In modern times the subject of algebra has been
widened to include ABSTRACT ALGEBRA, GROUP THEORY, and the study of alternative number systems
such as MODULAR ARITHMETIC. BOOLEAN ALGEBRA
looks at the algebra of logical inferences, matrix algebra the arithmetic of MATRIX operations, and vector
algebra the mechanics of VECTOR operations and
VECTOR SPACEs.
An algebraic structure is any set equipped with one
or more operations (usually BINARY OPERATIONs) satisfying a list of specified rules. For example, any group,
RING, FIELD, or vector space is an algebraic structure. In
advanced mathematics, a vector space that is also a
field is called an “algebra.”
See also BRACKETS; COMMUTATIVE PROPERTY; DISTRIBUTIVE PROPERTY; EXPANDING BRACKETS; FUNDAMENTAL THEOREM OF ALGEBRA; HISTORY OF EQUATIONS
AND ALGEBRA (essay); ORDER OF OPERATION.
algebraic number A number is called algebraic if it
is the root of a POLYNOMIAL with integer coefficients.
For example, (1/2) (5 + √
—
13) is algebraic since it is a
solution to the equation x
2 – 5x + 3 = 0. All RATIONAL
NUMBERS are algebraic (since a fraction a/b is the solution to the equation bx – a = 0), and all square, cube,
and higher roots of integers are algebraic (since
n
√
—
a is a
solution to x
n – a = 0).
At first thought it seems that all numbers are algebraic, but this is not the case. In 1844 French mathematician JOSEPH LIOUVILLE made the surprising
discovery that the following number, today called
“Liouville’s constant,” cannot be a solution to any integer polynomial equation:
Numbers that are not algebraic are called “transcendental.”
In 1873 French mathematician Charles Hermite
(1822–1901) proved that the number e is transcendental, and, nine years later in 1882 German mathematician
CARL LOUIS FERDINAND VON LINDEMANN established
that π is transcendental. In 1935 Russian mathematician
Aleksandr Gelfond (1906–68) proved that any number
of the form a
b is transcendental if a and b are both algebraic, with a different from 0 or 1, and b irrational.
(Thus, for example, 2 √
–
3 is transcendental.)
The German mathematician GEORG CANTOR
(1845–1918) showed that the set of algebraic numbers
is COUNTABLE. As the set of real numbers is uncountable, this means that most numbers are transcendental.
The probability that a real number chosen at random is
algebraic is zero. Although it was proven in 1929 that
e
π is transcendental, no one to this day knows whether
or not π
π is algebraic.
In analogy with algebraic numbers, a FUNCTION
y = f(x) is called “algebraic” if it can be defined by a
relation of the form
p n (x)y
n + p n – 1(x)y
n – 1+…+p 1 (x)y + p 0 (x) = 0
where the functions p i (x) are polynomials in x. For
example, the function y = √
—
x is an algebraic function,
since it is defined by the equation y
2 – x = 0. A transcendental function is a function that is not algebraic.
Mathematicians have shown that trigonometric, logarithmic, and exponential functions are transcendental.
See also CARDINALITY.
algorithm An algorithm is a specific set of instructions for carrying out a procedure or solving a mathematical problem. Synonyms include “method,”
“procedure,” and “technique.” One example of an
algorithm is the common method of LONG DIVISION.
Another is the EUCLIDEAN ALGORITHM for finding the
GREATEST COMMON DIVISOR of two positive integers.
The word algorithm is a distortion of “al-Khw – arizm – ı,”
the name of a Persian mathematician (ca. 820) who
wrote an influential text on algebraic methods.
See also BASE OF A NUMBER SYSTEM; MUHAMMAD
IBN M –
US
– A AL-KHW – ARIZM – ı.
alternating series A SERIES whose terms are alternately
positive and negative is called an alternating series. For
example, the GREGORY SERIES
is
an alternating series, as is the (divergent) series: 1 – 1 + 1
– 1 + 1 – 1 +… Alternating series have the form
, with each a i positive
number.
( )
−
= − + − +
−
=
∞
∑ 1
1
1
2
3
4
1
n
n
n
a a a a a K
1
1
3
1
5
1
7
4
− + − + =
K
π
L
n
n
=
=
=
∞
∑
1
10
0 11000100000000000000000100
1
!
.
...
alternating series 11
widened to include ABSTRACT ALGEBRA, GROUP THEORY, and the study of alternative number systems
such as MODULAR ARITHMETIC. BOOLEAN ALGEBRA
looks at the algebra of logical inferences, matrix algebra the arithmetic of MATRIX operations, and vector
algebra the mechanics of VECTOR operations and
VECTOR SPACEs.
An algebraic structure is any set equipped with one
or more operations (usually BINARY OPERATIONs) satisfying a list of specified rules. For example, any group,
RING, FIELD, or vector space is an algebraic structure. In
advanced mathematics, a vector space that is also a
field is called an “algebra.”
See also BRACKETS; COMMUTATIVE PROPERTY; DISTRIBUTIVE PROPERTY; EXPANDING BRACKETS; FUNDAMENTAL THEOREM OF ALGEBRA; HISTORY OF EQUATIONS
AND ALGEBRA (essay); ORDER OF OPERATION.
algebraic number A number is called algebraic if it
is the root of a POLYNOMIAL with integer coefficients.
For example, (1/2) (5 + √
—
13) is algebraic since it is a
solution to the equation x
2 – 5x + 3 = 0. All RATIONAL
NUMBERS are algebraic (since a fraction a/b is the solution to the equation bx – a = 0), and all square, cube,
and higher roots of integers are algebraic (since
n
√
—
a is a
solution to x
n – a = 0).
At first thought it seems that all numbers are algebraic, but this is not the case. In 1844 French mathematician JOSEPH LIOUVILLE made the surprising
discovery that the following number, today called
“Liouville’s constant,” cannot be a solution to any integer polynomial equation:
Numbers that are not algebraic are called “transcendental.”
In 1873 French mathematician Charles Hermite
(1822–1901) proved that the number e is transcendental, and, nine years later in 1882 German mathematician
CARL LOUIS FERDINAND VON LINDEMANN established
that π is transcendental. In 1935 Russian mathematician
Aleksandr Gelfond (1906–68) proved that any number
of the form a
b is transcendental if a and b are both algebraic, with a different from 0 or 1, and b irrational.
(Thus, for example, 2 √
–
3 is transcendental.)
The German mathematician GEORG CANTOR
(1845–1918) showed that the set of algebraic numbers
is COUNTABLE. As the set of real numbers is uncountable, this means that most numbers are transcendental.
The probability that a real number chosen at random is
algebraic is zero. Although it was proven in 1929 that
e
π is transcendental, no one to this day knows whether
or not π
π is algebraic.
In analogy with algebraic numbers, a FUNCTION
y = f(x) is called “algebraic” if it can be defined by a
relation of the form
p n (x)y
n + p n – 1(x)y
n – 1+…+p 1 (x)y + p 0 (x) = 0
where the functions p i (x) are polynomials in x. For
example, the function y = √
—
x is an algebraic function,
since it is defined by the equation y
2 – x = 0. A transcendental function is a function that is not algebraic.
Mathematicians have shown that trigonometric, logarithmic, and exponential functions are transcendental.
See also CARDINALITY.
algorithm An algorithm is a specific set of instructions for carrying out a procedure or solving a mathematical problem. Synonyms include “method,”
“procedure,” and “technique.” One example of an
algorithm is the common method of LONG DIVISION.
Another is the EUCLIDEAN ALGORITHM for finding the
GREATEST COMMON DIVISOR of two positive integers.
The word algorithm is a distortion of “al-Khw – arizm – ı,”
the name of a Persian mathematician (ca. 820) who
wrote an influential text on algebraic methods.
See also BASE OF A NUMBER SYSTEM; MUHAMMAD
IBN M –
US
– A AL-KHW – ARIZM – ı.
alternating series A SERIES whose terms are alternately
positive and negative is called an alternating series. For
example, the GREGORY SERIES
is
an alternating series, as is the (divergent) series: 1 – 1 + 1
– 1 + 1 – 1 +… Alternating series have the form
, with each a i positive
number.
( )
−
= − + − +
−
=
∞
∑ 1
1
1
2
3
4
1
n
n
n
a a a a a K
1
1
3
1
5
1
7
4
− + − + =
K
π
L
n
n
=
=
=
∞
∑
1
10
0 11000100000000000000000100
1
!
.
...
alternating series 11