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Is it blogspam when it's called syndication? Here's the "original" source of the column, which is on a less... determinedly fashionable website, so it might be friendlier to mobile users and people who dislike fixed headers: https://www.straightdope.com/columns/read/3330/how-did-anyon... Also, previously: https://news.ycombinator.com/item?id=14818633
The Greeks were really into geometry using the compass and straight edge so they actually did a lot of math without really needing numbers at all. They viewed calculation as less worthy of mathematicians and my understanding is that we don't have a lot of evidence for how merchants and engineers did basic calculations since most of the great Greek math texts ignored it.
Algorithms and algebra probably existed in some informal way but they weren't really formalized until the Arabs did it with the help of Arabic numerals.
So, while you can do some calculations in roman numerals or using an abacus, the interesting question to me is: Did the Greeks (and the Romans) not develop algebra or use Arabic numerals because they weren't that into numbers as compared to geometry? Or was it the other way around? Did the clumsiness of doing calculations in Roman numerals keep them from developing more complex systems of numerical calculation?
I'm not an expert on the subject at all but it's always interested me. It makes me think of Bret Victor's Media for Thinking the Unthinkable (http://worrydream.com/MediaForThinkingTheUnthinkable/)
Roman numerals are very economical. The base with best radix economy (except for e) is 3. But roman numerals are better still. For representing 0-999, you would need 19 base-3 digits, but only 15 roman numerals (plus a symbol for zero).
...in terms of digits, which is a very odd quality to optimize for.
Earlier quoted context omitted.
...in terms of digits, which is a very odd quality to optimize for.
...in terms of only the worst case numeral. Common numerals like II III VII VIII have worse length, while I IV VI are same, and V X IX are better. And factoring in per-digit cost, Roman numerals up to 999 have 5 distinct digits, 46% more cost per digit than base 3, making it worse than base 3 in almost every case, information theoretically. (You could win some back with a huffman encoding, though)
In An Introduction to Mathematics (1911) Alfred North Whitehead wrote: By relieving the brain of all unnecessary work, a good notation sets it free to concentrate on more advanced problems, and, in effect, increases the mental power of the race. Before the introduction of the Arabic notation, multiplication was difficult, and the division even of integers called into play the highest mathematical faculties. Probably…
I feel like the article misses the most interesting question about Roman numerals and Roman (Greek really) math. How did the numerical system influence the math that they developed and used? The Greeks were really into geometry using the compass and straight edge so they actually did a lot of math without really needing numbers at all. They viewed calculation as less worthy of mathematicians and my understanding is t…
Probably? I mean, look what the world achieved after it left roman numerals behind.
Earlier quoted context omitted.
...in terms of digits, which is a very odd quality to optimize for.
...in terms of only the worst case numeral. Common numerals like II III VII VIII have worse length, while I IV VI are same, and V X IX are better. And factoring in per-digit cost, Roman numerals up to 999 have 5 distinct digits, 46% more cost per digit than base 3, making it worse than base 3 in almost every case, information theoretically. (You could win some back with a huffman encoding, though)
You would need 19 base 3 digits and only 15 roman numerals (plus N for zero) in order to represent every number.
In base three, you sometimes have all three digits equal. With roman numerals, you can reuse the same digit in different positions.
In An Introduction to Mathematics (1911) Alfred North Whitehead wrote: By relieving the brain of all unnecessary work, a good notation sets it free to concentrate on more advanced problems, and, in effect, increases the mental power of the race. Before the introduction of the Arabic notation, multiplication was difficult, and the division even of integers called into play the highest mathematical faculties. Probably…