordinate clauses and in the infinitive with 'zu', the verb stays together.
Ich schreibe mir deine Nummer sofort auf.
I'll write down your number right away.
'Aufschreiben' (to write down) is very often used with a dative object that indicates the person for whom something is written down. The structure is: jemandem (dative) etwas (accusative) aufschreiben.
Der Polizist schrieb dem Zeugen seine Kontaktdaten auf.
The police officer wrote down his contact details for the witness.
In the meaning 'notieren' (to note down), the verb is often combined with information you don't want to forget: write down an address, a phone number, a thought, an idea, or a shopping list.
Er hat sich während des Vortrags viele Notizen aufgeschrieben.
He wrote down a lot of notes during the lecture.
When the verb is used in the context of doctors, patients, or pharmacies, it means 'ein Rezept ausstellen' (to issue a prescription) or 'verordnen' (to prescribe). Signal words are 'Arzt' (doctor), 'Rezept' (prescription), or names of medications.
Meine Ärztin hat mir Antibiotika aufgeschrieben.
My doctor prescribed me antibiotics.
In a shop, bar, or restaurant, 'etwas aufschreiben lassen' (to have something put on a tab) means that you don't pay immediately, but have the amount put on a bill. You often hear the question: 'Können Sie das aufschreiben?' (Can you put that on my tab?).
Ich habe mein Portemonnaie vergessen, können Sie die Getränke bitte aufschreiben?
I forgot my wallet; can you please put the drinks on my tab?
to write
Both words can mean recording something in writing. Schreiben describes the general activity of writing, while aufschreiben more clearly emphasizes the purpose of recording something so you can remember it or look it up later. Without this particular purpose, schreiben fits; for a short note, aufschreiben is usually the better choice.

Ich schreibe die Telefonnummer auf, damit ich sie nicht vergesse.
I write down the phone number so I won't forget it.
Ich schreibe meinem Bruder die Telefonnummer in einer E-Mail.
I write my brother the phone number in an email.