Published by the Applied Probability Trust
© Applied Probability Trust 2002
14
A Note on Roman Arithmetic
JOE GANI
How to make multiplication difficult.
Introduction
A recent review (reference 1) of Dirk Struik’s classical book
A Concise History of Mathematics encouraged me to re-read
the fourth revised edition of this work (reference 2).
In his Chapter II, ‘The Ancient Orient’, Struik discusses
the number systems used by the Mesopotamians, Egyptians
and Chinese. The complexity of simple arithmetic operations
can hardly be imagined by us today, when the Hindu
decimal position system and the use of Arabic numerals are
prevalent. The Roman system followed the Egyptian; let us,
for example, imagine a Roman child doing the following sum:
multiply XIV by IX. To simplify this and avoid the negative
values of I implied in IV and IX, we may rewrite this as XIIII
by VIIII. Let us assume that Roman children learned their
multiplication tables much as we do today, and proceed as
follows:
XIIII
× VIIII
I × XIIII = XIIII
XIIII
XIIII
XIIII
V × XIIII = LVVVV
XCXXXVI = CXXVI
where X + X + X + X + L = XC, IIII + IIII + IIII +
IIII + VVVV = XXXVI , and XCXXX = CXX , since the
X before the C cancels the first X after the C. If we compare
this with the current calculation, which we set out similarly,
14
×9
36
90
126
we see how greatly the Hindu–Arabic arithmetic system has
simplified our lives.
I cannot imagine how complicated the rules for division
must have been, even though many calculations would have
been carried out on the abacus. Remarkably, the Hindu–
Arabic system was resisted, and Struik writes: ‘In the statutes
of the “Arte del Cambio” of 1299 and even later the money
changers of Florence were forbidden to use Arabic numerals
and were obliged to use Roman ones.’ It was only in the
fourteenth century that Italian merchants began to use the
Hindu–Arabic numerals. And just as well!
References
1. D. E. Rowe, Looking back on a bestseller: Dirk Struik’s A
Concise History of Mathematics, Notices Amer. Math. Soc. 48
(2001), pp. 590–592.
2. D. J. Struik, A Concise History of Mathematics, 4th edn (Dover,
New York, 1987).
Joe Gani is a retired mathematician who lives in Canberra.