So the bronze denominations of Roman coinage heavily declined in quality in the late 4th century CE and for most of the 5th. They got smaller, lighter, and cruder, to the point where it's often hard to identify them with how badly they're made. The top left coin here is an example that I think is from the reign of Theodosius II (408-450 CE), though I'm not 100% certain of that. They were so worthless that they usually just distributed sealed sacks of them rather than bother with separating out the coins. The term for such a sack is a "follis" (plural "folles"), a term also used to mean scrotum among other things.
But then we get the beginning of "Byzantine coinage" with the reforms of Anastatius I (491-518).1 Anastasius was a bit conservative and put restrictions on the erotic pantomime performances that were popular in Constantinople, but he also made significant improvements in the functionality of the government and its finances. And one thing he did was turn the follis into a big coin. These early folles continued the standard side view portraiture of earlier coins along with standard Latin inscriptions, but on the reverse, they usually used a Greek numeral to tell you how many of those small, shitty old coins they were worth. Most commonly, you see M meaning 40 and K meaning 20. In the above examples, A half follis of Anastasius is second on that first row, followed by folles of Justin I (518-527) and an early example of Justinian I (527-565). I've rarely seen early folles like these with much surviving detail, so it seems they circulated for a long time.
During the reign of Justinian portraiture shifted to be frontal, and that's how it would stay on Byzantine coins from here on out. These next three coins depict Justin II (565-578) and his wife Sophia, Maurice (582-602), and Phocas (602-610). That Phocas coin is a flat, uneven strike, but it's supposed to have "XXXX" on the reverse. For whatever reason, values written in Roman numerals are common during his reign.
One thing I find interesting about these coins is that you can see the fortunes of the empire reflected in the size and quality of these coins. They peaked at 40mm wide during the reign of Justinian, but then shrank back down as he started having financial difficulties due to the plague and constant wars. And you can see things getting particularly bad in the last two coins of the second row. The first is from the reign of Heraclius (610-641), a man who reconquered the Levant and Egypt from the Persians only to lose it all permanently to the Arab conquests. It was also during the reign of Heraclius that they finally shifted the inscriptions to be all Greek. I've seen claim that this is a sign that Greek was now the "official language" of the empire, but the empire was always multilingual, and he probably just decided to switch coins to what happened to be the most popular language in the empire rather than one that was mostly only spoken outside its borders at this point. That last coin is from his grandson Constans II (641-668), a man who was not named Constans,2 and who famously looted the Italian territories he visited due to the empire being that short on money.
But the Byzantine Empire had plenty of ups along with their downs, and you can see that reflected on the bottom row coins, which aren't as big and heavy as the early ones, but definitely better in both weight and construction to what was made under Constans II. The fractions of the follis had long since ceased to be made by the time of this bottom row, leaving only one bronze denomination. The M still stuck around for a while, but even that vestigial thing was abandoned by this point. The first coin here is of Leo VI (886-912), and both the obverse and reverse just state that he's Leo, Emperor of the Romans, with the reverse using the extra space to say it's "by the grace of God." The next coin is from the reign of Constantine VII (913-959), and while it basically has the same message with a different name, you'll notice that it's a very messy looking coin. This is what's called an overstrike. What they would sometimes do in the Byzantine period when they received an old coin in payment was to then restrike the coin with the image and message of the current emperor. This usually left bits of the old design, in this case, I'm pretty sure it's Leo VI again.
Our final coin is what's called an "anonymous follis." During this period, they started using entirely religious imagery on the coins sometimes rather than depicting the emperor. This one is from the reign of Romanus IV (1068-1071). The obverse is a little double struck, so Jesus' head looks bent, and the reverse depicts Mary. It wasn't long after this coin that the empire's fortunes really took a bad turn with the Battle of Manzikert, and the resulting financial troubles resulted in a major coinage overhaul, but I already talked about that in another post.
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The entire distinction between the Roman Empire and Byzantine Empire is completely arbitrary. There was a continuous form of government and national identity from Augustus through Constantine XI that only ended with the fall of Constantinople in 1453. These people always called themselves Romans, as did any contemporary source from the Middle East. It was only when Western Europe started seeing themselves as the successors to Rome, particularly starting with Charlemagne, that they started referring to the empire as a bunch of Greeks, a term they not only hadn't used for themselves in centuries but used to describe an extinct people they descended from. You can't be the true successor of the glory of Rome if Rome is still around, so Western Europe decided it wasn't, and for some reason, significant pushback on that idea in the West is only fairly recent. There's also no clear consensus on when exactly the split between "Rome" and "Byzantium" happened, since the distinction isn't really based on a historical event, anyway, but coin collectors at least picked a point when there was a significant, sudden change in the coinage.
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The Byzantine Empire had twelve emperors named Constantine, and we number them from 1 to 11. This guy was the fourth guy with the name Constantine who reigned, but he was unpopular enough that people tended to instead refer to him by the more diminutive "Constans," and for some reason, that's stuck in most retrospective sources, and his son is called Constantine IV. It would probably be a pain to rectify that, since you'd then have to constantly be clarifying whether you were using the old or new Constantine numbering system.