n!

–

0!

fair division 187

1

–

2

=

Γ

When requals n(arrange all nobjects) this formula

reads . To coincide with our previously computed

answer of n!, it is natural to define 0! as equal to 1.

The exclamation-point notation for factorial was

first used by Christian Kramp in 1808, in his paper

“Élémens d’arithmétique universelle,” though other

notations for n! popular at the time, and later, included

n

_

|

, n′, Π(n) and Γ(n+1).

L

EONHARD

E

ULER

(1707–83) attempted to general-

ize the factorial function to noninteger values. At the

age of 22 he discovered the following

LIMIT

quantity

that helps achieve this:

(He called this expression the gamma function to

honor A

DRIEN

-M

ARIE

L

EGENDRE

’s use of the symbol Γ

for factorial.) This expression has the property, as you

may check, that Γ(x+ 1) = x· Γ(x) for all positive

real values x. Also .

Consequently, Γ(m) = (m– 1)! for all positive integers m.

Thus, for example, Γ(7) = 6! = 720. It is now possible to

also define quantities such as and (√

–

2)! using this

expression.

I

NTEGRATION BY PARTS

shows that the

IMPROPER

INTEGRAL

∫∞

0e–ttm–1 dt also has value (m– 1)!. Mathe-

maticians have shown that Euler’s gamma function and

the corresponding improper integral agree for all posi-

tive real values x:

Γ(x) = ∫+∞

0e–ttx–1 dt

This integral has the unexpected value √

–

πwhen x= .

Thus we may conclude, for example, that

, and .

See also

BINOMIAL THEOREM

;

PERMUTATION

;

S

TIRLING

’

S FORMULA

.

factorization The process or the result of writing a

number or a

POLYNOMIAL

as a product of terms is

called factorization. For example, the

FUNDAMENTAL

THEOREM OF ARITHMETIC

asserts that every whole num-

ber can be written as a product of

PRIME

numbers and,

up to the order of the terms, this factorization is unique.

(For instance, 132 = 2 ×2 ×3 ×11.) Thus every whole

number has a unique “prime factorization.” The

FUN

-

DAMENTAL THEOREM OF ALGEBRA

asserts that, in the

realm of

COMPLEX NUMBERS

, every polynomial factors

completely into linear terms. (For instance, 2x3– x2

–13x– 6 = (x– 3)(x+ 2)(2x+ 1) and x2– 4x+ 5 =

(x–2 + i)(x– 2 – i).) If one wishes to remain in the

realm of the

REAL NUMBERS

, then every polynomial with

real coefficients is guaranteed to factor into a product of

linear terms and irreducible

QUADRATIC

terms. (For

instance, x4– 1 = (x2+ 1)(x– 1)(x+ 1).)

See also

DECOMPOSITION

;

FACTOR THEOREM

.

factor theorem The

REMAINDER THEOREM

shows

that if a

POLYNOMIAL

p(x) is divided by a term of the

form x– afor some constant a, then the remainder

term is the constant p(a):

p(x) = (x– a)Q(x) + p(a)

Thus if the value of the polynomial is zero at x= a,

that is, p(a) = 0, then the polynomial factors as p(x) =

(x– a)Q(x).This leads to the following factor theorem:

A linear term x– ais a factor of a polynomial

p(x) if, and only if, p(a) = 0.

For example, for p(x) = 2x3– 4x2– 10x+ 12, we have

p(1) = 0, p(–2) = 0, and p(3) = 0. Consequently, x– 1, x

+ 2, and x– 3 are each factors of the polynomial. (In

this example we have p(x) = 2(x– 1)(x+ 2)(x– 3).)

Since 2 and –2 are clearly each zeros of x6–64, this

polynomial must be divisible by (x– 2)(x+ 2) = x2– 4.

See also

FUNDAMENTAL THEOREM OF ALGEBRA

.

fair division (cake cutting) A classic puzzle asks for

a fair way to divide a piece of cake between two greedy

brothers. The “you cut, I choose” scheme asks one

brother to slice the cake into what he believes to be

two equal parts and has the second brother choose one

of the two pieces. The first is then guaranteed to

receive, in his measure, precisely 50 percent of the cake

and the second brother, if he has a different estimation