The beginning of the Roman Empire was the annexation of Ostia, which (thanks to the Gauls burning the city and its written records) wound up being a prehistoric event, but it took place about 396 BC; Rome was sacked in 410 AD, but the last Emperor was deposed in 476 AD, a period of 872 years. This doesn't account for the various poorly documented territorial gains during the preceding generations, both before the last Etruscan king was booted around 509 BC and the period of slightly more than a century until Ostia's annexation.
Calling the realm of Constantinople an empire after the loss of Anatolia to the muzzie onslaught, and the piratical raids by the muzzies thereafter, isn't realistic imho. After the death of Boris II early in the 11th c, there was a more or less steady erosion of control over the eastern territories, mismanagement, tax increases, and the reemergence of what we might recognize as the Deep State.
Small successor states tottered on for a time (a decade?) even after the fall of Constantinople to the Turks, but they'd already gained territory in the 14th c via seagoing invasion of parts of the Adriatic coast of Balkan territory.
On a side note, I've often guffawed at the term "Byzantine" to refer to the eastern Roman Empire.
I agree with other observations on this thread that we in the West have tended to neglect study of the Empire after the fall of Rome. Whatever it’s called, I find it a fascinating world, lost to the predations of the barbarians from deserts and steppes. I also agree with the observation that by the time the Venetians sacked Constantinople the Empire was really not an Empire anymore. But, if you want to see some of the most beautiful objects from old Constantinople, they can be found in St. Marks, Venice.