Hair Color

Black Brown Blonde Red

Male 25 23 8 2

Female 18 16 14 5

98 contingency table

1,402

457

contingency table A table showing the number of

units from a sample having certain combinations of

attributes is called a contingency table. For example, a

marketing research project records the hair color of

participating men and women and presents the results

in a contingency table:

The

CHI

-

SQUARED TEST

can be used to look for correla-

tions between attributes displayed in such tables.

continued fraction A number that is an integer plus

a fraction with denominator that is itself an integer

plus a fraction—and continued this way—is called a

continued fraction. For example,

and

are continued-fraction representations of the quantities

1,402/457 and √

–

2 (as we shall establish below). The

first continued fraction stops after a finite number of

steps, and the second continues forever. A continued

fraction is said to be in standard form if, like the sec-

ond example, all the numerators are equal to one, and

all the integers involved are positive.

Every positive real number xcan be written as a

continued fraction in standard form. If xdenotes the

largest integer less than or equal to x, and {x} the frac-

tional part of xas given by the

FRACTIONAL PART

FUNCTION

, then:

As the quantity 1/{x} itself is a positive real number

greater than one, we can, in the same way, write it as

an integer plus another fraction with unit numerator.

Repeated application of this procedure produces a con-

tinued fraction in standard form.

For example, if x= , then we can write

, and ,

and so on. This produces the standard-form continued

fraction:

It is not difficult to show that if x= a/b is a frac-

tion, then 1/{x} is a new fraction with denominator

smaller than b. Repeated application of this procedure

must eventually produce a continued fraction with

denominator equal to one, so that the procedure ter-

minates. This shows that all numbers that are rational

(that is, equal to a fraction) have continued-fraction

representations that stop after a finite number of

steps. (And, conversely, any such continued fraction

“unravels” to produce a quantity that is rational.)

Consequently:

All quantities with infinitely long continued-

fraction representations are irrational.

For example, one can check that and

substituting this formula into itself gives the continued

fraction representation presented above: