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Arithmetic and Roman Numerals


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Roman Numerals present something of a mystery. While the Plebeians may have been too poor to have much use for numbers as large as 1000 or M by virtue of their poverty and limited technology, their system (excluding for the moment shorthand representations like IX for 10 - 1 = 9) is a descendant of the Egyptian system apparently without the shorthand convention, and with different characters. The beauty of it, though, was that Plebeians did not need schooling to count and learn some arithmetic tricks. The non-subtractive notation was a direct representation of results on an abacus.

So how did the accountants of Patricians operate with such a limited system during the glory days of Rome? How was ordinance requisitioned by the builders of the Colosseum? How did Generals count and supply their soldiers, and compare the strength of their armies with those they wished to conquer? We cannot avoid these questions and continue the myth that there was no larger number than 3999 = MMMCMXCIX available to them. Is an answer to this known in any quarter, I wonder? Is our history deficient with regard to this small matter?

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😡 No takers? I'll give it a day...and then present my SPECULATION on this subject. Why? Because someone may want to erect a monument and put a date on it in the year 4000 in Roman Numerals? No, there had do be a system that the Roman elite used simply to function. And believe it or no, evidence of such a system seems to survive in an unsuspected present day phenomenon. And it seems that SPECULATION is all that remains to resolve this matter of some interest (to other wackos like me).

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They had a couple of different systems they turned to.

 

Quote

Large numbers

During the centuries that Roman numerals remained the standard way of writing numbers throughout Europe, there were various extensions to the system designed to indicate larger numbers, none of which were ever standardised.

Apostrophus

220px-Westerkerk_MDCXXX.jpg
 
"1630" on the Westerkerk in Amsterdam. "M" and "D" are given archaic "apostrophus" form.

One of these was the apostrophus,[46] in which 500 was written as IↃ, while 1,000 was written as CIↃ.[20] This is a system of encasing numbers to denote thousands (imagine the Cs and s as parentheses), which has its origins in Etruscan numeral usage. The IↃ and CIↃ used to represent 500 and 1,000 most likely preceded, and subsequently influenced, the adoption of "D" and "M" in conventional Roman numerals.

In this system, an extra denoted 500, and multiple extra s are used to denote 5,000, 50,000, etc. For example:

Base number   CIↃ = 1,000 CCIↃↃ = 10,000 CCCIↃↃↃ = 100,000
1 extra IↃ = 500 CIↃↃ = 1,500 CCIↃↃↃ = 10,500 CCCIↃↃↃↃ = 100,500
2 extra s IↃↃ = 5,000   CCIↃↃↃↃ = 15,000 CCCIↃↃↃↃↃ = 105,000
3 extra s IↃↃↃ = 50,000     CCCIↃↃↃↃↃↃ = 150,000
220px-Roman_numerals_Bungus_1584-1585.pn
 
Page from a 16th-century manual, showing a mixture of apostrophus and vinculum numbers (see in particular the ways of writing 10,000).

Sometimes CIↃ was reduced to for 1,000. John Wallis is often credited for introducing the symbol for infinity (modern ∞), and one conjecture is that he based it on this usage, since 1,000 was hyperbolically used to represent very large numbers. Similarly, IↃↃ for 5,000 was reduced to ; CCIↃↃ for 10,000 to ; IↃↃↃ for 50,000 to ; and CCCIↃↃↃ for 100,000 to . [47]

Vinculum

Another system was the vinculum, in which conventional Roman numerals were multiplied by 1,000 by adding a "bar" or "overline".[47] It was a common alternative to the apostrophic ↀ during the Imperial era: both systems were in simultaneous use around the Roman world (M for '1000' was not in use until the Medieval period).[48] [49] The use of vinculum for multiples of 1,000 can be observed, for example, on the milestones erected by Roman soldiers along the Antonine Wall in the mid-2nd century AD.[50] There is some scope for confusion when an overline is meant to denote multiples of 1,000, and when not. The Greeks and Romans often overlined letters acting as numerals to highlight them from the general body of the text, without any numerical significance. This stylistic convention was, for example, also in use in the inscriptions of the Antonine Wall,[51] and the reader is required to decipher the intended meaning of the overline from the context. The vinculum for marking 1,000s continued in use in the Middle Ages, though it became known more commonly as titulus.[52]

Some modern sources describe Vinculum as if it were a part of the current "standard":[53] however this is purely hypothetical - since no common modern usage requires numbers larger than the current year (MMXXI). Nonetheless, for reference: here are some examples, to give an idea of how it might be used::

  • IV = 4,000
  • IVDCXXVII = 4,627
  • XXV = 25,000
  • XXVCDLIX = 25,459

https://en.wikipedia.org/wiki/Roman_numerals#Large_numbers

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Thanks for your interest and the content. The M features significantly in my solution, and may have had its interpretation changed among an intellectual elite to supply a concept foreign to "normal people" of the day, being originally used to "cross out" the place where overflow produced a zero and a carry to the next place in the representation. Thus misunderstood, M, a squiggle originally, may have been redeployed as the article describes in the middle ages. Further, the invention of our Hindu-Arabic numerals is historically coincident with the Fall of Rome, employing the concept of exponentiation. The Roman Numerals could have been more cleverly refined as our numerals today, continuing the slow evolution. Egyptians used patterns of 1 to 9 strokes as in tallying, and invented characters or hieroglyphs used likewise for further powers of 10 up to 10,000,000. This last number is the clue to what may have been the numerals of the learned elite. Compare the two equivalent representations below:

9,223,372,036,854,775, 808

IX  CCXXIII  CCCLXXII  XXVI  DCCCLIV  DCCLXXV  DCCCVIII

To the untrained eye, this would have only ever appeared to be 7 distinct Roman Numerals. The M, as zero, would have rendered 1,000,000 as I M M. It would not be too surprising that this knowledge escaped the history pages. It may have been way over the heads of the general rank and file. It may have incited revolution and maintained as covert as the military secret that destroyed Troy. The male population of Troy were put to the sword because the Trojan Horse was set on fire to decoy the guard on the city gate. Without this the alarm would have been raised to thwart the Greek plan. Surely this tactic would have been considered beforehand as absolutely necessary for success, and may possibly be used again. 

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