central-limit theorem 69

of degree n. Then the matrix Asatisfies this

particular polynomial equation.

For example, consider the 2 ×2 matrix .

Then the “characteristic polynomial” of this matrix is:

One now checks that

is indeed the zero matrix.

See also

IDENTITY MATRIX

;

LINEARLY DEPENDENT

AND INDEPENDENT

.

ceiling function See

FLOOR

/

CEILING

/

FRACTIONAL PART

FUNCTIONS

.

center of gravity (balance point) The location at

which the weight of the object held in space can be

considered to act is called the object’s center of gravity.

For example, a uniform rod balances at its midpoint,

and this is considered its center of gravity. A flat rect-

angular plate made of uniform material held parallel to

the ground balances at its center. This point is the fig-

ure’s center of gravity.

Archimedes’

LAW OF THE LEVER

finds the balance

point Pof a system of two masses m1and m2held in

space. The two-mass system can then be regarded as a

single mass m1+ m2located at P.

The center of gravity of a system of three masses in

the space can be found by finding the balance point of

just two masses, using

ARCHIMEDES

’ law of the lever,

and then applying the law a second time to find the

center of gravity of this balance point and the third

mass. This procedure can, of course, be extended to

find the center of gravity of any finite collection of

masses. (A location computed this way is technically

the center of mass of the system. If the force of gravity

is assumed to be uniform, then the center of mass coin-

cides with the center of gravity of the system.)

This principle can be extended to locate the center

of gravity of arbitrary figures in the plane (viewed as

flat, uniformly dense objects held parallel to the

ground). If the figure is composed of a finite collection

of rectangles glued together, one locates the center of

each rectangle, the mass of each rectangle, and then

regards the system as a collection of individual masses

at different locations. Applying Archimedes’ law of the

lever as above locates the figure’s center of gravity. If a

figure can only be approximated as a union of rectan-

gles, one can find the approximate location of the cen-

ter of gravity via this principle, and then improve the

approximation by taking the

LIMIT

result of using finer

and finer rectangles in the approximations. This

approach will yield an

INTEGRAL

formula for the loca-

tion of the center of gravity.

See also C

EVA

’

S THEOREM

;

SOLID OF REVOLUTION

.

central-limit theorem In the early 1700s scientists

from a wide range of fields began to notice the recur-

ring appearance of the

NORMAL DISTRIBUTION

in their

studies and experiments. Any measurement that repre-

sents an average value of a sample, or an aggregate

value of a series of results, tends to follow this classical

bell-shaped distribution. The work of M

ARQUIS DE

P

IERRE

-S

IMON

L

APLACE

in 1818 and Aleksandr

Mikhailovich Lyapunov in 1901, and others, led to the

establishment of the central-limit theorem:

If an experiment involves the repeated compu-

tation of the average value of Nmeasurements

(a different set of Nmeasurements each time),

then the set of average values obtained very

closely follows a normal distribution—even if

the original experiments do not. The larger the

value of N, the better the approximation to a

normal curve.

One can go further and say that if the original

experiments have mean µand standard deviation σ, then

the collection of average values also has mean µ, but