486 Stirling’s formula

n

appears no more important or special than any other

real number, 6.234 or 84.668, for instance. One scholar

at the time called all numbers consequently “equally

boring.”

Stevin is also remembered for introducing and pop-

ularizing the symbols “+,” “ –,” and “√

–

” for addition,

subtraction, and the square-root operation. He died in

The Hague, Netherlands, ca. March 1620.

Stirling’s formula The surprising formula:

shows that for large values of n, the function

is a very good approximation for the

FACTORIAL

n!. This

limit equation is called Stirling’s formula. Although

named after the Scottish mathematician James Stirling

(1692–1770), the formula was discovered by French

mathematician A

BRAHAM

D

E

M

OIVRE

(1667–1754)

while attempting to write a formula for the

NORMAL DIS

-

TRIBUTION

curve. Stirling wrote an influential paper on

infinite series that helped De Moivre discover this result.

The formula can be derived from W

ALLIS

’

S PROD

-

UCT

. We present here a simple argument that gives a

sense of how the result could be true. Noting that ln n!

can be written as a sum of logarithms, we have:

Evaluating the integral, using

INTEGRATION BY PARTS

,

yields:

∫n

1ln x dx = xln x– x

|

n

1= nln n– n+ 1 ≈nln n– n

Thus ln n! ≈ln(nn) – n, giving . A more

refined argument yields Stirling’s result.

Student’s t-test See

STATISTICS

:

INFERENTIAL

.

subfactorial In his 1878 study of

PERMUTATION

s,

mathematician W. Allen Whitworth introduced the

notion of a subfactorial. Now denoted n¡, the nth sub-

factorial of an integer nis given by

with 0¡set equal to 1. (See

FACTORIAL

.) The value n¡is

obtained by multiplying the previous subfactorial num-

ber by nand adding (–1)n. Thus:

1¡= 1 ×0¡– 1 = 1 ×1 – 1 = 0

2¡= 2 ×1¡+ 1 = 2 ×0 + 1 = 1

3¡= 3 ×2¡– 1 = 3 ×1 – 1 = 2

4¡= 4 ×3¡+ 1 = 4 ×2 + 1 = 9

for example. A study of the number eshows that the

ratio approaches the value as nbecomes large.

Subfactorials arise in the study of certain permuta-

tions called derangements.

See also

E

.

substitution The act of replacing all occurrences of a

variable by another variable, expression, or numerical

value is called substitution. For example, to evaluate the

expression x2– 2x+ 3 for x= 5, simply replace all

occurrences of xwith the value 5. This yields the corre-

sponding value of 18 for the expression. (We say that we

have “substituted the value 5 for x.”) One can also sub-

stitute other quantities for x. For instance, replacing

each occurrence of xwith the quantity y– 1 yields the

expression (y– 1)2+ 2(y– 1) + 3, which simplifies to

y2+ 2. (This particular substitution proves to be fruitful,

for it makes clear now that the expression x2– 2x+ 3

will never adopt a value smaller than 2, and that the

expression equals this minimal value only when y= 0,

that is, when x= 1.) Substitution techniques are used to

solve

SIMULTANEOUS LINEAR EQUATIONS

.

The method of substitution can be applied simulta-

neously to more than one variable in an equation. For

instance, the

AREA

Aof a triangle with side-lengths a,

b, and c, with angle θbetween sides of lengths aand b,

is given by A= absinθ. The

LAW OF COSINES

asserts

c2= a2+ b2– 2ab cos θ. Solving for sin θand cos θin

each expression and substituting the results into the

equation sin2θ+ cos2θ= 1 yields H

ERON

’

S FORMULA

.

The square root of 2 satisfies the equation:

1

–

2

1

–

e

n¡

–

n!