mathematician NIELS HENRIK ABEL (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 x
5 – a = 0, for
instance, have solutions x =
5
√
–
a.) In 1831 French mathematician ÉVARISTE GALOIS (1811–32) completely classified those equations that can be so solved. The work
he conducted gave rise to a whole new branch of mathematics, today called GROUP THEORY.
quartile See PERCENTILE.
quaternions The product of two nonzero REAL NUMBERS 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 R
2
, complex-number multiplication provides a way for defining
a multiplication in R
2 that makes R
2 a 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, WILLIAM ROWAN HAMILTON (1805–65) wondered whether there was a suitable
multiplication for R
3 making 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 R
4 via the
invention of a new number system called the quaternions. These consist of the real numbers together with
three new symbols i, j, k, which, like the square root of
–1, satisfying the relations:
i
2 = j
2 = k
2 = –1
along with:
i × j = k
j× i = –k
j × k = i
k× j = –i
k × i = j
i× k = –j
A typical element of the quaternions appears as:
a + bi + cj + dk
where a,b,c, and d are real numbers. Multiplication of
two quaternions is defined by the DISTRIBUTIVE PROPERTY 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 multiplication does affect the answer. For example, i × j is different from j × i.
In the 1950s mathematicians proved that R
n is a
division algebra only for n equal to 1, 2, 4, or 8. These
are the real numbers, the complex numbers, the quaternions, 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 mathematician 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 doctorate 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
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 x
5 – a = 0, for
instance, have solutions x =
5
√
–
a.) In 1831 French mathematician ÉVARISTE GALOIS (1811–32) completely classified those equations that can be so solved. The work
he conducted gave rise to a whole new branch of mathematics, today called GROUP THEORY.
quartile See PERCENTILE.
quaternions The product of two nonzero REAL NUMBERS 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 R
2
, complex-number multiplication provides a way for defining
a multiplication in R
2 that makes R
2 a 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, WILLIAM ROWAN HAMILTON (1805–65) wondered whether there was a suitable
multiplication for R
3 making 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 R
4 via the
invention of a new number system called the quaternions. These consist of the real numbers together with
three new symbols i, j, k, which, like the square root of
–1, satisfying the relations:
i
2 = j
2 = k
2 = –1
along with:
i × j = k
j× i = –k
j × k = i
k× j = –i
k × i = j
i× k = –j
A typical element of the quaternions appears as:
a + bi + cj + dk
where a,b,c, and d are real numbers. Multiplication of
two quaternions is defined by the DISTRIBUTIVE PROPERTY 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 multiplication does affect the answer. For example, i × j is different from j × i.
In the 1950s mathematicians proved that R
n is a
division algebra only for n equal to 1, 2, 4, or 8. These
are the real numbers, the complex numbers, the quaternions, 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 mathematician 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 doctorate 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