people of the ancient Shang dynasty had developed a
base-10 notational system utilizing place value. This
establishes that the Chinese were one of the first civiliza-
tions to invent a
DECIMAL REPRESENTATION
system
essentially equivalent to the one we use today. Like other
civilizations of the time, however, the Chinese had not
developed a notation for zero and so wrote, for exam-
ple, the numbers 43 and 403 the same way, namely as
“|||| |||,” relying on context to distinguish the two.
The most important early Chinese mathematical
text is Jiuzhang suanshu (Nine chapters on the mathe-
matical art) dating from the period of the Han dynasty
(206
B
.
C
.
E
. to 220
C
.
E
.). The author of the work is
unknown, but it is believed to be a summary of all
mathematical knowledge possessed in China up to the
third century
C
.
E
., and may well have been the result of
several authors contributing to the same work. The
text is a presentation of 246 problems replete with
solutions and general recipes for solving problems of a
similar type. The work is generally very practical in
nature, with three chapters devoted to issues of land
surveying and engineering, and three to problems in
taxation and bureaucratic administration. But the text
does describe sophisticated mathematical techniques of
an abstract nature, and it offers many problems of a
recreational flavor. The document thus also clearly
demonstrates that scholars of the time were also inter-
ested in the study of mathematics for its own sake.
Jiuzhang suanshu is clearly not written for begin-
ners in the art of mathematics: many basic arithmetic
processes are assumed known. Another text of the
same period, Chou pei suan ching (Arithmetic classic of
the gnomon and the circular paths of heaven),
describes basic mathematical principles such as work-
ing with fractions (and establishing common denomi-
nators), methods of extracting square roots, along with
basic principles and elements of geometry, and surpris-
ingly, what appears to be a proof of what we today call
P
YTHAGORAS
’
S THEOREM
. At the very least, the Chinese
of this period knew the theorem for right triangles with
sides of length 3, 4, and 5 as the diagram of hsuan-
thu—four copies of a 3–4–5 right triangle arranged in a
square—appears in the text. Although this diagram in
itself does not constitute a “proof” of Pythagoras’s the-
orem, the idea embodied in the diagram can nonethe-
less easily be expanded upon to establish a general
proof of the result. For this reason, it is believed that
the Chinese had independently established the same
famous result. This is certainly verified in the text of
Jiuzhang suanshu, as many problems posed in the piece
rely on the reader making use of the theorem.
Many scholars from the second to the 15th cen-
turies wrote commentaries on the work Jiuzhang suan-
shu and extended many of the results presented there.
Perhaps the most famous of these were the commen-
taries of Liu Hui who, in 263
C
.
E
., offered written
proofs of the formulae for the volumes of a square
pyramid and a tetrahedron presented in Jiuzhang suan-
shu, as well as developed a more precise value for π
than presented in the text. Liu Hui later went on to
write Haidao suanjing (Sea island mathematical man-
ual), in which he solved problems related to the survey-
ing and mapping of inaccessible objects using a refined
method of “double differences” arising from pairs of
similar triangles. This extended the work of propor-
tions presented in Jiuzhang suanshu. Liu Hui also
claimed that the material of Jiuzhang suanshu dates
back to 1100
B
.
C
.
E
., but added that the actual text was
not written until 100
B
.
C
.
E
. Historians today differ
about how seriously to take his claim.
The text Jiuzhang suanshu also contains recipes for
extracting square and cube roots, and methods for solv-
ing systems of linear equations using techniques very
similar to the methods of
LINEAR ALGEBRA
we use today.
Liu Hui’s commentary gives justification for many of the
rules presented. Although not formal proofs based on
axioms, it is fair to describe Liu Hui’s justifications as
valid informal proofs. It seems that mathematicians for
the centuries that followed remained satisfied with sim-
ple informal arguments and justifications, and no formal
rigor was deemed necessary.
Chinese scholars also made significant contribu-
tions to the study of
COMBINATORICS
,
NUMBER THEORY
,
and
ALGEBRA
. Mathematician C
H
’
IN
C
HIU
-
SHAO
(ca.
1200
C
.
E
.) developed inventive methods for evaluating
polynomial expressions and solving polynomial equa-
tions. He also established the famous “Chinese remain-
der theorem” of number theory. The work of L
I
Y
E
of
the same period also establishes an algebra of polyno-
mials. Although Chinese mathematicians of the time
were familiar with negative numbers, they ignored neg-
ative solutions to equations, deeming them absurd.
The famous text Su-yuan yu-chien (The precious
mirror of the four elements) written by the scholar C
HU
S
HIH
-C
HIEH
(ca. 1300
C
.
E
.) contains a diagram of what
has in the West become known as P
ASCAL
’
S TRIANGLE
.
Chinese mathematics 73