Chebyshev’s theorem (Chebyshev’s inequality) This
result, due to the Russian mathematician P
AFNUTY
L
VOVICH
C
HEBYSHEV
(1821–94), can be thought of as
an extension of the 68-95-99.7 rule for the
NORMAL
DISTRIBUTION
to one applicable to all distributions. It
states that if an arbitrary
DISTRIBUTION
has mean μand
standard deviation σ, then the probability that a mea-
surement taken at random will have value differing
from μby more than k standard deviations is at most
1/k2. This shows that if the value σis small, then all
DATA
values taken in an experiment are likely to be
tightly clustered around the value μ.
Manufacturers make use of this result. For exam-
ple, suppose a company produces pipes with mean
diameter 9.57 mm, with a standard deviation of 0.02
mm. If manufacturer standards will not tolerate a pipe
more than four standard deviations away from the
mean (0.08 mm), then Chebyshev’s theorem implies
that on average about 1/16, that is 6.25 percent, of the
pipes produced per day will be unusable.
The
LAW OF LARGE NUMBERS
follows as a conse-
quence of Chebyshev’s theorem.
See also
STATISTICS
:
DESCRIPTIVE
.
chicken See
PRISONER
’
S DILEMMA
.
Ch’in Chiu-shao (Qin Jiushao) (1202–1261) China
Algebra Born in Szechwan (now Sichuan), China,
mathematician and calendar-maker Ch’in Chiu-shao is
remembered for his 1243 text Shushu jiuzhang (Mathe-
matical treatise in nine sections), which contains,
among many methods, an effective technique of iter-
ated multiplication for evaluating polynomial equa-
tions of arbitrary degree. (In modern notation, this
technique is equivalent to replacing a polynomial such
as 4x3+ 7x2– 50x+ 9, for instance, with its equivalent
form as a series of nested parentheses: ((4x+ 7)x–
50)x+ 9. In this example, only three multiplications
are needed to evaluate the nested form of the polyno-
mial compared with the six implied by the first form of
the expression. In practice, this technique saves a con-
siderable amount of time.) This approach was discov-
ered 500 years later in the West independently by
Italian mathematician Paolo Ruffini (1765–1822) and
English scholar William George Horner (1786–1837).
Ch’in Chiu-shao also extended this method to find
solutions to polynomial equations.
His text is also noted for its development of
MODU
-
LAR ARITHMETIC
. In particular, Ch’in Chiu-shao proved
the following famous result, today known as the Chi-
nese remainder theorem:
If a set of integers miare pair-wise
COPRIME
,
then any set of equations of the form
x≡ai(mod mi) has a unique solution modulo
the product of all the mi.
For example, this result establishes that there is
essentially only one integer xthat leaves remainders
of 1, 11, and 6, respectively, when divided by 5, 13,
and 16 (namely, 726, plus or minus any multiple of
5×13 ×16 = 1040).
After serving in the army for 14 years, Ch’in Chiu-
shao entered government service in 1233 to eventually
become provincial governor of Qiongzhou. His 1243
piece Shushu jiuzhang was his only mathematical work.
Chinese mathematics Unfortunately, very little is
known about early Chinese mathematics. Before the
invention of paper around 1000
C
.
E
., the Chinese
wrote on bark or bamboo, materials that were far
more perishable than clay tablets or papyrus. To make
matters worse, just after the imperial unification of
China of around 215
B
.
C
.
E
., Emperor Shi Huang-ti of
the Ch’ih dynasty ordered that all books from earlier
periods be burned, along with the burying alive of any
scholars who protested. Only documents deemed “use-
ful,” such as official records and texts on medicine,
divination, and agriculture were exempt. Consequently
very little survived beyond this period, although some
scholars did try to reconstruct lost materials from
memory.
The art of mathematics was defined by ancient
Chinese scholars as suan chu, the art of calculation.
Often the mathematics studied was extremely practical
in nature, covering a wide range of applications,
including engineering, flood control, and architecture,
as well astronomy and divination. Practitioners of the
art were capable scientists. Records show, for example,
that the Chinese had invented seismographs to measure
earthquakes by the year 1000
C
.
E
., and used compasses
made with magnetic needles a century later.
Evidence of mathematical activity in China can be
dated back to the 14th century
B
.
C
.
E
.. Tortoise shells and
cattle bones inscribed with tally marks indicate that the
72 Chebyshev’s theorem