COMPLEX NUMBERS
into the study of
TRIGONOMETRY
,
leading to the famous formula that now bears his
name. He was also the first to describe and use the nor-
mal frequency curve in statistics.
Immigrating to England in 1685, De Moivre
worked as a private tutor in mathematics. He had
hoped to receive a faculty position in mathematics but,
as a foreigner, was never offered such an appointment.
He remained a private tutor all his life, despite the rep-
utation he had garnered as a capable and influential
scholar. He was elected a fellow of the R
OYAL
S
OCIETY
in 1697 and, in 1710, was appointed to a commission
set up by the society to adjudicate on the rival claims of
S
IR
I
SAAC
N
EWTON
and G
OTTFRIED
W
ILHELM
L
EIBNIZ
as the discoverers of
CALCULUS
.
De Moivre published two notable texts. The first,
Doctrine of Chance (1718), carefully examined the
underlying principles of
PROBABILITY
theory and soundly
developed fundamental notions such as “statistical inde-
pendence” and the “probability product law,” as well as
established foundations for applications to the theory of
annuities. The second, Miscellanea analytica (1730), suc-
cessfully identified the principles that later allowed him
to write down a formula for the
NORMAL DISTRIBUTION
,
a task that had stymied scholars before this time. This
second work also contained the mathematics necessary
to establish S
TIRLING
’
S FORMULA
.
It is said in all seriousness that De Moivre correctly
predicted the day of his own death. Noting that he was
sleeping 15 minutes longer each day, De Moivre sur-
mised that he would die on the day he would sleep for
24 hours. A simple mathematical calculation quickly
yielded the date, November 27, 1754. He did indeed
pass away on that day.
See also D
E
M
OIVRE
’
S FORMULA
.
De Moivre’s formula (De Moivre’s identity) In
1707 French mathematician A
BRAHAM
D
E
M
OIVRE
discovered the following formula, now called De
Moivre’s formula:
(cos θ+ isin θ)n= cos(nθ) + isin(nθ)
For positive integers nthe formula can be proved by
INDUCTION
, making use of the addition formulae for
the sine and cosine functions from
TRIGONOMETRY
. A
much simpler approach follows by making use of
E
ULER
’
S FORMULA
cosθ+isinθ= eiθand realizing that De
Moivre’s result is nothing more than a restatement of
the exponent rule:
(eiθ)n= einθ
This shows that the De Moivre’s formula actually holds
for any real value for n.
De Morgan, Augustus (1806–1871) British Alge-
bra, Logic Born on June 27, 1806, in Madura,
India, English citizen Augustus De Morgan is remem-
bered in mathematics for his considerable contribu-
tions to
FORMAL LOGIC
and
ALGEBRA
. In 1847 he
developed a formal system of symbolic manipulations
that encapsulated the principles of Aristotelian logic
and included the famous laws that now bear his
name. He is also remembered for properly defining
the process of mathematical
INDUCTION
and setting
this method of proof in a rigorous context.
De Morgan entered Trinity College, Cambridge, at
the age of 16, and, at the completion of his bachelor’s
degree, applied for the chair of mathematics at the newly
founded University College, London, at the young age of
21. Despite having no mathematical publications at the
time, he was awarded the position in 1827.
De Morgan became a prolific writer in mathemat-
ics. His first text, Elements of Arithmetic, published in
1830, was extremely popular and saw many improved
editions. He later wrote pieces on the topics of
CALCU
-
LUS
and algebra, and his 1849 text Trigonometry and
Double Algebra was also extremely influential. This
latter piece contained a useful geometric interpretation
of
COMPLEX NUMBERS
. De Morgan also wrote literally
hundreds of articles for the Penny Cyclopedia, a publi-
cation put out by the Society for the Diffusion of Use-
ful Knowledge. He presented many original pieces as
entries in this work. His precise definition of induction,
for instance, appears in an article in the 1838 edition.
Taking an active interest in the general dissemina-
tion of mathematical knowledge, De Morgan cofounded
in 1866 an academic society, the London Mathematical
Society, and became its first president. The society still
exists today and works to facilitate and promote mathe-
matical research.
As a collector of odd numerical facts, De Morgan
noted that being 43 in the year 1849 was a curious
122 De Moivre’s formula