As I understand it, we adopted a decimal coin system in 1792 which set the foundation for the cash system we have used our entire nations history. One question I have always held and I hope this isn't as inept as it sounds..... We identified "cents" and "half cents" on the appropriate coins of the day. Why then did we not identify quarter dollars, half dollars and dollars? Was it that we just expected commerce to recognize the sizes as being similar to the reale's that were circulating at the time?
I've wondered that myself. You can include dismes and half dismes too. It's hard to tell the difference just looking at a photo.
The idea for the decimal system of coinage and money was actually hatched by Thomas Jefferson and Robert Morris circa 1782. Fortunately they held on to that idea, and we decided to make it a part of the Coinage Act of 1792. As for the issue as to why the silver and gold coins did not have denominations on them, that goes back to the practice of the period. I have some British coins from the era, and there is no mark of face value on them. I don't have any regular British Mint George III silver coins, but here a George II shilling. Note that it does not say "shilling" or "12 pence" on any part of the coin. People went by the weight and size of the piece. Here is a 1776 dated George III guinea. This coin was worth 21 shillings, but nowhere does it say that. Here is coin from a bit later, an 1817 George III half sovereign, which was worth 10 shillings. There is no mark of value. Here is an 1804 George III Dollar which was also given a value of five shillings. This was issued by the Bank of England. The trouble is by 1811 the war with France had upped the value to five shillings and six pence, even though the stated value was five shillings. That's a bit confusing. The trouble was the value of these coins was fluid because of the changes in the political situation and the relative cost of gold and silver. Therefore during this period, the mints often didn't put a value on the piece. It was traded "by tail" or by the its metal content. The U.S. followed suit until John Reich started re-designing all of the U.S. coins in 1807. He started to put values on them, although it was something modest like "5 D." or "50 C." Only the copper coins had "ONE CENT" or "HALF CENT" boldly shown on their designs. Those coins didn't have enough metal in them to merit that face value. People used them for what they stated they were worth.
BTW there were a few modest value markings on the early U.S. coins. Note the tiny "1/2" on the reverse of this 1796 half dollar. It didn't say that it half a dollar, but there it is. The trouble is, these coins are so rare, few people saw them. That includes modern coin collectors.
And Russia adopted a decimal system in 1710. They had the first decimal coinage system, we had the second.