mathematician N
IELS
H
ENRIK
A
BEL
(1802–29). Thus—
although there is a quadratic formula for solving
degree-two equations, Cardano’s formula for solving
degree-three equations, and Ferrari’s and Descarte’s
methods for solving degree-four equations—there will
never be a general formula for solving all equations of
degree five or higher.
Of course some degree-five equations can be solved
algebraically. (Equations of the form x5– a= 0, for
instance, have solutions x= 5
√
–
a.) In 1831 French math-
ematician É
VARISTE
G
ALOIS
(1811–32) completely clas-
sified those equations that can be so solved. The work
he conducted gave rise to a whole new branch of math-
ematics, today called
GROUP THEORY
.
quartile See
PERCENTILE
.
quaternions The product of two nonzero
REAL NUM
-
BERS
is never zero, nor is the product of two nonzero
COMPLEX NUMBERS
. Both these sets of numbers are said
to be division algebras. Since each complex number
a+ib corresponds to a point (a,b) in the plane R2, com-
plex-number multiplication provides a way for defining
a multiplication in R2that makes R2a division algebra:
(a,b) ×(c,d) = (ac – bd, ad + bc)
(Notice that the “obvious” multiplication given by
(a,b) ×(c,d) = (ac,bd) is not satisfactory: (1,0) ×(0,4),
for example, gives the zero answer.)
In the early 19th century, W
ILLIAM
R
OWAN
H
AMIL
-
TON
(1805–65) wondered whether there was a suitable
multiplication for R3making it, too, a division algebra.
Despite his best efforts, he was never able to propose a
suitable definition.
Along the way, however, Hamilton was able to
provide a suitable multiplication rule for R4via the
invention of a new number system called the quater-
nions. These consist of the real numbers together with
three new symbols i, j, k, which, like the square root of
–1, satisfying the relations:
i2= j2= k2= –1
along with:
i×j= kj×i= –k
j×k= ik×j= –i
k×i= ji×k= –j
A typical element of the quaternions appears as:
a+ bi + cj + dk
where a,b,c, and dare real numbers. Multiplication of
two quaternions is defined by the
DISTRIBUTIVE PROP
-
ERTY
and the relations outlined above. For example,
(2 + i) ×(3 + 3j+ k) = 6 + 6j+ 2k+ 3i+ 3k– j
= 6 + 3i+ 5j+ 5k
Hamilton was able to show that the quaternions
are a division algebra. Notice that they do not satisfy
the
COMMUTATIVE PROPERTY
: the order of multiplica-
tion does affect the answer. For example, i×jis differ-
ent from j×i.
In the 1950s mathematicians proved that Rnis a
division algebra only for nequal to 1, 2, 4, or 8. These
are the real numbers, the complex numbers, the quater-
nions, and another system called the Cayley numbers
(also known as the octonions). These extended number
systems are sometimes called hypercomplex numbers.
Quételet, Lambert Adolphe Jacques (1796–1874)
Flemish Statistics, Astronomy Born on February 22,
1796, in Ghent, Belgium, Adolphe Quételet is often
referred to by mathematical historians as the father of
modern statistics. Although trained as a mathemati-
cian and astronomer, Quételet is remembered for his
pioneering work in collecting statistical data and
using it to test traditional views on issues of medicine
and criminology. His new field of “social mechanics”
profoundly influenced European thinking in the social
sciences.
Raised in Ghent, Belgium, Quételet received a doc-
torate in mathematics from the University of Ghent in
1819, having written a dissertation on the theory of
CONIC SECTIONS
. After teaching mathematics for four
years, he moved to Paris in 1823 to begin a study of
astronomy. During the course of this work he was
introduced to the discipline of
PROBABILITY
theory and
the particular statistical methods astronomers were using
to gain accurate measurements of physical phenomena.
He began to wonder whether the same techniques
apply to human affairs. To test this idea, he undertook
a study of data from government records to analyze the
numerical consistency of crimes. This work garnered
432 quartile