There are several intermediate steps which connect Urnary and Roman.
visually highlight precounted intervals. This creates a new symbol out of 5. This system is well suited for incremental counting of potentially large numbers - such as counting days on a prison wall.
As the numbers grow large, a similar concept can be applied to group blocks of 50 or 500. Mayan, Egyptian, and other historical solutions exist.
We’re not limited to groups of 5, that’s just ( pun) handy. When playing Cricket (Darts) you’re using a base 3 symbology
While the Babylonians used a base 60. This number is usually chosen since it divides evenly in so many ways. This allows even partitioning among 2 to 6 people.
As the numbers grow larger we need groups of groups, and a new symbol for each. Roman numerals assigned a unique symbol for 1, 5, 10, 50, 100, 500, 1000.
This system can compactly represent very large numbers, in a very small space. But making the numbers follow a more uniform progression simplifies math. By standardizing on a 1,10,100,1000,… pattern all sorts of nice mathematical regularities pop out. Suddenly a mathematical operation such as multiplication that was difficult, could instead be calculated using a 10x10 lookup table. By choosing a binary system that table becomes 2x2.
I recall sitting through 3/4th of a semester of linear algebra. While studying for a test I finally had all the pieces in front of me, having that eureka moment. I’d wondered ‘why didn’t he just say that?!’ Having since done some teaching, I can appreciate that even when you know it, it can be very hard to convey. But I think too often we do fail to mention the forest while teaching the trees.
If I had ever previously encountered the Mobius Transformation, I suspect this youtube video might have been similarly insightful
Visual understanding, while not the only path, is often a key insight.
ps. loving the blog.