{
{
Short
description|Area
of
mathematics
}
}
{
{
About||the
kind
of
algebraic
structure|Algebra
over
a
field|other
uses
}
}
{
{
Pp-move
}
}
{
{
Pp-semi-indef
}
}
[
[
File
:
Quadratic
formula.svg|thumb|The
[
[
quadratic
formula
]
]
expresses
the
solution
of
the
equation
{
{
math|1=
''
ax
''
<
sup
>
2
<
/sup
>
+
``
bx
''
+
``
c
''
=
0
}
}
,
where
{
{
mvar|a
}
}
is
not
zero
,
in
terms
of
its
coefficients
{
{
math|
''
a
''
,
``
b
''
}
}
and
{
{
mvar|c
}
}
.|249x249px
]
]
'
''
Algebra
''
'
(
{
{
Lang-ar|الجبر
}
}
{
{
Transliteration|ar|Al-Jabr
}
}
,
{
{
Translation|'reunion
of
broken
parts
'
<
ref
name=oed
>
{
{
Cite
encyclopedia
|url=http
:
//www.lexico.com/definition/algebra
|archive-url=https
:
//web.archive.org/web/20131120000000/http
:
//www.lexico.com/definition/algebra
|url-status=dead
|archive-date=2013-11-20
|title=algebra
|dictionary=
[
[
Lexico|Oxford
Dictionaries
]
]
UK
English
Dictionary
|publisher=
[
[
Oxford
University
Press
]
]
}
}
{
{
Cite
web
|url=http
:
//www.oxforddictionaries.com/us/definition/english/algebra
|title=Algebra
:
Definition
of
algebra
in
Oxford
dictionary
–
British
&
World
English
(
US
)
|access-date=2013-11-20
|archive-date=2013-12-31
|archive-url=https
:
//web.archive.org/web/20131231173558/http
:
//www.oxforddictionaries.com/us/definition/english/algebra
|url-status=dead
}
}
.
<
/ref
>
or
'
[
[
Traditional
bone-setting|bone-setting
]
]
'
}
}
;
<
ref
name=
''
CRC
Press
''
>
{
{
cite
book
|last1=Menini
|first1=Claudia
|url=https
:
//books.google.com/books
?
id=3mlQDwAAQBAJ
&
q=bonesetting+algebra
&
pg=PA722
|title=Abstract
Algebra
:
A
Comprehensive
Treatment
|last2=Oystaeyen
|first2=Freddy
Van
|date=2017
|publisher=
[
[
CRC
Press
]
]
|isbn=978-1-4822-5817-2
|language=en
|access-date=2020-10-15
|archive-url=https
:
//web.archive.org/web/20210221075950/https
:
//books.google.com/books
?
id=3mlQDwAAQBAJ
&
q=bonesetting+algebra
&
pg=PA722
|archive-date=2021-02-21
|url-status=live
}
}
<
/ref
>
{
{
IPA|
[
ʔldʒbr
]
}
}
{
{
pronunciation|Q3968-ar.wav|listen|help=no
}
}
)
is
the
study
of
[
[
Variable
(
mathematics
)
|variables
]
]
and
the
rules
for
manipulating
these
variables
in
[
[
formula
]
]
s.
<
ref
>
See
{
{
harvnb|Herstein|1964
}
}
,
page
1
:
``
An
algebraic
system
can
be
described
as
a
set
of
objects
together
with
some
operations
for
combining
them
''
.
<
/ref
>
Originating
in
ancient
[
[
Babylonian
mathematics|Babylonian
techniques
of
calculation
]
]
,
it
is
now
a
way
of
thinking
that
appears
throughout
almost
all
areas
of
[
[
mathematics
]
]
.
<
ref
>
See
{
{
harvnb|Herstein|1964
}
}
,
page
1
:
``
...
it
also
serves
as
the
unifying
thread
which
interlaces
almost
all
of
mathematics
''
.
<
/ref
>
[
[
Elementary
algebra
]
]
deals
with
the
manipulation
of
variables
(
commonly
represented
by
[
[
Roman
letters
]
]
)
as
if
they
were
numbers
and
is
therefore
essential
in
all
applications
of
mathematics
.
[
[
Abstract
algebra|
''
Higher
''
or
``
abstract
''
algebra
]
]
,
which
professional
mathematicians
typically
just
call
``
algebra
''
,
is
the
study
of
[
[
algebraic
structure
]
]
s
that
generalize
the
operations
familiar
from
ordinary
[
[
arithmetic
]
]
.
For
example
,
a
[
[
group
(
mathematics
)
|group
]
]
is
a
set
with
a
[
[
binary
operation
]
]
,
a
rule
for
combining
two
members
of
that
set
to
produce
a
third
,
which
satisfies
some
of
the
same
basic
properties
as
addition
of
integers
.
Other
algebraic
structures
include
[
[
ring
(
mathematics
)
|rings
]
]
and
[
[
field
(
mathematics
)
|fields
]
]
.
[
[
Linear
algebra
]
]
,
which
deals
with
[
[
linear
equation
]
]
s
and
[
[
linear
mapping
]
]
s
,
is
used
for
modern
presentations
of
[
[
geometry
]
]
,
<
!
--
Berger
's
``
Geometry
''
must
be
cited
here
--
>
and
has
many
practical
applications
(
in
[
[
weather
forecasting
]
]
,
for
example
)
.
There
are
many
areas
of
mathematics
that
belong
to
algebra
,
some
having
``
algebra
''
in
their
name
,
such
as
[
[
commutative
algebra
]
]
,
and
some
not
,
such
as
[
[
Galois
theory
]
]
.
The
word
``
algebra
''
is
not
only
used
for
naming
an
area
of
mathematics
and
some
subareas
;
it
is
also
used
for
naming
some
sorts
of
algebraic
structures
,
such
as
an
[
[
algebra
over
a
field
]
]
,
commonly
called
an
``
algebra
''
.
Sometimes
,
the
same
phrase
is
used
for
a
subarea
and
its
main
algebraic
structures
.
For
example
,
the
subject
known
as
[
[
Boolean
algebra
]
]
studies
[
[
Boolean
algebra
(
structure
)
|structures
called
Boolean
algebras
]
]
.
A
mathematician
specialized
in
algebra
is
called
an
algebraist
.
==
Etymology
==
[
[
File
:
Muḥammad
ibn
Mūsā
al-Khwārizmī.png|thumb|upright=0.8|The
word
``
algebra
''
comes
from
the
title
of
a
book
by
[
[
Muhammad
ibn
Musa
al-Khwarizmi
]
]
.
<
ref
>
Esposito
,
John
L.
(
2000
)
.
''The
Oxford
History
of
Islam
''
.
Oxford
University
Press
.
p.
188
.
{
{
ISBN|978-0-19-988041-6
}
}
.
<
/ref
>
]
]
The
word
``
algebra
''
comes
from
the
{
{
lang-ar|الجبر|lit=reunion
of
broken
parts
,
<
ref
name=
''
oed
''
/
>
[
[
bonesetting
]
]
<
ref
name=
''
CRC
Press
''
/
>
|translit=al-jabr
}
}
from
the
title
of
the
early
9th
century
book
``
[
[
The
Compendious
Book
on
Calculation
by
Completion
and
Balancing|ʿIlm
al-jabr
wa
l-muqābala
]
]
''
``
The
Science
of
Restoring
and
Balancing
''
by
the
[
[
Persian
people|Persian
]
]
mathematician
and
astronomer
[
[
Muḥammad
ibn
Mūsā
al-Khwārizmī|al-Khwarizmi
]
]
.
In
his
work
,
the
term
``
al-jabr
''
referred
to
the
operation
of
moving
a
term
from
one
side
of
an
equation
to
the
other
,
المقابلة
``
al-muqābala
''
``
balancing
''
referred
to
adding
equal
terms
to
both
sides
.
Shortened
to
just
``
algeber
''
or
``
algebra
''
in
Latin
,
the
word
eventually
entered
the
English
language
during
the
15th
century
,
from
either
Spanish
,
Italian
,
or
[
[
Medieval
Latin
]
]
.
It
originally
referred
to
the
surgical
procedure
of
setting
[
[
Broken
bone|broken
]
]
or
[
[
Dislocated|dislocated
bones
]
]
.
The
mathematical
meaning
was
first
recorded
{
{
in
lang|en
}
}
in
the
16th
century.
<
ref
>
{
{
cite
encyclopedia|title=Algebra|editor=T
.
F.
Hoad|encyclopedia=The
Concise
Oxford
Dictionary
of
English
Etymology|publisher=Oxford
University
Press|location=Oxford|year=2003|url=https
:
//archive.org/details/conciseoxforddic00tfho|url-access=subscription|doi=10.1093/acref/9780192830982.001.0001|isbn=978-0-19-283098-2
}
}
<
/ref
>
==
Different
meanings
of
``
algebra
''
==
The
word
``
algebra
''
has
several
related
meanings
in
mathematics
,
as
a
single
word
or
with
qualifiers
.
*
As
a
single
word
without
an
[
[
Article_
(
grammar
)
|article
]
]
,
``
algebra
''
names
a
broad
part
of
mathematics
.
*
As
a
single
word
with
an
article
or
in
the
plural
,
``
an
algebra
''
or
``
algebras
''
denotes
a
specific
mathematical
structure
,
whose
precise
definition
depends
on
the
context
.
Usually
,
the
structure
has
an
addition
,
multiplication
,
and
scalar
multiplication
(
see
[
[
Algebra
over
a
field
]
]
)
.
When
some
authors
use
the
term
``
algebra
''
,
they
make
a
subset
of
the
following
additional
assumptions
:
[
[
Associative
property|associative
]
]
,
[
[
Commutative
property|commutative
]
]
,
[
[
Unital
algebra|unital
]
]
,
and/or
finite-dimensional
.
In
[
[
universal
algebra
]
]
,
the
word
``
algebra
''
refers
to
a
generalization
of
the
above
concept
,
which
allows
for
[
[
Operation
(
mathematics
)
|n-ary
operations
]
]
.
*
With
a
qualifier
,
there
is
the
same
distinction
:
**
Without
an
article
,
it
means
a
part
of
algebra
,
such
as
[
[
linear
algebra
]
]
,
[
[
elementary
algebra
]
]
(
the
symbol-manipulation
rules
taught
in
elementary
courses
of
mathematics
as
part
of
[
[
primary
education|primary
]
]
and
[
[
secondary
education
]
]
)
,
or
[
[
abstract
algebra
]
]
(
the
study
of
the
algebraic
structures
for
themselves
)
.
**
With
an
article
,
it
means
an
instance
of
some
algebraic
structure
,
like
a
[
[
Lie
algebra
]
]
,
an
[
[
associative
algebra
]
]
,
or
a
[
[
vertex
operator
algebra
]
]
.
**
Sometimes
both
meanings
exist
for
the
same
qualifier
,
as
in
the
sentence
:
``
[
[
Commutative
algebra
]
]
is
the
study
of
[
[
commutative
ring
]
]
s
,
which
are
[
[
algebra
(
ring
theory
)
|commutative
algebras
]
]
over
the
integers
''
.
==
Algebra
as
a
branch
of
mathematics
==
Algebra
began
with
computations
similar
to
those
of
[
[
arithmetic
]
]
,
with
letters
standing
for
numbers.
<
ref
name=citeboyer
/
>
This
allowed
proofs
of
properties
that
are
true
no
matter
which
numbers
are
involved
.
For
example
,
in
the
[
[
quadratic
equation
]
]
:
$
a
x
^2+
b
x
+
c
=0,
$
$
a
,
b
,
c
$
can
be
any
numbers
whatsoever
(
except
that
$
a
$
can
not
be
$
0
$
)
,
and
the
[
[
quadratic
formula
]
]
can
be
used
to
quickly
and
easily
find
the
values
of
the
unknown
quantity
$
x
$
which
satisfy
the
equation
.
That
is
to
say
,
to
find
all
the
solutions
of
the
equation
.
Historically
,
and
in
current
teaching
,
the
study
of
algebra
starts
with
the
solving
of
equations
,
such
as
the
quadratic
equation
above
.
Then
more
general
questions
,
such
as
``
does
an
equation
have
a
[
[
Solution
(
equation
)
|solution
]
]
?
``
,
``
how
many
[
[
Zero
of
a
function|solutions
]
]
does
an
equation
have
?
``
,
``
what
can
be
said
about
the
nature
of
the
solutions
?
''
are
considered
.
These
questions
led
extending
algebra
to
non-numerical
objects
,
such
as
[
[
permutation
]
]
s
,
[
[
vector
(
mathematics
)
|vectors
]
]
,
[
[
matrix
(
mathematics
)
|matrices
]
]
,
and
[
[
polynomial
]
]
s.
The
structural
properties
of
these
non-numerical
objects
were
then
formalized
into
[
[
algebraic
structure
]
]
s
such
as
[
[
group
(
mathematics
)
|groups
]
]
,
[
[
ring
(
mathematics
)
|rings
]
]
,
and
[
[
field
(
mathematics
)
|fields
]
]
.
Before
the
16th
century
,
mathematics
was
divided
into
only
two
subfields
,
[
[
arithmetic
]
]
and
[
[
geometry
]
]
.
Even
though
some
methods
,
which
had
been
developed
much
earlier
,
may
be
considered
nowadays
as
algebra
,
the
emergence
of
algebra
and
,
soon
thereafter
,
of
[
[
infinitesimal
calculus
]
]
as
subfields
of
mathematics
only
dates
from
the
16th
or
17th
century
.
From
the
second
half
of
the
19th
century
on
,
many
new
fields
of
mathematics
appeared
,
most
of
which
made
use
of
both
arithmetic
and
geometry
,
and
almost
all
of
which
used
algebra
.
Today
,
algebra
has
grown
considerably
and
includes
many
branches
of
mathematics
,
as
can
be
seen
in
the
[
[
Mathematics
Subject
Classification
]
]
<
ref
>
{
{
cite
web|url=https
:
//www.ams.org/mathscinet/msc/msc2010.html|title=2010
Mathematics
Subject
Classification|access-date=2014-10-05|archive-date=2014-06-06|archive-url=https
:
//web.archive.org/web/20140606010248/http
:
//www.ams.org/mathscinet/msc/msc2010.html|url-status=live
}
}
<
/ref
>
where
none
of
the
first
level
areas
(
two
digit
entries
)
are
called
``
algebra
''
.
Today
algebra
includes
section
08-General
algebraic
systems
,
12-
[
[
Field
theory
(
mathematics
)
|Field
theory
]
]
and
[
[
polynomial
]
]
s
,
13-
[
[
Commutative
algebra
]
]
,
15-
[
[
Linear
algebra|Linear
]
]
and
[
[
multilinear
algebra
]
]
;
[
[
matrix
theory
]
]
,
16-
[
[
associative
algebra|Associative
rings
and
algebras
]
]
,
17-
[
[
Nonassociative
ring
]
]
s
and
[
[
Non-associative
algebra|algebras
]
]
,
18-
[
[
Category
theory
]
]
;
[
[
homological
algebra
]
]
,
19-
[
[
K-theory
]
]
and
20-
[
[
Group
theory
]
]
.
Algebra
is
also
used
extensively
in
11-
[
[
Number
theory
]
]
and
14-
[
[
Algebraic
geometry
]
]
.
==
History
==
{
{
Main|History
of
algebra|Abstract
algebra
#
History|Timeline
of
algebra
}
}
The
use
of
the
word
``
algebra
''
for
denoting
a
part
of
mathematics
dates
probably
from
the
16th
century
.
{
{
citation
needed|date=July
2022
}
}
The
word
is
derived
from
the
[
[
Arabic
language|Arabic
word
]
]
``
al-jabr
''
that
appears
in
the
title
of
the
treatise
``
[
[
Al-Kitab
al-muhtasar
fi
hisab
al-gabr
wa-l-muqabala
]
]
''
(
``
The
Compendious
Book
on
Calculation
by
Completion
and
Balancing
''
)
,
written
in
circa
820
by
[
[
Al-Kwarizmi
]
]
.
''Al-jabr
''
referred
to
a
method
for
transforming
[
[
equation
(
mathematics
)
|equation
]
]
s
by
subtracting
[
[
like
terms
]
]
from
both
sides
,
or
passing
one
term
from
one
side
to
the
other
,
after
changing
its
sign
.
Therefore
,
``
algebra
''
referred
originally
to
the
manipulation
of
equations
,
and
,
by
extension
,
to
the
[
[
theory
of
equations
]
]
.
This
is
still
what
historians
of
mathematics
generally
mean
by
the
term
``
algebra
''
.
{
{
citation
needed|date=July
2022
}
}
In
mathematics
,
the
meaning
of
``
algebra
''
has
evolved
after
the
introduction
by
[
[
François
Viète
]
]
of
symbols
(
[
[
variable
(
mathematics
)
|variable
]
]
s
)
for
denoting
unknown
or
incompletely
specified
numbers
,
and
the
resulting
use
of
the
[
[
mathematical
notation
]
]
for
equations
and
[
[
formula
]
]
s.
So
,
algebra
became
essentially
the
study
of
the
action
of
[
[
operation
(
mathematics
)
|operation
]
]
s
on
[
[
expression
(
mathematics
)
|expression
]
]
s
involving
variables
.
This
includes
but
is
not
limited
to
the
theory
of
equations
.
At
the
beginning
of
the
20th
century
,
algebra
evolved
further
by
considering
operations
that
act
not
only
on
numbers
but
also
on
elements
of
so-called
[
[
mathematical
structure
]
]
s
such
as
[
[
group
(
mathematics
)
|group
]
]
s
,
[
[
field
(
mathematics
)
|field
]
]
s
and
[
[
vector
space
]
]
s.
This
new
algebra
was
called
``
[
[
Moderne
Algebra|modern
algebra
]
]
''
by
[
[
van
der
Waerden
]
]
in
his
eponymous
treatise
,
whose
name
has
been
changed
to
``
Algebra
''
in
later
editions
.
===
Early
history
===
[
[
File
:
Image-Al-Kitāb
al-muḫtaṣar
fī
ḥisāb
al-ğabr
wa-l-muqābala.jpg|thumb|upright=0.8|A
page
from
[
[
:
en
:
Muhammad
ibn
Musa
al-Khwarizmi|Al-Khwārizmī
]
]
's
``
[
[
The
Compendious
Book
on
Calculation
by
Completion
and
Balancing|al-Kitāb
al-muḫtaṣar
fī
ḥisāb
al-ğabr
wa-l-muqābala
]
]
''
]
]
The
roots
of
algebra
can
be
traced
back
to
the
ancient
[
[
Babylonian
mathematics|Babylonians
]
]
,
<
ref
>
{
{
cite
book
|last=Struik
|first=Dirk
J
.
|year=1987
|title=A
Concise
History
of
Mathematics
|location=New
York
|publisher=Dover
Publications
|isbn=978-0-486-60255-4
|url-access=registration
|url=https
:
//archive.org/details/concisehistoryof0000stru_m6j1
}
}
<
/ref
>
who
developed
an
advanced
arithmetical
system
with
which
they
were
able
to
do
calculations
in
an
[
[
algorithm
]
]
ic
fashion
.
The
Babylonians
developed
formulas
to
calculate
solutions
for
problems
typically
solved
today
by
using
[
[
linear
equation
]
]
s
,
[
[
quadratic
equation
]
]
s
,
and
[
[
indeterminate
equation|indeterminate
linear
equations
]
]
.
By
contrast
,
most
[
[
Ancient
Egyptian
mathematics|Egyptians
]
]
of
this
era
,
as
well
as
[
[
Greek
mathematics|Greek
]
]
and
[
[
Chinese
mathematics
]
]
in
the
1st
millennium
BC
,
usually
solved
such
equations
by
geometric
methods
,
such
as
those
described
in
the
``
[
[
Rhind
Mathematical
Papyrus
]
]
''
,
[
[
Euclid
's
Elements|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
,
although
this
would
not
be
realized
until
[
[
Mathematics
in
medieval
Islam|mathematics
developed
in
medieval
Islam
]
]
.
<
ref
>
See
{
{
harvnb|Boyer|1991
}
}
.
<
/ref
>
By
the
time
of
[
[
Plato
]
]
,
Greek
mathematics
had
undergone
a
drastic
change
.
The
Greeks
created
a
[
[
Greek
geometric
algebra|geometric
algebra
]
]
where
terms
were
represented
by
sides
of
geometric
objects
,
usually
lines
,
that
had
letters
associated
with
them.
<
ref
name=citeboyer
>
See
{
{
harvnb|Boyer|1991
}
}
,
``
Europe
in
the
Middle
Ages
''
,
p.
258
:
``
In
the
arithmetical
theorems
in
Euclid
's
``
Elements
''
VII–IX
,
numbers
had
been
represented
by
line
segments
to
which
letters
had
been
attached
,
and
the
geometric
proofs
in
al-Khwarizmi
's
``
Algebra
''
made
use
of
lettered
diagrams
;
but
all
coefficients
in
the
equations
used
in
the
``
Algebra
''
are
specific
numbers
,
whether
represented
by
numerals
or
written
out
in
words
.
The
idea
of
generality
is
implied
in
al-Khwarizmi
's
exposition
,
but
he
had
no
scheme
for
expressing
algebraically
the
general
propositions
that
are
so
readily
available
in
geometry
.
``
<
/ref
>
[
[
Diophantus
]
]
(
3rd
century
AD
)
was
an
[
[
Alexandria
]
]
n
Greek
mathematician
and
the
author
of
a
series
of
books
called
``
[
[
Arithmetica
]
]
''
.
These
texts
deal
with
solving
[
[
algebraic
equation
]
]
s
,
<
ref
>
{
{
cite
book
|author-link=Florian
Cajori
|first=Florian
|last=Cajori
|year=2010
|url=https
:
//books.google.com/books
?
id=gZ2Us3F7dSwC
&
pg=PA34
|title=A
History
of
Elementary
Mathematics
–
With
Hints
on
Methods
of
Teaching
|page=34
|publisher=Read
Books
Design
|isbn=978-1-4460-2221-4
|access-date=2020-10-15
|archive-date=2021-02-21
|archive-url=https
:
//web.archive.org/web/20210221075950/https
:
//books.google.com/books
?
id=gZ2Us3F7dSwC
&
pg=PA34
|url-status=live
}
}
<
/ref
>
and
have
led
,
in
[
[
number
theory
]
]
,
to
the
modern
notion
of
[
[
Diophantine
equation
]
]
.
Earlier
traditions
discussed
above
had
a
direct
influence
on
the
[
[
History
of
Iran|Persian
]
]
mathematician
[
[
Muhammad
ibn
Musa
al-Khwarizmi|Muḥammad
ibn
Mūsā
al-Khwārizmī
]
]
(
{
{
circa|780
}
}
–850
)
.
He
later
wrote
``
[
[
The
Compendious
Book
on
Calculation
by
Completion
and
Balancing
]
]
''
,
which
established
algebra
as
a
mathematical
discipline
that
is
independent
of
[
[
geometry
]
]
and
[
[
arithmetic
]
]
.
<
ref
>
{
{
Cite
book|title=Al
Khwarizmi
:
The
Beginnings
of
Algebra|author=Roshdi
Rashed|publisher=Saqi
Books|date=
2009|isbn=978-0-86356-430-7
}
}
<
/ref
>
The
[
[
Hellenistic
period|Hellenistic
]
]
mathematicians
[
[
Hero
of
Alexandria
]
]
and
Diophantus
<
ref
>
{
{
cite
web|url=http
:
//library.thinkquest.org/25672/diiophan.htm
|title=Diophantus
,
Father
of
Algebra
|access-date=2014-10-05
|url-status=dead
|archive-url=https
:
//web.archive.org/web/20130727040815/http
:
//library.thinkquest.org/25672/diiophan.htm
|archive-date=2013-07-27
}
}
<
/ref
>
as
well
as
[
[
Indian
mathematics|Indian
mathematicians
]
]
such
as
[
[
Brahmagupta
]
]
,
continued
the
traditions
of
Egypt
and
Babylon
,
though
Diophantus
'
``
Arithmetica
''
and
Brahmagupta
's
``
[
[
Brāhmasphuṭasiddhānta
]
]
''
are
on
a
higher
level.
<
ref
>
{
{
cite
web|url=http
:
//www.algebra.com/algebra/about/history/|title=History
of
Algebra|access-date=2014-10-05|archive-date=2014-11-11|archive-url=https
:
//web.archive.org/web/20141111040653/http
:
//www.algebra.com/algebra/about/history/|url-status=live
}
}
<
/ref
>
{
{
Better
source
needed|date=October
2017
}
}
For
example
,
the
first
complete
arithmetic
solution
written
in
words
instead
of
symbols
,
<
ref
>
{
{
Cite
book
|last=Mackenzie
|first=Dana
|url=https
:
//www.worldcat.org/oclc/761851013
|title=The
universe
in
zero
words
:
the
story
of
mathematics
as
told
through
equations
|date=2012
|publisher=Princeton
University
Press
|isbn=978-0-691-15282-0
|location=Princeton
,
N.J.
|pages=61
|oclc=761851013
}
}
<
/ref
>
including
zero
and
negative
solutions
,
to
quadratic
equations
was
described
by
Brahmagupta
in
his
book
``
Brahmasphutasiddhanta
,
''
published
in
628
AD.
<
ref
name=
''
Bradley
''
>
{
{
Cite
book
|last=Bradley
|first=Michael
J
.
|url=https
:
//www.worldcat.org/oclc/465077937
|title=The
birth
of
mathematics
:
ancient
times
to
1300
|date=2006
|publisher=Chelsea
House
|isbn=978-0-7910-9723-6
|location=New
York
|page=86
|oclc=465077937
}
}
<
/ref
>
Later
,
Persian
and
[
[
Arabs|Arab
]
]
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-Khwarizmi
's
contribution
was
fundamental
.
He
solved
linear
and
quadratic
equations
without
algebraic
symbolism
,
[
[
negative
numbers
]
]
or
[
[
zero
]
]
,
thus
he
had
to
distinguish
several
types
of
equations.
<
ref
name=
''
Meri2004
''
>
{
{
cite
book|first=Josef
W.|last=Meri|title=Medieval
Islamic
Civilization|url=https
:
//books.google.com/books
?
id=H-k9oc9xsuAC
&
pg=PA31|access-date=2012-11-25|year=2004|publisher=Psychology
Press|isbn=978-0-415-96690-0|page=31|archive-date=2013-06-02|archive-url=https
:
//web.archive.org/web/20130602195207/http
:
//books.google.com/books
?
id=H-k9oc9xsuAC
&
pg=PA31|url-status=live
}
}
<
/ref
>
In
the
context
where
algebra
is
identified
with
the
[
[
theory
of
equations
]
]
,
the
Greek
mathematician
Diophantus
has
traditionally
been
known
as
the
``
father
of
algebra
''
and
in
the
context
where
it
is
identified
with
rules
for
manipulating
and
solving
equations
,
Persian
mathematician
al-Khwarizmi
is
regarded
as
``
the
father
of
algebra
''
.
<
ref
>
{
{
Cite
book|last=Corona|first=Brezina|title=Al-Khwarizmi
:
The
Inventor
Of
Algebra|publisher=Rosen
Pub
Group|date=
2006|isbn=978-1404205130|location=New
York
,
United
States
}
}
<
/ref
>
<
ref
>
See
{
{
harvnb|Boyer|1991
}
}
,
page
181
:
``
If
we
think
primarily
of
the
matter
of
notations
,
Diophantus
has
good
claim
to
be
known
as
the
'father
of
algebra
'
,
but
in
terms
of
motivation
and
concept
,
the
claim
is
less
appropriate
.
The
Arithmetica
is
not
a
systematic
exposition
of
the
algebraic
operations
,
or
of
algebraic
functions
or
of
the
solution
of
algebraic
equations
''
.
<
/ref
>
<
ref
>
See
{
{
harvnb|Boyer|1991
}
}
,
page
230
:
``
The
six
cases
of
equations
given
above
exhaust
all
possibilities
for
linear
and
quadratic
equations
...
In
this
sense
,
then
,
al-Khwarizmi
is
entitled
to
be
known
as
'the
father
of
algebra
{
{
'
''
}
}
.
<
/ref
>
<
ref
>
See
{
{
harvnb|Boyer|1991
}
}
,
page
228
:
``
Diophantus
sometimes
is
called
the
father
of
algebra
,
but
this
title
more
appropriately
belongs
to
al-Khowarizmi
''
.
<
/ref
>
<
ref
name=
''
Gandz
''
>
See
{
{
harvnb|Gandz|1936
}
}
,
pp
.
263–277
:
``
In
a
sense
,
al-Khwarizmi
is
more
entitled
to
be
called
``
the
father
of
algebra
''
than
Diophantus
because
al-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
''
.
<
/ref
>
<
ref
>
{
{
Cite
journal
|last=Christianidis
|first=Jean
|date=August
2007
|title=The
way
of
Diophantus
:
Some
clarifications
on
Diophantus
'
method
of
solution|journal=
[
[
Historia
Mathematica
]
]
|volume=34|issue=3|pages=289–305|quote=It
is
true
that
if
one
starts
from
a
conception
of
algebra
that
emphasizes
the
solution
of
equations
,
as
was
generally
the
case
with
the
Arab
mathematicians
from
al-Khwārizmī
onward
as
well
as
with
the
Italian
algebraists
of
the
Renaissance
,
then
the
work
of
Diophantus
appears
indeed
very
different
from
the
works
of
those
algebraists|doi=10.1016/j.hm.2006.10.003|doi-access=
}
}
<
/ref
>
<
ref
>
{
{
cite
journal
|first=G
.
C.
|last=
Cifoletti
|title=
La
question
de
l'algèbre
:
Mathématiques
et
rhétorique
des
homes
de
droit
dans
la
France
du
16e
siècle
|journal=
Annales
de
l'École
des
Hautes
Études
en
Sciences
Sociales
,
50
(
6
)
|year=
1995
|pages=
1385–1416
|quote=
Le
travail
des
Arabes
et
de
leurs
successeurs
a
privilégié
la
solution
des
problèmes.Arithmetica
de
Diophantine
ont
privilégié
la
théorie
des
equations
}
}
<
/ref
>
It
is
open
to
debate
whether
Diophantus
or
al-Khwarizmi
is
more
entitled
to
be
known
,
in
the
general
sense
,
as
``
the
father
of
algebra
''
.
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.
<
ref
>
See
{
{
harvnb|Boyer|1991
}
}
,
page
228.
<
/ref
>
Those
who
support
Al-Khwarizmi
point
to
the
fact
that
he
introduced
the
methods
of
``
[
[
Reduction
(
mathematics
)
|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
,
<
ref
name=Boyer-229
>
See
{
{
harvnb|Boyer|1991
}
}
,
``
The
Arabic
Hegemony
''
,
p.
229
:
``
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
''
.
<
/ref
>
and
that
he
gave
an
exhaustive
explanation
of
solving
quadratic
equations
,
<
ref
>
See
{
{
harvnb|Boyer|1991
}
}
,
``
The
Arabic
Hegemony
''
,
p.
230
:
``
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
''
.
<
/ref
>
supported
by
geometric
proofs
while
treating
algebra
as
an
independent
discipline
in
its
own
right.
<
ref
name=
''
Gandz
''
/
>
His
algebra
was
also
no
longer
concerned
``
with
a
series
of
problems
to
be
resolved
,
but
an
[
[
Expository
writing|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
''
.
<
ref
name=Rashed-Armstrong
>
{
{
Cite
book
|last1=
Rashed
|first1=
R.
|last2=
Armstrong
|first2=
Angela
|year=
1994