prime 409

postulate Another name for

AXIOM

. It is customary

to call the axioms of E

UCLIDEAN GEOMETRY

and of

ARITHMETIC

postulates rather than axioms.

See also P

EANO

’

S POSTULATES

.

power series Any

SERIES

of the form = a0+

a1x+ a2x2+ a3x3+… where the coefficients anare con-

stant and xis a variable is called a power series. For

example, the T

AYLOR SERIES

of any function is a power

series, as is the

GEOMETRIC SERIES

1+ x+ x2+ x3+….

Whether or not a power series converges depends on

which value is chosen for the variable x. For example,

the geometric series converges if –1 < x< 1 and

diverges otherwise. At the very least, every power series

converges for x= 0.

Mathematicians have proved the following result:

Given a power series , precisely one of

the following is true:

1. The series converges only for x= 0.

2. The series converges (absolutely) for all val-

ues of x.

3. There is a positive real number Rsuch that

the series converges (absolutely) for all val-

ues –R< x< Rand diverges for |x| > R.

(The series might, or might not, converge

for each of the values x= Rand x= –R.)

The value Ris called the radius of convergence of the

power series. (If the series converges only for x= 0, one

usually says that the radius of convergence is zero. If

the series converges for all values of x, then the radius

of convergence is infinite.)

One often makes use of the ratio test from the study

of

CONVERGENT SERIES

to find the radius of conver-

gence of a power series. For example, the ratio test

shows that the series converges if

is less than 1, and diverges

if greater than 1. Thus the radius of convergence

for this power series is R= 1. (Going further, not-

ice that, for x= 1 the series is the

HARMONIC

SERIES

, which diverges, and for

it x= –1 it is the alternating harmonic series

, which converges. The power

series in question thus converges for –1 ≤x< 1.)

See also

ABSOLUTE CONVERGENCE

.

precision The total number of digits recorded while

taking a measurement is called the precision of the

measurement. For example, a recorded length of 3.650

meters is precise to four

SIGNIFICANT FIGURES

. (The

final digit zero indicates that the measurement was

indeed made to the nearest millimeter.) The number of

digits to the right of the decimal point is called the

accuracy of the measurement. Thus, for example, the

figure 3.650 is accurate to three decimal places.

See also

ERROR

.

premise A statement that is known or is assumed to

be true and on which a logical

ARGUMENT

is based is

called the premise of the argument. A premise could be

an

AXIOM

of a particular mathematical theory, or

merely an assumption taken to be true for the purposes

of discovering its consequences. For example, the

premise that “parallel lines do meet at a point” led

mathematicians to discover the realm of

PROJECTIVE

GEOMETRY

, a valid and fruitful new approach to under-

standing ordinary geometry.

prime A whole number possessing just two positive

factors is called a prime number, or simply a prime. For

example, 7 has only two positive factors, namely 1 and 7,

and so is prime. The number 24 has eight positive factors

and so is not prime, and the number 1 has only one fac-

tor and is not prime. The term composite is used to

describe numbers greater than 1 that are not prime.

(Medieval mathematician F

IBONACCI

(1170–1250) called

prime numbers “incomposite.”) It is vital that the num-

ber 1 be considered neither prime nor composite for the

FUNDAMENTAL THEOREM OF ARITHMETIC

to hold true.

The first 25 prime numbers are: 2, 3, 5, 7, 11, 13,

17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71,

73, 79, 83, 89, 97. These are all the primes smaller than