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What is the base for Roman Numerals? It starts off with unary then goes back and forth between multiples of 5 and 10.

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Sometimes an "I" could stand for $(-1)$ as well, e.g.: "XIX" for $19$. :) –  Srivatsan Sep 24 '11 at 15:44
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There isn't a concept of "place value" with Roman numerals, so you can't really speak of a base... –  J. M. Sep 24 '11 at 15:45
    
As a note: IX is relatively recent; classically it'd be VIIII... –  J. M. Sep 24 '11 at 15:46
    
Base $10$, but not positional. –  André Nicolas Sep 24 '11 at 15:47
    
Base 10, but relies on symbol-values to indicate position. –  wendy.krieger May 17 '13 at 9:16

4 Answers 4

I disagree with Henning's and J.M.'s identification of positional systems and systems with a base. There are examples of non-positional systems with a single base (10 in both cases): Egyptian numerals and Chinese numerals. The first footnote in the Wikipedia article on Roman numerals calls them "a decimal system in which the number 5 is an auxiliary base".

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Interesting that you consider the concepts of "base" and "place value" decoupled. I suppose you'd raise similar objections to the answers in this question? –  J. M. Sep 30 '11 at 16:44
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And there is at least one positional system that does not (quite) have a base: the Mayan system, at least when applied to the calendar, was vigesimal except in the third position, which represented $360$, not $20^2$. (The only recorded numbers large enough to show the difference are calendrical; it’s supposed that the basic system was actually vigesimal, and that the calendrical usage is the exception, but I don’t think that this has actually been demonstrated.) –  Brian M. Scott Apr 12 '13 at 8:10

Since it is not a positional system, it does not really make sense to speak of a unique "base" for Roman numerals, mathematically. One could describe them as "mixed base 5-10" (most commonly) or even "mixed base 2-5", but neither of those are formal descriptions.

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There's also such a thing as a "subbase". The base-60 numeral system used in Ptolemy's Almagest in the trigonometric tables and elsewhere had digits (actually Greek letters) for 1 through 10, and then digits for 20, 30, 40, and 50. So the number 43 had the 40 digit followed by the 3 digit. But suppose you had fourty-three 60s plus 25. You'd have the 40-digit, then the 3-digit, making up the 60's place, then the 20-digit followed by the 5-digit in the unit's place. So 60 was the base and 10 was the subbase. Ptolemy did some computations with five or more two-digit places. –  Michael Hardy Sep 24 '11 at 21:50
    
It is the periodic nature of the underlying abacus, not the written notation, that sets the base. One can directly compare LXIV to 44, faster than 3s 8d, to 44d. A stone in a given column is 10 times that of a lower column. Just because we represent 100p by a £1 coin, it does not mean that decimal currency is in use. –  wendy.krieger May 17 '13 at 8:31

Roman numbers represent the two-on-five abacus, so the columns process by tens, but it's a cooperation between a count of 5 and a division of 2.

The actual numbers are token-weights (like money). You don't need a zero-cent coin to separate the $1 coin and the 1c coin. The idea of representing four and nine as one-before-five and one-before-ten is a later invention, but the sumerians had a one-before-ten rune for nine too. You don't write though, 49 as IL, but XL IX, which proves that there is token-weighted decimal digits in use.

       D    L     V       Decimal tokens.
  -------------------        is base 10, because any symbol is 10 times
  M    C    X     I       the value to the right.  

 £50    £5   50p    5p     Decimal currency.
 £20    £2   20p    2p        I don't need to have any 10p, 20p or 50p
 £10    £1   10p    1p     coins to fill out the 'empty column'


1/4  £100   £5   5s   3d   1/4d    Base 20 abacus, for money.
    ---------------------------       the last column is broken into 3
1     £20   £1   1s   1d              rather than five, without fault.

One can see that there is indeed a notion of place in the roman column. One writes a unit-stone as C if it's in the second column, or 'M' if it's in the third column, or I if it's in the first column. It's probably like saying that you have no notion of place of money, because the £100 note in your pocket isn't supported by zero-coins to fill the £10, the £1 and 10p and 1p columns.

Alternating Bases

By Neugebauer 'the exact sciences in antiquity'

The Sumerian number system for ordinary numbers, is a variety of hodgepodge of decimal, sixty and twelfty symbols. A number like dec 192, might be written as 100 + 60 + 30 + 2, might be represented in the matching tables as 3A2 (ie 3 10 2).

The correct style to represent sumerian numbers, is not the manner of our base 60 time, but to use rune for rune: ie A = 10, B=20, C=30, D=40, E=50, and numbers 1-9 as such.

The runes A to E start off life as fractions, ie 1/6 to 5/6. So one might write 1 1/2, as 1C. It's kind of like having numbers fro 0/12 to 11/12 for fractions of a dozen, or uncia, like the romans do. But where the romans ultimately use weight-fractions, the sumerian identification of A mina = 10 shekels, (ie 1/6 mina = 10 shekels), and that one can write numbers as eg 1C mina 2A shekels ie 1 3/6 mina, 2 1/6 shekels. (where we would write 1:32:10 mina).

In any case, the implementation of the abacus everywhere is that the units are in the low column, and the divisions start in the high column, so

                  C = 3/6 mina      A = 1/6 shekel.
    ----------------------------------------------
     1 = 1 mina   2 = 2 shekels    

Sumerian numbers are written as 'alternating divisions and multiples', rather than a tight sixty-number. The number we write as 1.24 51 10, is written as 1B4E1A without any spacing, and its recriprocal (also given) is .42 25 35 as D2B5C5, again without any spacing. One does not really need spacing when the high and low digits are different.

It's also a division system. That is 1B4E1A is not to be read as 1 25 51 10 = dec 212570, but as 1:25 51 10, = 1.41421356. It's like our time and angle. A three-number angle is degrees, a five-number angle is degrees and minutes, and a seven-figure angle is degrees, minutes and seconds. Also 1500 hours is not 62 1/2 days, but 15 o'clock. denoted as HHMM. The hundred is not 5 score but three score.

The sumerian numbers were constructed as a division system to avoid division. We note this from the appearence of tables of pairs of numbers whose product is 60, eg 1C D (1.5 * 40 = 60), sorted in increasing size. So if one wanted to divide by 72, one sees in the table 1A2 E. The second table to consult is a set of ready-reckners that gives multiples of E for 1 to 20, and then 40. Such tables were still in commercial use until the 1970s, and might still be found in book-sales.

There is reference in Neugebauer, of 'the problem of seven brothers', since 7 does not appear in the recriprocal tables. The thing is referred to as the equal of a scientific paper, that eg 8C4A6 < '7 < 8C4A8 (where '7 is the egyptian style of writing 1/7). The actual result is to repeat 8C4A7.

There is also examples of e v e n l y s p a c e d numbers, like writing 8 C 4 A 6. This is little pussle to me, since i often do this to effect criss-cross multiplication in base 120.

Alternating bases demand alternating arithmetic. This is not too hard. It's just that you have to detangle this nonsence that a base requires digits from 0 to b-1, and that the multiplication table is of the order of $O(b^2)$. I don't use much more than 12 * 12 = 1.24 in base 120. You just don't need to multiply 77 * 88 = 56.56, for example.

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The concept of a numeral base is a characteristic radix numeral systems. This includes the familiar systems such as base 10 and base 2, but also includes base -2 (negative), base 2i (imaginary), base i-1 (complex) or even base e (transcendental) for that matter.

The Roman numeral system is not a radix system and therefore has no corresponding base characteristic.

There are many online references that discuss various aspects of this. Here is one.

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