In the middle ages, there was greater Persia and two Iraqs (Iraq al-Arab to the west and Iraq al-Ajman to the east which is today's western and central Iran).
If we are being pedantic, either the polymaths in question were Persian (which they are) or Iraqi Arabs, not Iranians. In fact, Persian empire also encompassed parts of modern day Uzbekistan, Kyrgyzstan, Turkmenistan, Afghanistan, Pakistan, and Kazakhstan, until they came under the Turcko-Mongol rule.
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Anyway, “Persia” (Persis) comes from the Greek version of “Pars”. Persians call their language “Farsi”, from the Arabic version, “Fars”. These all have the same origin, and date back millennia.