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Persian or Farsi?
In short, Persian language is the official language in Iran. The Iranians call this language 'Farsi'. There is one more thing to add to this short explanation and that is about 'Persian' and 'Iranian'. Remember that all Persians are Iranian, but not all Iranians are Persian!
I hope this definition is clear enough and is a good answer to 'Persian or Farsi?' discussions.
Actually, it is more correct to call it Parsi.
It was Arabs who named it Farsi.