b

––

a+ b

a

––

a+ b

b

–

a

a

–

b

Omar Khayyám 363

octant The xy-, yz-, and xz-planes in a three-dimen-

sional C

ARTESIAN COORDINATE

system divide space

into eight regions called “octants.” The set of points

(x,y,z) satisfying x> 0, y> 0, and z> 0 constitute the

first or “positive” octant. The remaining octants are

labeled the second, third, and fourth octants counter-

clockwise around the positive z-axis, with the fifth,

sixth, seventh, and eighth octants underneath these

four, again in a counterclockwise sense, with the fifth

octant directly under the first.

The two-dimensional analog of an octant is a

QUAD

-

RANT

. For ngreater than three, the n-dimensional ana-

log is called an orthant. Points in the positive orthant

have all coordinates positive. Popular science fiction

novels often describe space as divided into quadrants.

This is the incorrect term for three-dimensional space.

An

ANGLE

of 45°is also sometimes called an octant.

odds If in a game of chance there are aoutcomes

that are deemed favorable (thereby constituting a

desired

EVENT

E), and boutcomes that are unfavorable

(not in E), then the odds in favor of E is the ratio ,

usually written a:b or “ato b”, and the odds against E

is the ratio , written b:a or “bto a.”

For example, in casting a die, the odds in favor of

rolling a 2 are 1:5. There is just one favorable outcome,

namely a 2, and five unfavorable outcomes: {1,3,4,5,6}.

Odds are usually written in reduced form. For exam-

ple, if the odds in favor of an event are 12 to 4, one

typically writes 3:1 rather than 12:4.

Odds are closely connected to

PROBABILITY

compu-

tations. If the odds in favor of an event E are ato b,

then one is informed that aof a total of a+ bpossible

outcomes belong to the set E. Thus the probability of

event E occurring is given by P(E) = , and that of

it not occurring by P(not E) = . Conversely, if the

probability of an event E is known, then the odds in

favor of E can be computed as a ratio of probabilities:

, and the odds against as the

inverse ratio of probabilities.

Odds are typically used at horse races—usually

“odds against”—to tell gamblers the payoffs on vari-

ous bets. For example, advertised odds of 5 to 3 on a

particular horse indicates that a $3 bet on that horse

will win $5 (plus the return of the original three dol-

lars) if that horse wins the race. It also tells the gam-

bler that this horse is believed to have only a 3/8

chance of winning.

Omar Khayyám (Umar al-Kh–

ayamm–

ı) (1048–1122)

Arab Algebra, Astronomy Born on May 18, 1048,

Persian scholar Omar Khayyám is remembered in

mathematics for his significant contributions to the

advancement of

ALGEBRA

. In his famous text, Treatise on

Demonstration of Problems of Algebra, Khayyám devel-

oped both geometric and algebraic rules for solving

QUADRATIC

and

CUBIC EQUATION

s, making clever use of

CONIC SECTIONS

. He was the first mathematician to ever

conceive of a general theory for solving cubics and was

the first to recognize that ruler-and-compass construc-

tions alone would never suffice as a geometric approach

for this goal. Outside of mathematics, Omar Khayyám is

best known for his poems, which were freely translated

by Edward Fitzgerald in 1859 in the text The Rubáiyát

of Omar Khayyám.

Khayyám studied philosophy and mathematics at

his birthplace of Nishapur, Persia (now Iran), and

quickly became an expert scholar. By age 25 he had

written three significant texts: one on music, one on

arithmetic, and one a first text on the topic of algebra.

In 1073 Khayyám moved to the city of Esfah–

an to help

with the construction of an observatory and to head a

team of scientists studying astronomy. Over the course

of his 18 years at the observatory Khayyám completed

extraordinarily accurate tables of astronomical data and

tables of trigonometric values. He also computed the

length of the year to be 365.24219858156 days, which,

at the time, was correct to the sixth decimal place. (The

length of a year is changing over time. Today its value is

365.242190 days. The value of the sixth decimal place

changes over the course of a century.)

Later in life Khayyám moved to the cultural center of

Merv (now Mary in Turkmenistan) and returned to his

interests in algebra. He wrote his famous treatise on the

topic while there. Khayyám noted that the great Greek

scholars of antiquity made no serious study of the theory

of cubic equations and decided to take it upon himself to

develop this work appropriately. Khayyám created inge-

nious geometric methods for finding cubic equations, but