postulate Another name for AXIOM. It is customary
to call the axioms of EUCLIDEAN GEOMETRY and of
ARITHMETIC postulates rather than axioms.
See also PEANO’S POSTULATES.
power series Any SERIES of the form
= a 0 +
a 1 x + a 2 x
2 + a 3 x
3 +… where the coefficients a n are constant and x is a variable is called a power series. For
example, the TAYLOR SERIES of any function is a power
series, as is the GEOMETRIC SERIES 1 + x + x
2 + x
3 +….
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 values of x.
3. There is a positive real number R such that
the series converges (absolutely) for all values –R < x < R and diverges for |x| > R.
(The series might, or might not, converge
for each of the values x = R and x = –R.)
The value R is 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 convergence 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, notice 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 understanding 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 factor and is not prime. The term composite is used to
describe numbers greater than 1 that are not prime.
(Medieval mathematician FIBONACCI (1170–1250) called
prime numbers “incomposite.”) It is vital that the number 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
( )
−
= − + −
=
∞
∑
1
1
1
2
1
3
0
n
n
n
L
1
1
1
2
1
3
0
n
n
n
= + + +
=
∞
∑
L
lim
| | | |
n
n
n
x
x
→∞
=
+
=
1
lim
|
|
|
|
n
n
n
x
n
x
n
→∞
+
+
1
1
x
n
n
n=
∞
∑
0
a x
n
n
n=
∞
∑
0
a x
n
n
n=
∞
∑
0
prime 409
to call the axioms of EUCLIDEAN GEOMETRY and of
ARITHMETIC postulates rather than axioms.
See also PEANO’S POSTULATES.
power series Any SERIES of the form
= a 0 +
a 1 x + a 2 x
2 + a 3 x
3 +… where the coefficients a n are constant and x is a variable is called a power series. For
example, the TAYLOR SERIES of any function is a power
series, as is the GEOMETRIC SERIES 1 + x + x
2 + x
3 +….
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 values of x.
3. There is a positive real number R such that
the series converges (absolutely) for all values –R < x < R and diverges for |x| > R.
(The series might, or might not, converge
for each of the values x = R and x = –R.)
The value R is 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 convergence 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, notice 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 understanding 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 factor and is not prime. The term composite is used to
describe numbers greater than 1 that are not prime.
(Medieval mathematician FIBONACCI (1170–1250) called
prime numbers “incomposite.”) It is vital that the number 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
( )
−
= − + −
=
∞
∑
1
1
1
2
1
3
0
n
n
n
L
1
1
1
2
1
3
0
n
n
n
= + + +
=
∞
∑
L
lim
| | | |
n
n
n
x
x
→∞
=
+
=
1
lim
|
|
|
|
n
n
n
x
n
x
n
→∞
+
+
1
1
x
n
n
n=
∞
∑
0
a x
n
n
n=
∞
∑
0
a x
n
n
n=
∞
∑
0
prime 409