It depends on what you consider the fall of Rome.
It is conventionally remembered that Rome fell in the year 476, but that isn’t the whole story.
First, a bit of background:
Prior to the rise of the Roman Empire, Alexander the Great conquered the middle east and the eastern mediterranean. Alexander died and his empire fractured as his four generals fought over his territorial conquests. For centuries after Alexander’s conquest, the eastern mediterranean was ruled by Greeks, and became linguistically Greek speaking and heavily Greek influenced.
When the Romans eventually conquered the eastern Mediterranean, the lingua franca of the region was already Greek. Whereas the Romans considered other people groups they conquered to be barbarians, they never called the Greeks ‘barbarians’, because they respected the Greeks, and were heavily influenced by Greek culture, religion, philosophy, mathematics, and aesthetics. Even centuries after Rome conquered the eastern mediterranean, the eastern half of Rome spoke Greek. (This is why the books of the New Testament, which were written in the eastern mediterranean in the period when Rome ruled that region, are all written in Greek.) But the western half of Rome was culturally dominated by the Romans and used Latin as their lingua franca.
By the time of the emperor Diocletian (283-305), the empire was having trouble administering its two halves, so he divided the empire into two administrative halves: the Latin speaking west, and the Greek speaking east.
The Latin west gradually collapsed due to corruption and barbarian invasions in the face of Roman in-fighting and gradual loss of military power (which was expensive and difficult to maintain), and the city of Rome itself fell in the year 476. Several pandemic plagues ripping through that region didn’t help, and certainly contributed to its f all. But this doesn’t mean the Roman empire fell at that time. The eastern half, which spoke Greek, and which had Byzantium (re-named Constantinople in 330) as its capital, didn’t conclusively end until the Ottomans conquered Constantinople in 1453. But it did gradually lose bits and pieces of its territory.
The eastern Roman empire is referred to as the Byzantine empire to differentiate what era of Roman history is being referred to, but they themselves just identified as Romans, and as the Roman empire. But they were distinct in that
* they spoke Greek rather than Latin
* In the period which Historians refer to the as the Byzantine empire, they were religiously Christian
* their capital was Constantinople, not Rome.
Later on during the Byzantine era, when they had an empress, Irene, the royal court had a huge dispute with the Pope in Rome, and the Pope crowned Charlemagne (a germanic warlord) as the “Roman Emperor” with the justification that the title “Roman Emperor” wasn’t being used at the time (because they had an empress), which pissed off the Byzantines. This spawned the ironically named “Holy Roman Empire” in the 800’s, which is a whole different entity, which wasn’t holy, wasn’t Roman, and wasn’t an empire, but was a confederation of Germanic kings and princes.
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