pi 393

In 1882 German mathematician F

ERDINAND VON

L

INDEMANN

proved that πis a

TRANSCENDENTAL NUM

-

BER

, establishing once and for all that the problem of

SQUARING THE CIRCLE

cannot be solved.

With the advent of computing machines in the 20th

century, mathematicians could compute more and more

digits of π. In 1949 J

OHN VON

N

EUMANN

used the U.S.

government’s ENIAC computer to compute πto the

2,037th decimal place. (It took 70 hr of machine time.)

In 1981 Japanese scientists Kazunori Miyoshi and

Kazuhiko Nakayama evaluated 2 million decimal places

of π, and in 1991, using a homebuilt supercomputer in a

New York City apartment, brothers Gregory and David

Chudnovsky calculated πto 2,260,321,366 decimal

places. Today over 1.24 ×1012 digits of πare known.

The sequence of digits 0123456789 appears in

the decimal expansion of πbeginning at the

17,387,594,880th decimal place. This is the first, but

not the only, appearance of this sequence. The

9876543210 first appears at the 21,981,157,633rd

decimal place.

There are many beautiful formulae for π. For

instance, V

IÈTE

’

S FORMULA

, the G

REGORY SERIES

, the

ZETA FUNCTION

, and W

ALLIS

’

S PRODUCT

show, respec-

tively, that:

The B

UFFON NEEDLE PROBLEM

also provides another

surprising appearance of the π. The Swiss mathemati-

cian L

EONHARD

E

ULER

(1707–83) also showed:

(Similar formulae follow from the general identity:

for suitable choices of xand y.) Hungarian mathemati-

cian P

AUL

E

RDÖS

(1913–96) established the following

remarkable result:

Beginning with a positive integer n, round it up

to the nearest multiple of n– 1, and then

round the result up to the nearest multiple of

n– 2, and so on, up until the nearest multiple

of 2. Call the result f(n).(We have, for

instance, f(3) = 4, f(5) = 10, and f(7) = 18.)

Then the

LIMIT

of the ratio of n2to f(n),as n

becomes large, is π:

A number of basic questions about the interplay

between πand Euler’s number eremain unanswered.

For instance, no one yet knows whether the numbers

π+ e, , or loge(π) are rational or irrational (nor

whether ππis algebraic or transcendental). It is curious

that the quantity eπ– πhas a value extraordinarily

close to 20.

Our choice to use the symbol πto denote the ratio

of the circumference of a circle to its diameter is due to

British mathematician William Jones (1675–1749),

who first used it in his 1706 publication Synopsis pal-

mariorum matheseos. It is believed that he chose it

because πis the initial letter of the Greek word

περιϕ′

ερεια for “periphery.” Euler followed Jones’s

choice and popularized the use of this symbol in his

influential 1736 text Mechanica.

The number πhas captured the interest of many

mathematical enthusiasts. There are clubs across the

globe for those who can recite, from memory, the first

100 and even the first 1,000 digits of π. Some people

declare March 14 “pi-day,” and deem the time 1:59 of

that day significant. (This matches the decimal expan-

sion 3.14159…) There is a popular mnemonic for

memorizing the first 12 digits of π:

See. I have a rhyme assisting my feeble brain,

its tasks ofttimes resisting.

π

–

e