1

–

q

p

–

q

p

–

q

least squares method 305

false, and undecided. A further generalization, called

FUZZY LOGIC

, treats truth as a continuous quantity

with values ranging from 0 (utterly false) to 1 (utterly

true). As an example, in this theory the sentence:

This sentence is false.

is assigned a truth value of of 1/2, and is deemed half

true and half false. (One arrives at this value as fol-

lows: first note that if a statement pis assigned a truth-

value v, which is either 0 or 1, then the statement

¬

p

has opposite value 1 – v, 1 or 0. We assume that this

remains true even if vis of fractional value. If prepre-

sents the sentence: “This sentence is false,” then its

truth establishes its falsehood, p→

¬

p, and its false-

hood its truth,

¬

p→p. The truth-value vof the state-

ment oscillates between the values vand 1 – v. The

only stable value for voccurs when v= 1 – v, yielding

the appropriate truth-value v= 1/2.)

See also

ARGUMENT

;

SELF

-

REFERENCE

.

leading coefficient For a

POLYNOMIAL

p(x) = anxn+

an–1xn–1 +…+ a1x+ a0, the coefficient anof the highest

power of the variable is called the leading coefficient of

the polynomial. For example, the leading coefficient of

the polynomial 2x3– 3x+ 6 is 2, and that of x+ 3 is 1. A

polynomial is called monic if its leading coefficient is 1.

In solving a polynomial equation anxn+ an–1xn–1

+…+ a1x+ a0= 0, it is often convenient to assume that

the polynomial in question is monic. One achieves this

by dividing the equation through by an. Solving the

equation 2x4– 8x3+ 2x– 6 = 0, for instance, is equiva-

lent to solving x4– 4x3+ x– 3 = 0.

If all the coefficients of a monic polynomial are

integers, then any rational

ROOT

to the polynomial

must itself be an integer. For example, if the fraction

x= (written in reduced form) were a solution to the

polynomial equation x4– 4x3+ x– 3 = 0, then, substi-

tuting in this value for xand multiplying through by

q4yields:

p4– 4p3q+ pq3– 3q4= 0

This shows that p4is a multiple of q. Since pand q

share no common factors, this is only possible if q

equals 1. Thus the root x= = pis an integer.

A similar argument shows that if the

CONSTANT

term a0of a polynomial is 1, then any rational root of

the polynomial (if it has one) must be a fraction of the

form x= . Consequently, any monic polynomial with

a constant term of 1 can possess at most one rational

(or integer) root, namely, x= 1. This proves, for exam-

ple, that x= 1 is the only rational root of the equation:

x7– x5+ 2x2– 3x+ 1 = 0

(One checks that x= 1 is indeed a solution to this

equation.)

least common multiple A number that is a multiple

of two or more other numbers is called a

COMMON

MULTIPLE

of those numbers. The smallest common mul-

tiple is called their least common multiple, written as

“lcm.” For example, the least common multiple of 10,

12, and 15 is 60. There is no smaller number that is

evenly divisible by each of these numbers. We have

lcm(10, 12, 15) = 60.

The least common multiple of a set of integers can

be found by splitting each number into prime factors.

For example, to find the lcm of 180 and 378, write:

180 = 2 ×2 ×3 ×3 ×5

378 = 2 ×3 ×3 ×3 ×7

The lcm is then found by multiplying the prime factors

together, taking each the maximum number of times it

appears in any of the numbers. In our case: lcm(180,

378) = 2 ×2 ×3 ×3 ×3 ×5 ×7 = 3,780. This method

shows that any common multiple of a collection of

integers is a multiple of the least common multiple. It

also shows that, for two positive integers aand b:

where “gcd” denotes the

GREATEST COMMON FACTOR

(divisor) of the two numbers. The analogous relation-

ship for three or more integers, however, does not hold

in general.

See also

FUNDAMENTAL THEOREM OF ARITHMETIC

.

least squares method If a

SCATTER DIAGRAM

indi-

cates a linear correlation between the two variables of