much attention and stimulated general discussions on
the nature of free will versus social determinism. He
continued his work in analyzing crime and then
extended it to analyze mortality, human physique, and
social issues. This work produced much controversy
among social scientists of the time.
Quételet came to perceive the average value of a
human physical or mental quality, or of a general
societal characteristic, as a tangible and real “ideal”
that nature is trying to create, and he argued that the
fact that specific individuals rarely follow these
norms is to be expected as usual statistical deviation.
His analyses of social phenomena were certainly
mathematically sophisticated and gained him much
respect as a researcher. Nonetheless, many of his
ideas were controversial. In 1835 he published a
detailed account of his new social science in Sur
l’homme et le developpement de ses facultés, essai
d’une physique sociale (A treatise on man, and the
development of his faculties), which, among other
things, detailed the “average man.”
The Quételet index (QI) used today, also known
as the body mass index (BMI), is derived from
Quételet’s work. This index of obesity is defined by
the formula:
An individual with QI-value greater than 30 is considered obese.
Quételet died in Brussels, Belgium, on February 17,
1874. He had a profound effect on the development of
working statistics and the shaping of the study of social
science. Florence Nightingale considered his work of
the stature of a “new bible.”
See also HISTORY OF PROBABILITY AND STATISTICS
(essay).
quotient In NUMBER THEORY and ARITHMETIC, the
result of dividing one integer by another is called the
quotient. There might or might not be a remainder.
For example, 13 divided by 4 gives a quotient of 3
and a remainder of 1. In general, the quotient of an
integer a divided by b is an integer q such that a = bq
+ r, where r is a whole number with 0 ≤ r < q. The
quotient can be computed with the aid of the floor
function:
Outside of the arithmetic of integers, a quotient is
taken to mean simply the result of dividing a by b (and
so the issue of remainders is moot). For example, the
quotient of √
–
48 and 3 is
.
The quotient polynomial of two polynomials p(x)
and q(x) is a polynomial Q(x) such that p(x) =
Q(x)q(x) + R(x) for some polynomial R(x) of degree
smaller than that of q(x). For example, dividing p(x) =
x
4 + 2x
3 + 3x
2 + 5x + 5 by q(x) = x
2 + 2x + 1 gives a
quotient of Q(x) = x
2 + 1 and a remainder of R(x) =
x + 3:
x
4 + 2x
3 + 3x
2 + 5x + 5 = (x
2 + 1) (x
2 + 2x + 1) + (x + 3)
Such a computation can be accomplished by the process of LONG DIVISION.
In calculus, the QUOTIENT RULE provides a means
for differentiating the ratio of two functions.
See also EUCLIDEAN ALGORITHM; FLOOR/CEILING/
FRACTIONAL PART FUNCTIONS; MODULAR ARITHMETIC;
REMAINDER THEOREM; SYNTHETIC DIVISION.
quotient rule The DERIVATIVE of the quotient of two
functions is given by the quotient rule:
For example, we have
=
. The rule can be proved by making use of
both the PRODUCT RULE and the CHAIN RULE: simply
think of the quantity
as the product: f(x)·(g(x))
–1 .
Consequently:
d
dx
f x
g x
f x g x
f x g x
f x
g x
( )
( )
( ) ( )
( ) ( )
( )
( )
⎛
⎝
⎜
⎞
⎠
⎟ =
⋅ ( )
⎛
⎝
⎞
⎠
′
= ′ ⋅ ( ) + ⋅ ( )
⎛
⎝
⎞
⎠
′
−
−
−
1
1
1
f (x)
––
g(x)
x
x
x
x
cos
sin
− 2
d
dx
x
x
x x
x
x
sin
cos
sin
⎛
⎝
⎜
⎞
⎠
⎟ =
⋅ −
⋅1
2
d
dx
f x
g x
f x g x f x g x
g x
( )
( )
( ) ( ) ( ) ( )
( )
⎛
⎝
⎜
⎞
⎠
⎟ =
′ ⋅
−
⋅ ′
( )
2
48 3
48
3
2 309
÷ =
≈ .
q
a
b
=
⎢
⎣
⎢
⎥
⎦
⎥
QI =
(
)
=
×
(
)
weight in kilograms
height in meters
704
weight in pounds
height in inches
2
2
quotient rule 433
the nature of free will versus social determinism. He
continued his work in analyzing crime and then
extended it to analyze mortality, human physique, and
social issues. This work produced much controversy
among social scientists of the time.
Quételet came to perceive the average value of a
human physical or mental quality, or of a general
societal characteristic, as a tangible and real “ideal”
that nature is trying to create, and he argued that the
fact that specific individuals rarely follow these
norms is to be expected as usual statistical deviation.
His analyses of social phenomena were certainly
mathematically sophisticated and gained him much
respect as a researcher. Nonetheless, many of his
ideas were controversial. In 1835 he published a
detailed account of his new social science in Sur
l’homme et le developpement de ses facultés, essai
d’une physique sociale (A treatise on man, and the
development of his faculties), which, among other
things, detailed the “average man.”
The Quételet index (QI) used today, also known
as the body mass index (BMI), is derived from
Quételet’s work. This index of obesity is defined by
the formula:
An individual with QI-value greater than 30 is considered obese.
Quételet died in Brussels, Belgium, on February 17,
1874. He had a profound effect on the development of
working statistics and the shaping of the study of social
science. Florence Nightingale considered his work of
the stature of a “new bible.”
See also HISTORY OF PROBABILITY AND STATISTICS
(essay).
quotient In NUMBER THEORY and ARITHMETIC, the
result of dividing one integer by another is called the
quotient. There might or might not be a remainder.
For example, 13 divided by 4 gives a quotient of 3
and a remainder of 1. In general, the quotient of an
integer a divided by b is an integer q such that a = bq
+ r, where r is a whole number with 0 ≤ r < q. The
quotient can be computed with the aid of the floor
function:
Outside of the arithmetic of integers, a quotient is
taken to mean simply the result of dividing a by b (and
so the issue of remainders is moot). For example, the
quotient of √
–
48 and 3 is
.
The quotient polynomial of two polynomials p(x)
and q(x) is a polynomial Q(x) such that p(x) =
Q(x)q(x) + R(x) for some polynomial R(x) of degree
smaller than that of q(x). For example, dividing p(x) =
x
4 + 2x
3 + 3x
2 + 5x + 5 by q(x) = x
2 + 2x + 1 gives a
quotient of Q(x) = x
2 + 1 and a remainder of R(x) =
x + 3:
x
4 + 2x
3 + 3x
2 + 5x + 5 = (x
2 + 1) (x
2 + 2x + 1) + (x + 3)
Such a computation can be accomplished by the process of LONG DIVISION.
In calculus, the QUOTIENT RULE provides a means
for differentiating the ratio of two functions.
See also EUCLIDEAN ALGORITHM; FLOOR/CEILING/
FRACTIONAL PART FUNCTIONS; MODULAR ARITHMETIC;
REMAINDER THEOREM; SYNTHETIC DIVISION.
quotient rule The DERIVATIVE of the quotient of two
functions is given by the quotient rule:
For example, we have
=
. The rule can be proved by making use of
both the PRODUCT RULE and the CHAIN RULE: simply
think of the quantity
as the product: f(x)·(g(x))
–1 .
Consequently:
d
dx
f x
g x
f x g x
f x g x
f x
g x
( )
( )
( ) ( )
( ) ( )
( )
( )
⎛
⎝
⎜
⎞
⎠
⎟ =
⋅ ( )
⎛
⎝
⎞
⎠
′
= ′ ⋅ ( ) + ⋅ ( )
⎛
⎝
⎞
⎠
′
−
−
−
1
1
1
f (x)
––
g(x)
x
x
x
x
cos
sin
− 2
d
dx
x
x
x x
x
x
sin
cos
sin
⎛
⎝
⎜
⎞
⎠
⎟ =
⋅ −
⋅1
2
d
dx
f x
g x
f x g x f x g x
g x
( )
( )
( ) ( ) ( ) ( )
( )
⎛
⎝
⎜
⎞
⎠
⎟ =
′ ⋅
−
⋅ ′
( )
2
48 3
48
3
2 309
÷ =
≈ .
q
a
b
=
⎢
⎣
⎢
⎥
⎦
⎥
QI =
(
)
=
×
(
)
weight in kilograms
height in meters
704
weight in pounds
height in inches
2
2
quotient rule 433