utes in an hour and partially in the number of degrees for
an equilateral triangle and in the 360 degrees in a circle.
Th e Babylonians were able to use their system for reciprocal
numbers, square roots, cubes, and cube roots of numbers.
Th ere is evidence that mathematics was taught in schools,
and some tablets survive with mathematical problems and
with the working out of pure mathematical theories.
Th e Assyrians followed the same system as the Babylo-
nians; indeed, they seem to have added nothing to the Baby-
lonian advances in mathematics. Th ey used the numbers for
purely applied purposes, such as for building temples and
palaces, for computing taxation owed to their rulers, and for
military reasons. As a result, when the Achaemenid Persian
Empire (538–331 b.c.e.) was established, it was from Baby-
lonian science and numbering that it drew its inspiration.
Although the Persians did embark on a number of prema-
thematical inquiries, their primary focus was applied math-
ematics.
Th ere were many other numeral systems that operated in
the Near East at the time, including that used by the Jews. Th e
Hebraic numeral system uses the additive principle, in which
the numeric values of the individual letters are added together
to form the whole. Diff erent numerals denote each number
from 1 through 10 and then 20, 30, 40, and so on until 100; dif-
ferent numbers are used for 200, 300, 400, 500, and so forth.
Th ere are 27 Hebrew numerals in all. Th e Hittites also had
their own numbering system, as did the Phoenicians.
By the time Alexander the Great conquered the Ach-
aemenid Persian Empire in the 330s b.c.e., Greek numerals
tended to dominate notation in the ancient world. In many
ways it had come about as Greek learning permeated much
of the Near East, even before Alexander’s conquest. Th is re-
mained the case throughout the rest of the ancient world, so
much so that Greek mathematicians dominated the fi eld dur-
ing the Roman Empire.
ASIA AND THE PACIFIC
BY FRANK J. SWETZ
Th e development of Asian counting systems and mathemat-
ics was greatly infl uenced by two pervasive civilizations: the
Hindu empires of the Indian subcontinent and the dynastic
empires of imperial China. Th rough conquest, trade, and
religious evangelization these civilizations spread their cus-
toms, beliefs, and rituals over much of the Asian region. In
particular, Korea, Annam (northern Vietnam), and Japan be-
came the intellectual heirs of China. Contacts resulted in the
adoption of Chinese mathematical practices.
From the Shang Dynasty (ca. 1500–ca. 1045 b.c.e.) on-
ward the Chinese had been using a decimal system of count-
ing and recording numbers. Th e earliest evidence of such
numbers has been found on Shang oracle shells and bones
used for fortune-telling. More formal systems of numeration
appear in bronze inscriptions of the following Zhou Dynasty
(ca. 1045–ca. 256 b.c.e.). Finally, during the Han Dynasty
(ca. 202–ca. 220 b.c.e.), this decimal numeration system was
standardized. By the early centuries of the Common Era,
Chinese use of number symbols had evolved into four dis-
tinct systems: a form for common use, a more elaborate form
for legal and administrative documents, a commercial form
that allowed for quick recording, and a scientifi c, counting-
rod, or rod-numeral, form that lent itself to calculations on a
computing board. It was the latter set, the rod numerals, that
was most infl uential in terms of mathematical facility and
cultural transmission and eventual adoption in the territories
infl uenced by China.
Rod numerals actually represented confi gurations of a
set of wooden comput i ng rods employed on a cou nt i ng boa rd.
Th ey were basically tally symbols representing numbers; thus
3 would be represented by three vertical rods, recorded as a
numeral by three vertical strokes. In designating digits, this
process continued up to recording 5 by using fi ve vertical
strokes; for numbers above 5 a horizontal stroke would be
added, serving as a cap over the vertical strokes. Th is hori-
zontal stroke represented a count of 5. Th e vertical strokes
added below it would then represent units to be added to the- So a horizontal stroke covering one vertical stroke would
represent the number 6 and a horizontal stroke with four ver-
tical strokes would stand for the number 9. Th ese symbols
were used to designate collections of units—100s, 10,000s,
and so on (alternating powers of 10). For the remaining alter-
nating powers of 10—10s, 1,000s, 100,000s, and so on—the
orientation of the strokes was changed; vertical strokes were
replaced by horizontal strokes and horizontal by vertical.
Th us, alternating 10’s positions in a decimal-based numeral
would be symbolized by contrasting sets of strokes.
In this system numerals would be written from left to
right, with the highest power of 10 occupying the left -most
position in the numeral. An empty space in the numeral in-
dicated an empty position on a counting board (a “zero”).
A detailed description of Chinese counting rods and their
computational procedures is given in Sun zi suan jing (Th e
Mathematical Classic of Master Sun), written about the year
400 c.e. Th is rod counting and numeration system was most
effi cient and was used throughout Asia for many years until it
was replaced by the abacus.
Common Chinese numerals were (and still are) written
in a vertical column with the highest place value at the top.
Th ese columns of characters are unique in that they serve as
numerals, counting symbols, and number words or phrases.
Th ere are separate characters for each grouping of 10s—10,
100s, 1,000s—and, in a numeral, each 10’s grouping would be
preceded by a counting character; thus a reader would note
two characters (words), such as three hundreds or fi ve tens. In
such a numeration system there is no need for a “zero” sym-
bol as a placeholder, the name of the particular 10’s place is
just missing from the designation.
Th e positional value scheme of the Chinese numerals
was very effi cient and fl exible, allowing for the recording of
large numbers, up to 10^7 ; it could even, conceptually, express
numbers and counting: Asia and the Pacific 801