The chi-squared distribution is a DISTRIBUTION representing the values one would expect χ
2 to 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 χ
2 value 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 correlation detected, only that it seems to exist. Further
examination of the data by alternative methods may
provide details of the association.
See also KARL PEARSON.
chord If A and B are two points on a continuous
curve, then a straight-line segment connecting A to B is
called a chord to the curve. This is not to be confused
with the ARC of the curve connecting A to B.
The CIRCLE THEOREMS show that any chord of a
circle is bisected by a radius that is perpendicular to
it. BERTRAND’S PARADOX shows that the act of selecting chords of a circle at random can lead to philosophical difficulties.
See also CIRCLE.
Chu Shih-Chieh (Zhu Shijie) (ca. 1270–1330) Chinese Algebra Often regarded as one of China’s greatest 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
PASCAL’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 presents 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 n terms 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 provides methods for solving equations via a process of
successive approximations.
Many historians claim that Chu Shih-Chieh’s impressive work represents the peak of ancient Chinese mathematics, 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
r from a given point O forms a closed curve in the
plane called a circle. The length r is called the radius of
the circle, and the point O its center. If the center point
has CARTESIAN COORDINATES 0 = (a,b), then the DISTANCE FORMULA shows that any point (x,y) on the circle satisfies the equation:
(x – a)
2 + (y – b)
2 = r
2
If the point (x,y) on this circle makes an angle θ with
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 r and 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 π is not immediately obvious. This is a property of EUCLIDEAN
GEOMETRY of the plane, and a careful study of SCALE
explains why it must be true. (The value of π varies
n(n + 1)(n + 2)
6
n(n + 1)
2
n(n + 1)
2
circle 75
2 to 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 χ
2 value 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 correlation detected, only that it seems to exist. Further
examination of the data by alternative methods may
provide details of the association.
See also KARL PEARSON.
chord If A and B are two points on a continuous
curve, then a straight-line segment connecting A to B is
called a chord to the curve. This is not to be confused
with the ARC of the curve connecting A to B.
The CIRCLE THEOREMS show that any chord of a
circle is bisected by a radius that is perpendicular to
it. BERTRAND’S PARADOX shows that the act of selecting chords of a circle at random can lead to philosophical difficulties.
See also CIRCLE.
Chu Shih-Chieh (Zhu Shijie) (ca. 1270–1330) Chinese Algebra Often regarded as one of China’s greatest 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
PASCAL’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 presents 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 n terms 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 provides methods for solving equations via a process of
successive approximations.
Many historians claim that Chu Shih-Chieh’s impressive work represents the peak of ancient Chinese mathematics, 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
r from a given point O forms a closed curve in the
plane called a circle. The length r is called the radius of
the circle, and the point O its center. If the center point
has CARTESIAN COORDINATES 0 = (a,b), then the DISTANCE FORMULA shows that any point (x,y) on the circle satisfies the equation:
(x – a)
2 + (y – b)
2 = r
2
If the point (x,y) on this circle makes an angle θ with
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 r and 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 π is not immediately obvious. This is a property of EUCLIDEAN
GEOMETRY of the plane, and a careful study of SCALE
explains why it must be true. (The value of π varies
n(n + 1)(n + 2)
6
n(n + 1)
2
n(n + 1)
2
circle 75