There was a time in history when paper money was mistrusted. Coins made of precious metal (gold and silver were the best); one could travel in a foreign land where one's coins were welcome, for their intrinsic worth.
When I was growing up in the U.K, we had farthings (becoming rare), ha'pennies, threepenny bits (tickeys, in South Africa), tanners (sixpenny coins), bobs (shillings), florins (two shillings), and half-crowns (two shillings and sixpence). Sovereigns and half-sovereigns were rare, because the gold was worth more than the face value.
The main paper money was the "quid" (one pound note, greenish in color) and the "ten bob note" (ten shillings.) These were the highest denominations in general use, and weer similar in size to US currency. There were higher denomination notes, but most people never saw one higher than the "fiver", a beautiful white banknote, larger than the "quid", that crackled when folded.
I still remember the occasion in the early 1940s when I lost a five-pound note, and really grieved over the loss. It felt much like losing a hundred dolllar bill would feel today.
As in most developed countries, we kept a supply of change in our pockets or coin purses. Paper money belonged in a wallet. In countries where sterling, the Euro, or the Canadian
"looney" rule, this logical pattern is followed: coins for small amounts, notes for larger amounts.
Not in the USA.
Within the memory of most of us, we have largely limited the metal dollar to casinos, and stashed away large amounts of 50 cent pieces, requiring us to keep quantities of quarters for meters. etc. Twice we have introduced dollar coins, which are not widely circulated, and are quite unpopular. Most of these are held by banks, So we continue to spend millions of dollars printing up one dollar bills, which have a much shorter life than the metal equivalents. The penny may be on its way out; many don't think it worthwhile to pick up a penny fallen on a sidewalk..The greenback is welcomed abroad, and many travelers like to carry wads around to use as tips.
And to think that paper money was once despised...