Algebra is the branch of
mathematics concerning the study of the rules of
operations and the things
which can be constructed from them, including
terms,
polynomials,
equations
and
algebraic structures.
Together with
geometry,
analysis,
topology,
combinatorics, and
number theory, algebra is one of the main
branches of
pure mathematics.
Elementary algebra is often part
of the curriculum in
secondary
education and introduces the concept of
variables representing
numbers. Statements based on these variables are
manipulated using the rules of operations that apply to numbers,
such as
addition. This can be done for a
variety of reasons, including
equation
solving.
Algebra is much broader than elementary algebra and studies what
happens when different rules of operations are used and when
operations are devised for things other than numbers. Addition and
multiplication can be generalised and
their precise definitions lead to
structures such as
groups,
rings and
fields.
History
While the word
algebra comes from the
Arabic language (al-jabr,
الجبر literally,
restoration), its origins can be traced to ancient
Indian mathematics. Muhammad ibn
Musa al-Khwarizmi learned the technique of Indian mathematics and
introduced it to the world through his famous book on arithmetic
text, the
Kitab al-jam’wal tafriq bi hisab al-Hindi ("Book
on Addition and Subtraction after the Method of the Indians"),
around 780-850 A.D. The first use of algebraic transformation was
descibed by
Brahmagupta in his book
Brahmasphutasiddhanta, where
first proposed solution of
Linear
Algebra and
Quadratic
Equation.
Algebra is also linked to the ancient
Babylonians, who developed an
advanced
arithmetical system with which
they were able to do calculations in an algorithmic fashion. The
Babylonians developed formulas to calculate solutions for problems
typically solved today by using
linear
equations,
quadratic
equations, and
indeterminate
linear equation. By contrast, most
Egyptians of this era
Greek and
Chinese mathematicians in the
first millennium BC, usually solved such
equations by
geometric methods, such as
those described in the
Rhind Mathematical Papyrus,
Euclid's Elements, and
The Nine
Chapters on the Mathematical Art. The geometric work of
the Greeks, typified in the
Elements, provided the
framework for generalizing formulae beyond the solution of
particular problems into more general systems of stating and
solving equations, though this would not be realized until the
medieval Muslim
mathematicians.
The
Hellenistic
mathematicians
Hero of Alexandria
and
Diophantus as well as
Indian mathematicians such as
Brahmagupta continued the traditions of Egypt
and Babylon, though Diophantus'
Arithmetica and Brahmagupta's
Brahmasphutasiddhanta are on a
higher level. Later, Arab and Muslim mathematicians developed
algebraic methods to a much higher degree of sophistication.
Although Diophantus and the Babylonians used mostly special ad hoc
methods to solve equations,
Al-Khowarazmi was the first to solve equations
using general methods. He solved the linear indeterminate
equations, quadratic equations, second order indeterminate
equations and equations with multiple variable.
The word "algebra" is named after the
Arabic
word "
al-jabr , الجبر" from the title of the book ,
meaning
The book of Summary Concerning Calculating by
Transposition and Reduction, a book written by the
Islamic Persian mathematician,
(considered the "father of algebra"), in 820. The word
Al-Jabr means
"reunion".The Hellenistic
mathematician
Diophantus has
traditionally been known as the "father of algebra" but in more
recent times there is much debate over whether al-Khwarizmi, who
founded the discipline of
al-jabr, deserves that title
instead. Those who support Diophantus point to the fact that the
algebra found in
Al-Jabr is slightly more elementary than
the algebra found in
Arithmetica and that
Arithmetica is syncopated while
Al-Jabr is fully
rhetorical. Those who support Al-Khwarizmi point to the fact that
he introduced the methods of "
reduction" and "balancing" (the
transposition of subtracted terms to the other side of an equation,
that is, the cancellation of
like terms
on opposite sides of the equation) which the term
al-jabr
originally referred to, "It is not certain just what the terms
al-jabr and
muqabalah mean, but the usual
interpretation is similar to that implied in the translation above.
The word
al-jabr presumably meant something like
"restoration" or "completion" and seems to refer to the
transposition of subtracted terms to the other side of an equation;
the word
muqabalah is said to refer to "reduction" or
"balancing" - that is, the cancellation of like terms on opposite
sides of the equation." and that he gave an exhaustive explanation
of solving quadratic equations, supported by geometric proofs,
while treating algebra as an independent discipline in its own
right. His algebra was also no longer concerned "with a series of
problems to be resolved, but an
exposition which starts with primitive
terms in which the combinations must give all possible prototypes
for equations, which henceforward explicitly constitute the true
object of study." He also studied an equation for its own sake and
"in a generic manner, insofar as it does not simply emerge in the
course of solving a problem, but is specifically called on to
define an infinite class of problems."
The Persian mathematician
Omar Khayyam
is credited with identifying the foundations of
algebraic geometry and found the general
geometric solution of the
cubic
equation. Another Persian mathematician,
Sharaf al-Dīn al-Tūsī,
found algebraic and numerical solutions to various cases of cubic
equations. He also developed the concept of a
function. The Indian mathematicians
Mahavira and
Bhaskara II, the Persian mathematician
Al-Karaji, "Abu'l Wefa was a capable algebraist as
well as a trigonometer. [...] His successor al-Karkhi evidently
used this translation to become an Arabic disciple of Diophantus -
but without Diophantine analysis! [...] In particular, to al-Karkhi
is attributed the first numerical solution of equations of the form
ax
^{2n} + bx
^{n} = c (only equations with positive
roots were considered)," and the Chinese mathematician
Zhu Shijie, solved various cases of cubic,
quartic,
quintic and higher-order
polynomial equations using numerical methods. In
1637
Rene Descartes published
La Géométrie, inventing
analytic geometry and introducing
modern algebraic notation.
Another key event in the further development of algebra was the
general algebraic solution of the cubic and quartic equations,
developed in the mid-16th century. The idea of a
determinant was developed by
Japanese mathematician Kowa Seki in the 17th century, followed by
Gottfried Leibniz ten years later,
for the purpose of solving systems of simultaneous linear equations
using
matrices.
Gabriel Cramer also did some work on matrices
and determinants in the 18th century.
Abstract algebra was developed in the 19th
century, initially focusing on what is now called
Galois theory, and on
constructibility issues.
Classification
Algebra may be divided roughly into the following categories:
In some directions of advanced study, axiomatic algebraic systems
such as groups, rings, fields, and algebras over a field are
investigated in the presence of a
geometric
structure (a
metric or a
topology) which is compatible with the
algebraic structure. The list includes a number of areas of
functional analysis:
Elementary algebra
Elementary algebra is the most basic form of
algebra. It is taught to students who are presumed to have no
knowledge of
mathematics beyond the
basic principles of
arithmetic. In
arithmetic, only
numbers and their
arithmetical operations (such as +, −, ×, ÷) occur. In algebra,
numbers are often denoted by symbols (such as
a,
x, or
y). This is useful because:
- It allows the general formulation of arithmetical laws (such as
a + b = b + a for all
a and b), and thus is the first step to a
systematic exploration of the properties of the real number system.
- It allows the reference to "unknown" numbers, the formulation
of equations and the study of how to solve
these (for instance, "Find a number x such that
3x + 1 = 10").
- It allows the formulation of function relationships (such as "If
you sell x tickets, then your profit will be 3x −
10 dollars, or f(x) = 3x − 10, where
f is the function, and x is the number to which
the function is applied.").
Polynomials
A
polynomial is an
expression that is constructed from
one or more
variables and
constants, using only the operations of addition, subtraction, and
multiplication (where repeated multiplication of the same variable
is standardly denoted as exponentiation with a constant
non-negative whole number exponent). For example,
x^{2} + 2
x − 3 is a polynomial in the
single variable
x.
An important class of problems in algebra is
factorization of polynomials, that is,
expressing a given polynomial as a product of other polynomials.
The example polynomial above can be factored as (
x −
1)(
x + 3). A related class of problems is finding
algebraic expressions for the
root of a polynomial in a single
variable.
Abstract algebra
Abstract algebra extends the familiar concepts
found in elementary algebra and
arithmetic of
numbers to
more general concepts.
Sets: Rather than
just considering the different types of
numbers, abstract algebra deals with the more general
concept of
sets: a collection of all objects (called
elements) selected by
property, specific for the set. All collections of the familiar
types of numbers are sets. Other examples of sets include the set
of all two-by-two
matrices, the
set of all second-degree
polynomials
(
ax^{2} +
bx +
c), the set of
all two dimensional
vectors in
the plane, and the various
finite
groups such as the
cyclic groups
which are the group of integers
modulo n.
Set theory is a branch of
logic and not technically a branch of algebra.
Binary operations:
The notion of
addition (+) is abstracted to
give a
binary operation, ∗ say. The notion of binary
operation is meaningless without the set on which the operation is
defined. For two elements
a and
b in a set
S,
a ∗
b is another element in the set;
this condition is called
closure.
Addition (+),
subtraction (-),
multiplication (×), and
division (÷) can be binary operations
when defined on different sets, as is addition and multiplication
of matrices, vectors, and polynomials.
Identity elements:
The numbers zero and one are abstracted to give the notion of an
identity element for an operation. Zero is the identity
element for addition and one is the identity element for
multiplication. For a general binary operator ∗ the identity
element
e must satisfy
a ∗
e =
a and
e ∗
a =
a. This holds for
addition as
a + 0 =
a and 0 +
a =
a and multiplication
a × 1 =
a and 1 ×
a =
a. Not all set and operator combinations have
an identity element; for example, the positive natural numbers (1,
2, 3, ...) have no identity element for addition.
Inverse elements:
The negative numbers give rise to the concept of
inverse
elements. For addition, the inverse of
a is
−
a, and for multiplication the inverse is 1/
a. A
general inverse element
a^{−1} must satisfy the
property that
a ∗
a^{−1} =
e and
a^{−1} ∗
a =
e.
Associativity:
Addition of integers has a property called associativity. That is,
the grouping of the numbers to be added does not affect the sum.
For example: In general, this becomes (
a ∗
b) ∗
c =
a ∗ (
b ∗
c). This property
is shared by most binary operations, but not subtraction or
division or
octonion
multiplication.
Commutativity: Addition of
integers also has a property called commutativity. That is, the
order of the numbers to be added does not affect the sum. For
example: 2+3=3+2. In general, this becomes
a ∗
b
=
b ∗
a. Only some binary operations have this
property. It holds for the integers with addition and
multiplication, but it does not hold for
matrix multiplication or
quaternion multiplication
.
Groups – structures of a set with a single binary
operation
Combining the above concepts gives one of the most important
structures in mathematics: a
group. A group is a combination
of a set
S and a single
binary
operation ∗, defined in any way you choose, but with the
following properties:
- An identity element e exists, such that for every
member a of S, e ∗ a and
a ∗ e are both identical to a.
- Every element has an inverse: for every member a of
S, there exists a member a^{−1} such that
a ∗ a^{−1} and a^{−1} ∗
a are both identical to the identity element.
- The operation is associative: if a, b and
c are members of S, then (a ∗
b) ∗ c is identical to a ∗ (b ∗
c).
If a group is also
commutative—that
is, for any two members
a and
b of
S,
a ∗
b is identical to
b ∗
a—then the group is said to be
Abelian.
For example, the set of integers under the operation of addition is
a group. In this group, the identity element is 0 and the inverse
of any element
a is its negation, −
a. The
associativity requirement is met, because for any integers
a,
b and
c, (
a +
b) +
c =
a + (
b +
c)
The nonzero
rational numbers form a
group under multiplication. Here, the identity element is 1, since
1 ×
a =
a × 1 =
a for any rational
number
a. The inverse of
a is 1/
a, since
a × 1/
a = 1.
The integers under the multiplication operation, however, do not
form a group. This is because, in general, the multiplicative
inverse of an integer is not an integer. For example, 4 is an
integer, but its multiplicative inverse is ¼, which is not an
integer.
The theory of groups is studied in
group
theory. A major result in this theory is the
classification of finite
simple groups, mostly published between about 1955 and 1983,
which is thought to classify all of the
finite simple groups
into roughly 30 basic types.
Semigroups,
quasigroups, and
monoids
are structures similar to groups, but more general. They comprise a
set and a closed binary operation, but do not necessarily satisfy
the other conditions. A
semigroup has an
associative binary operation, but might not have an
identity element. A
monoid is a semigroup
which does have an identity but might not have an inverse for every
element. A
quasigroup satisfies a
requirement that any element can be turned into any other by a
unique pre- or post-operation; however the binary operation might
not be associative.
All groups are monoids, and all monoids are semigroups.
Rings and fields—structures of a set with two particular binary
operations, (+) and (×)
Groups just have one binary operation. To fully explain the
behaviour of the different types of numbers, structures with two
operators need to be studied. The most important of these are
rings, and
fields.
Distributivity
generalised the
distributive law for numbers, and
specifies the order in which the operators should be applied,
(called the
precedence). For the
integers and and × is said to be
distributive over
+.
A
ring has two
binary operations (+) and (×), with × distributive over +. Under
the first operator (+) it forms an
Abelian group. Under
the second operator (×) it is associative, but it does not need to
have identity, or inverse, so division is not allowed. The additive
(+) identity element is written as 0 and the additive inverse of
a is written as −
a.
The integers are an example of a ring. The integers have additional
properties which make it an
integral domain.
A
field is a
ring with the additional property that all the elements
excluding 0 form an
Abelian group under ×. The
multiplicative (×) identity is written as 1 and the multiplicative
inverse of
a is written as
a^{−1}.
The rational numbers, the real numbers and the complex numbers are
all examples of fields.
Objects called algebras
The word
algebra is also used for various
algebraic structures:
See also
Notes
- http://www.brusselsjournal.com/node/4107/print
- A History of Mathematics: An Introduction (2nd Edition)
(Paperback) Victor J katz Addison Wesley; 2 edition (March 6,
1998)
- Struik, Dirk J. (1987). A Concise History of
Mathematics. New York: Dover Publications.
- Diophantus, Father of Algebra
- History of Algebra
- Or rather restoration, according to RH Webster's 2nd
ed.
- Carl B. Boyer, A History of Mathematics, Second
Edition (Wiley, 1991), pages 178, 181
- Carl B. Boyer, A History of Mathematics, Second
Edition (Wiley, 1991), page 228
- "The six cases of equations given above exhaust all
possibilities for linear and quadratic equations having positive
root. So systematic and exhaustive was al-Khwarizmi's exposition
that his readers must have had little difficulty in mastering the
solutions."
- Gandz and Saloman (1936), The sources of al-Khwarizmi's
algebra, Osiris i, p. 263–277: "In a sense, Khwarizmi is more
entitled to be called "the father of algebra" than Diophantus
because Khwarizmi is the first to teach algebra in an elementary
form and for its own sake, Diophantus is primarily concerned with
the theory of numbers".
References
- Donald R. Hill, Islamic Science and Engineering
(Edinburgh University Press, 1994).
- Ziauddin Sardar, Jerry Ravetz, and Borin Van Loon,
Introducing Mathematics (Totem Books, 1999).
- George Gheverghese Joseph, The Crest of the Peacock:
Non-European Roots of Mathematics (Penguin Books, 2000).
- John J
O'Connor and Edmund F Robertson, MacTutor History of
Mathematics archive (University of St Andrews, 2005).
- I.N. Herstein: Topics in Algebra. ISBN
0-471-02371-X
- R.B.J.T. Allenby: Rings, Fields and Groups. ISBN
0-340-54440-6
- L. Euler:
Elements of Algebra, ISBN 978-1-89961-873-6
- Isaac Asimov Realm of Algebra (Houghton Mifflin),
1961
External links