quotient rule 433

⎠

⎝

f(x)

––

g(x)

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 consid-

ered 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 adivided by bis an integer qsuch that a= bq

+ r, where ris 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 aby 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) =

x4+ 2x3+ 3x2+ 5x+ 5 by q(x) = x2+ 2x+ 1 gives a

quotient of Q(x) = x2+ 1 and a remainder of R(x) =

x+ 3:

x4+ 2x3+ 3x2+ 5x+ 5 = (x2+ 1) (x2+ 2x+ 1) + (x+ 3)

Such a computation can be accomplished by the pro-

cess of

LONG DIVISION

.

In calculus, the

QUOTIENT RULE

provides a means

for differentiating the ratio of two functions.

See also E

UCLIDEAN 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: