Shannon, Claude Elwood 461

1

––

239

1

–

5

x7

––

3

x5

––

3

x3

––

3

stunned the mathematical community with his con-

struction of a simple paradox, today called R

USSELL

’

S

PARADOX

, that shows that our naïve understanding of

the notion of set is fundamentally flawed. Although

Cantor believed that set theory is the foundation on

which all of mathematics is built, it became clear to

mathematicians that the concept of a set and what it

means to be an “element of” must remain as unde-

fined terms. In the decades that followed, mathemati-

cians such as E

RNST

F

RIEDRICH

F

ERDINAND

Z

ERMELO

(1871–1953) attempted to develop an axiomatic the-

ory of sets (based on undefined terms) that success-

fully avoids Russell’s paradox. To this day, not all

mathematicians agree that this goal has yet been

achieved.

See also

ARGUMENT

;

AXIOM OF CHOICE

;

CARDINAL

-

ITY

; C

ARTESIAN PRODUCT

;

DIFFERENCE

;

FINITE

.

seven bridges of Königsberg problem See

GRAPH

THEORY

.

sexagesimal See

BASE OF A NUMBER SYSTEM

.

Shanks, William (1812–1882) British Computation

Born on January 25, 1812, in Northumberland, Eng-

land, William Shanks is remembered for extraordinar-

ily accurate computations of mathematical constants,

all done by hand. Shanks evaluated the natural

LOGA

-

RITHM

s of the numbers 2, 3, 5, and 10, each to 137

decimal places, published a table of all the prime num-

bers up to 60,000, and evaluated all the powers of two

of the form 212n+1 for nfrom 1 to 60. He also evalu-

ated eand E

ULER

’

S CONSTANT

γto a large number of

decimal places, but he is best remembered for his com-

putation of πto many hundreds of decimal places. He

published this result in 1873.

Little is known of Shanks’s life. His methods of

computation were essentially nothing more than a

matter of patience and

BRUTE FORCE

. For example,

Shanks used the following identity to compute 707

digits of π:

using G

REGORY SERIES

tan–1(x) = x– + –

+…. By substituting in the values and , one can

begin computing the digits of π.

In 1946 English mathematician D. F. Ferguson

used the alternate identity to also compute π:

He noted that Shanks had made an error in the 528th

place and went on to correctly compute πto 808 deci-

mal places.

Shanks died in Houghton-le-Spring, England, in

1882. The exact date of death is not known.

See also

E

;

PI

.

Shannon, Claude Elwood (1916–2001) American

Information theory Born on April 30, 1916, in

Michigan, Claude Elwood Shannon is remembered as

the founder of the field of

INFORMATION THEORY

. His

seminal article, “The Mathematical Theory of Commu-

nication,” published with Warren Weaver in 1949, laid

down the principles of communication science and

introduced the revolutionary idea that information

(pictures, words, sounds) could successfully be trans-

mitted through a wire as a sequence of 0s and 1s. (Up

to then it was thought that it would be necessary to

transmit electromagnetic waves through wires to

accomplish this feat.) This paper introduced the term

bit for the first time. Shannon also developed a mathe-

matical means for measuring the information content

of a message.

Shannon was awarded an undergraduate degree in

mathematics and electrical engineering from the Uni-

versity of Michigan in 1936. He completed doctoral

work at the Massachusetts Institute of Technology in

1940 and worked on the construction of an early type

of mechanical computer. He demonstrated, for the first

time, that it is possible to combine B

OOLEAN ALGEBRA

with electrical relays and switching circuits to create a

machine that can “do” mathematical logic.

In 1941 Shannon joined AT&T Bell Telephones in

New Jersey as a research mathematician and published

his famous work 7 years later. This piece also included

the mathematical basis of error detection: by adding