The Analytic Atavar

Idiosyncratic Musings of a Retrograde Technophile

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Thursday, February 15, 2007

Romanorum computus

"... it is easy to understand that although addition of Roman numerals is quite satisfactory, multiplication and division are essentially impossible." - An overview of Egyptian mathematics
This is the conventional wisdom concerning Roman arithmetic, but it can not be correct since the Romans were great engineers and administrators of a large empire, and these tasks are impossible without efficient methods of multiplication and division. In fact, the previous great empire of Egypt had efficient methods prior to 1850 BCE - a fact known from the Rhind papyrus which is the earliest extant description of their arithmetical methods. It is likely that the Greeks and Romans were aware of these methods and adopted some form of them, since the Egyptian numerals were similar to the later Roman numerals.

SymbolValue
I1 (one) (unus)
V5 (five) (quinque)
X10 (ten) (decem)
L50 (fifty) (quinquaginta)
C100 (one hundred) (centum)
D500 (five hundred) (quingenti)
M1000 (one thousand) (mille)
Basically, a Roman number is an abbreviated hash count, where letters are used to stand for a group of hashes as shown in the table. Numerals larger than 5,000 are constructed by placing a bar above these symbols to indicate multiplication of the symbol's value by 1,000, i.e. V(overbar) = 5,000 &tc. A Roman number consists of a string of symbols beginning with the largest symbols followed by symbols of smaller and smaller value. The Romans did not use the modern subtractive notation, so 4 was written as IIII, not IV, 9 as VIIII, not IX, and so on. Addition is then a simple process: simply combine the two number strings, re-arrange the numerals from high to low, and combine any possible groups of symbols and replace by higher value symbols. For example:
CXVI (116) + XXIIII (24) = CXVIXXIIII = CXXXVIIIII = CXXXX (140)
where the base ten values are show in parenthesis, and the group VIIIII has been replaced by X. Subtraction is not much more difficult: eliminate common symbols in both the minuend and subtrahend (indicated below by the strike-thrus), expand symbols in the minuend as necessary to produce the remaining symbols in the subtrahend, complete the elimination, and combine any possible group of symbols, e.g.:
DCCXXVIII (728) - LI (51) = DCCXXVIII - LI = DCLLXXVII - L = DCLXXVII (677)
It is evident that these processes are applicable to any numbers, no matter how large.

Neither of these operations provides any clue as to how multiplication and division could be done. It was certainly not done by the algorithms here, for the number of additions or subtractions for any large multiplier is prohibitive. A Different Kind of Multiplication suggests that they used a form of peasant multiplication.

(1)(2)
XXVII(27)*LXXXII(82)
XIII(13)*CLXIIII(164)
VI(6)CCCXXVIII(328)
III(3)*DCLVI(656)
I(1)*MCCCXII(1312)
Consider the problem XXVII (27) x LXXXII (82). Form two columns with the two numbers as the first line (the table will be smaller if the smaller number is in the first column). For the next line, in the first column halve the number above, ignoring any remainder, and in the second column double the number above. Continue this process until only 1 remains in the first column. For all lines for which the number in column one is odd (indicated by the *), add together the corresponding numbers in column two, thusly:
LXXXII+CLXIIII+DCLVI+MCCCXII = MDCCCCCLLLXXXXXVIIIIIIIII = MMCCXIIII (2214)
That this is the correct answer can be easily verified by the reader, and the labor involved is only slightly more than the modern manual method of multiplication using arabic numerals. Dr. David P. Stern, the author of A Different Kind of Multiplication, opines:
"It was probably discovered by trial and error, and it always worked, though the Romans did not know why."
That they didn't know why is probably true, but it is more likely that they learned the method from some other group - perhaps the Greeks - though not the Egyptians, because they used a slightly different method discussed below.

The reason it always works is as follows. Many will recognize that repeated division by 2, keeping track of the remainders, is the method to convert any number to base-2 notation. If the number is even, there is no remainder (i.e., a remainder of 0), while if odd there is a remainder of 1. Reading from the bottom up in column one of the previous table and writing a 1 for the odd numbers and 0 for the even:
110112 = 1x24+1x23+1x21+1x20 = 16+8+2+1 = 27
we see that 110112 is the correct base-2 representation of 27. If we now multiply by 82:
82x(1x24+1x23+1x21+1x20) = 82x24+82x23+82x21+82x20
because multiplication is distributive. Now:
82x24 = 82x16 = 1312
82x23 = 82x8 = 656
82x21 = 82x2 = 164
82x20 = 82x1 = 82
It is clear that the doubling in column two produces precisely the power of 2 factors that when added together give the correct result. Unfortunately, this does not explain how the Romans would have divided.

(1)(2)
1*82
2*164
4328
8*656
16*1312
322624
The answer is, I believe, a feature of the method used by the Egyptians, which allows for both multiplication and division. The Egyptian method again used two columns, but started with 1 in the first column, the larger number in the second, and doubled both until the result in column one was greater than the smaller factor (in what follows, base-10 notation will be used). Starting with the smaller factor, we repeatedly subtract the largest number in column one less than the remainder until we get 0: 27-16=11, 11-8=3, 3-2=1, 1-1=0 keeping track of the numbers used (marked by the *), and then add the corresponding numbers from column two to get the product as before. This seems little different from and slightly more work than peasant multiplication, but there is a difference - it also allows division. Consider the problem 2250/82 - I have selected a dividend slightly larger than 2214 so there will be a remainder. If from the dividend we repeatedly subtract the largest number in column two less than the remainder: 2250-1312=938, 938-656=282, 282-164=118, 118-82=36, and then add the corresponding numbers from column one, 16+8+2+1=27, then 2250/82 = 27 with a remainder of 36. The suggestion that the Egyptian method is equally useful for division is an original one which I have seen nowhere else.

As to what method the Romans actually used, the answer is no one knows for sure. The age of the peasant method and the presence of the 5 factor symbols in the Roman numerals, facilitating halving and doubling, argues for the use of the peasant method, but leaves the explanation of division unresolved. What is amazing is the fact that nearly 4,000 or more years ago some unknown mathematical genius discovered that any number can be represented as a sum of powers of two, that multiplication is distributive, and combined these insights into efficient algorithms for multiplication and division, reducing them to the simpler operations of doubling, adding, and subtracting.

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