- EUROPE 143
7 8 3
425FIGURE 3. Multiplication with Hindu-Arabic numerals.be inclined to think that the phrase myrias diploos meant 2 · 10,000 rather than
ÉÏ,ÏÏÏ^2.
As you can see from this example, calculating (logistike) was not quite so trivial
for the Greeks as for us.
6. The Islamic world
It is well known that the numerals used all over the world today are an inheritance
from both the Hindu and Arabic mathematicians of 1000 years ago. The Hindu
idea of using nine symbols in a place-value system was known in what is now Iraq
in the late seventh century, before that area became part of the Muslim Empire.
In the late eighth century a scholar from India came to the court of Caliph al-
Mansur with a work on Hindu astronomy using these numerals, and this work was
translated into Arabic. An Arabic treatise on these numbers, containing the first
known discussion of decimal fractions, was written by al-Uqlidisi (ca. 920-ca. 980).
Having inherited works from the time of Mesopotamia and also Greek and
Hindu works that used the sexagesimal system in astronomy, the Muslim mathe-
maticians of a thousand years ago also used that system. The sexagesimal system
did not yield immediately to its decimal rival, and the technique of place-value
computation developed in parallel in the two systems. Ifrah (2000, pp. 539-555)
gives a detailed description of the long resistance to the new system. The sexagesi-
mal system is mentioned in Arabic works of Abu'l-Wafa (940 988) and Kushar ben
Laban (ca. 971 1029). It continued to appear in Arabic texts through the time of
al-Kashi (1427), although the decimal system also occurs in the work of al-Kashi.
In addition to the sexagesimal and decimal systems, the Muslim mathematicians
used an elaborate system of finger reckoning. Some implementations of the deci-
mal system require crossing out or erasing in the process of computation, and that
was considered a disadvantage. Nevertheless, the superiority of decimal notation in
computation was recognized early. For example, al-Daffa (1973, pp. 56-57) men-
tions that there there are manuscripts still extant dating to the twelfth century, in
which multiplication is performed by the very efficient method illustrated in Fig. 3
for the multiplication 524 · 783 = 410,292.
7. Europe
The system of Roman numerals that now remains in countries settled by Europeans
is confined to a few cases where numbers have only to be read, not computed with.