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| Lo Shu magic square, with its traditional graphical representation
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And Ancient Chinese Numbering System
CHINESE
MATHEMATICS
There was a pervasive fascination with numbers and mathematical
patterns in ancient China, and different numbers were believed to have cosmic
significance.
Even
as mathematical developments in the ancient Greek world were beginning to
falter during the final centuries BCE, the burgeoning trade empire of China was
leading Chinese mathematics to ever greater heights.
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| Ancient Chinese number system |
The
simple but efficient ancient Chinese numbering system, which dates back to at
least the 2nd millennium BCE, used small bamboo rods arranged to represent the
numbers 1 to 9, which were then places in columns representing units, tens,
hundreds, thousands, etc.
It
was therefore a decimal place value system, very similar to the one we use
today - indeed it was the first such number system, adopted by the Chinese over
a thousand years before it was adopted in the West - and it made even quite
complex calculations very quick and easy.
Written
numbers, however, employed the slightly less efficient system of using a
different symbol for tens, hundreds, thousands, etc.
This
was largely because there was no concept or symbol of zero, and it had the
effect of limiting the usefulness of the written number in Chinese.
The
use of the abacus is often thought of as a Chinese idea, although some type of
abacus was in use in Mesopotamia, Egypt and Greece, probably much earlier
than in China (the first Chinese abacus, or “suanpan”, we know of dates to
about the 2nd Century BCE).
There
was a pervasive fascination with numbers and mathematical patterns in ancient
China, and different numbers were believed to have cosmic significance.
In
particular, magic squares - squares of numbers where each row, column and diagonal
added up to the same total - were regarded as having great spiritual and
religious significance.
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| Lo Shu magic square, with its traditional graphical representation . |
The
Lo Shu Square, an order three square where each row, column and diagonal adds
up to 15, is perhaps the earliest of these, dating back to around 650 BCE (the
legend of Emperor Yu’s discovery of the the square on the back of a turtle is
set as taking place in about 2800 BCE).
But
soon, bigger magic squares were being constructed, with even greater magical
and mathematical powers, culminating in the elaborate magic squares, circles
and triangles of Yang Hui in the 13th Century (Yang Hui also produced a triangular
representation of binomial coefficients identical to the later Pascals’
Triangle, and was perhaps the first to use decimal fractions in the modern
form).
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| Early Chinese method of solving equations |
But
the main thrust of Chinese mathematics developed in response to the empire’s
growing need for mathematically competent administrators.
A
textbook called “Jiuzhang Suanshu” or “Nine Chapters on the Mathematical Art”
(written over a period of time from about 200 BCE onwards, probably by a
variety of authors) became an important tool in the education of such a civil
service, covering hundreds of problems in practical areas such as trade,
taxation, engineering and the payment of wages.
It
was particularly important as a guide to how to solve equations - the deduction
of an unknown number from other known information - using a sophisticated
matrix-based method which did not appear in the West until Carl Friedrich Gauss re-discovered
it at the beginning of the 19th Century (and which is now known as Gaussian
elimination).
Among
the greatest mathematicians of ancient China was Liu Hui, who produced a
detailed commentary on the “Nine Chapters” in 263 CE, was one of the first
mathematicians known to leave roots unevaluated, giving more exact results
instead of approximations.
By
an approximation using a regular polygon with 192 sides, he also formulated an
algorithm which calculated the value of πas 3.14159 (correct to five decimal places), as well as
developing a very early forms of both integral and differential calculus.
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| The Chinese Remainder Theorem |
The
Chinese went on to solve far more complex equations using far larger numbers
than those outlined in the “Nine Chapters”, though.
They
also started to pursue more abstract mathematical problems (although usually
couched in rather artificial practical terms), including what has become known
as the Chinese Remainder Theorem.
This
uses the remainders after dividing an unknown number by a succession of smaller
numbers, such as 3, 5 and 7, in order to calculate the smallest value of the
unknown number.
A
technique for solving such problems, initially posed by Sun Tzu in the 3rd
Century CE and considered one of the jewels of mathematics, was being used to
measure planetary movements by Chinese astronomers in the 6th Century AD, and
even today it has practical uses, such as in Internet cryptography.
By
the 13th Century, the Golden Age of Chinese mathematics, there were over 30 prestigious
mathematics schools scattered across China.
Perhaps
the most brilliant Chinese mathematician of this time was Qin Jiushao, a rather
violent and corrupt imperial administrator and warrior, who explored solutions
to quadratic and even cubic equations using a method of repeated approximations
very similar to that later devised in the West by Sir Isaac Newton in the
17th Century.
Qin
even extended his technique to solve (albeit approximately) equations involving
numbers up to the power of ten, extraordinarily complex mathematics for its
time.
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| Lo Shu magic square, with its traditional graphical representation |
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