The chi-squared distribution is a
DISTRIBUTION
repre-
senting the values one would expect χ2to adopt given
the assumption that the two features being studied are
independent. There is one distribution for each table of
given dimensions. Statistics texts usually present lists of
values for this statistic. It turns out that the χ2value in
this example is extraordinarily high, suggesting that
this study indicates a strong correlation between milk
choice and body piercings. The chi-squared test does
not give any information about the nature of the corre-
lation detected, only that it seems to exist. Further
examination of the data by alternative methods may
provide details of the association.
See also K
ARL
P
EARSON
.
chord If Aand Bare two points on a continuous
curve, then a straight-line segment connecting Ato Bis
called a chord to the curve. This is not to be confused
with the
ARC
of the curve connecting Ato B.
The
CIRCLE THEOREMS
show that any chord of a
circle is bisected by a radius that is perpendicular to
it. B
ERTRAND
’
S PARADOX
shows that the act of select-
ing chords of a circle at random can lead to philo-
sophical difficulties.
See also
CIRCLE
.
Chu Shih-Chieh (Zhu Shijie) (ca. 1270–1330) Chi-
nese Algebra Often regarded as one of China’s great-
est mathematicians, Chu Shih-Chieh is remembered for
his influential 1303 text Su-yuan yu-chien (Precious
mirror of the four elements). It contains a diagram of
P
ASCAL
’
S TRIANGLE
, as it has become known in the
West, representing one of the earliest appearances of
the figure in the history of mathematics. In the work,
Chu Shih-Chieh uses the triangle to describe a general
method for extracting roots to equations. He also pre-
sents a system of notation for polynomials in four
unknowns, which he calls the four elements—namely,
the celestial, the earthly, the material, and the human—
and provides effective techniques for manipulating
them to solve problems.
His section on
FINITE DIFFERENCES
gives formulae
for the sums of the first nterms of each diagonal of
Pascal’s triangle (for instance, that 1 + 2 + 3 + … + n
equals , and that 1 + 3 + 6 + 10 + … +
equals ), and also provides general
techniques for summing arbitrary series. He also pro-
vides methods for solving equations via a process of
successive approximations.
Many historians claim that Chu Shih-Chieh’s impres-
sive work represents the peak of ancient Chinese mathe-
matics, noting that relatively little progress was made for
a long time after the publication of this piece. Four years
before the release of Su-yuan yu-chien, Chu Shih-Chieh
wrote a mathematical text intended to help beginners in
the subject. Extremely little is known of his personal life.
circle The set of all points in a plane a fixed distance
rfrom a given point Oforms a closed curve in the
plane called a circle. The length ris called the radius of
the circle, and the point Oits center. If the center point
has C
ARTESIAN COORDINATES
0 = (a,b),then the
DIS
-
TANCE FORMULA
shows that any point (x,y) on the cir-
cle satisfies the equation:
(x– a)2+ (y– b)2= r2
If the point (x,y) on this circle makes an angle θwith
a horizontal line through the center of the circle, then
we have:
x= a+ rcosθ
y= b+ rsinθ
These are the
PARAMETRIC EQUATIONS
of a circle of
radius rand center (a,b).
The
DIAMETER
of a circle is the maximal distance
between two points on the circle. It equals twice the
radius of the circle. A circle is a figure of
CONSTANT
WIDTH
.
The length of the curve closed to form a circle is
called the circumference of the circle. Scholars since
the time of antiquity have observed that the ratio of
the circumference of a circle to its diameter is the same
for all circles. This constant value is called
PI
, denoted
π. We have:
π= 3.14159265…
That all circles yield the same value for πis not imme-
diately obvious. This is a property of E
UCLIDEAN
GEOMETRY
of the plane, and a careful study of
SCALE
explains why it must be true. (The value of πvaries
n(n+ 1)(n+ 2)
6
n(n+ 1)
2
n(n+ 1)
2
circle 75