<s>
A	O
computer	B-General_Concept
algebra	I-General_Concept
system	I-General_Concept
(	O
CAS	O
)	O
or	O
symbolic	B-General_Concept
algebra	I-General_Concept
system	I-General_Concept
(	O
SAS	O
)	O
is	O
any	O
mathematical	B-Application
software	I-Application
with	O
the	O
ability	O
to	O
manipulate	O
mathematical	O
expressions	O
in	O
a	O
way	O
similar	O
to	O
the	O
traditional	O
manual	O
computations	O
of	O
mathematicians	O
and	O
scientists	O
.	O
</s>
<s>
The	O
development	O
of	O
the	O
computer	B-General_Concept
algebra	I-General_Concept
systems	I-General_Concept
in	O
the	O
second	O
half	O
of	O
the	O
20th	O
century	O
is	O
part	O
of	O
the	O
discipline	O
of	O
"	O
computer	B-Algorithm
algebra	I-Algorithm
"	O
or	O
"	O
symbolic	B-Algorithm
computation	I-Algorithm
"	O
,	O
which	O
has	O
spurred	O
work	O
in	O
algorithms	O
over	O
mathematical	O
objects	O
such	O
as	O
polynomials	O
.	O
</s>
<s>
Computer	B-General_Concept
algebra	I-General_Concept
systems	I-General_Concept
may	O
be	O
divided	O
into	O
two	O
classes	O
:	O
specialized	O
and	O
general-purpose	O
.	O
</s>
<s>
General-purpose	O
computer	B-General_Concept
algebra	I-General_Concept
systems	I-General_Concept
aim	O
to	O
be	O
useful	O
to	O
a	O
user	O
working	O
in	O
any	O
scientific	O
field	O
that	O
requires	O
manipulation	O
of	O
mathematical	O
expressions	O
.	O
</s>
<s>
To	O
be	O
useful	O
,	O
a	O
general-purpose	O
computer	B-General_Concept
algebra	I-General_Concept
system	I-General_Concept
must	O
include	O
various	O
features	O
such	O
as	O
:	O
</s>
<s>
a	O
user	B-Application
interface	I-Application
allowing	O
a	O
user	O
to	O
enter	O
and	O
display	O
mathematical	O
formulas	O
,	O
typically	O
from	O
a	O
keyboard	O
,	O
menu	O
selections	O
,	O
mouse	O
or	O
stylus	O
.	O
</s>
<s>
a	O
programming	O
language	O
and	O
an	O
interpreter	B-Application
(	O
the	O
result	O
of	O
a	O
computation	O
commonly	O
has	O
an	O
unpredictable	O
form	O
and	O
an	O
unpredictable	O
size	O
;	O
therefore	O
user	O
intervention	O
is	O
frequently	O
needed	O
)	O
,	O
</s>
<s>
a	O
memory	B-General_Concept
manager	I-General_Concept
,	O
including	O
a	O
garbage	B-General_Concept
collector	I-General_Concept
,	O
needed	O
by	O
the	O
huge	O
size	O
of	O
the	O
intermediate	O
data	O
,	O
which	O
may	O
appear	O
during	O
a	O
computation	O
,	O
</s>
<s>
an	O
arbitrary-precision	B-Algorithm
arithmetic	I-Algorithm
,	O
needed	O
by	O
the	O
huge	O
size	O
of	O
the	O
integers	O
that	O
may	O
occur	O
,	O
</s>
<s>
For	O
example	O
,	O
the	O
computation	O
of	O
polynomial	B-Algorithm
greatest	I-Algorithm
common	I-Algorithm
divisors	I-Algorithm
is	O
systematically	O
used	O
for	O
the	O
simplification	O
of	O
expressions	O
involving	O
fractions	O
.	O
</s>
<s>
This	O
large	O
amount	O
of	O
required	O
computer	O
capabilities	O
explains	O
the	O
small	O
number	O
of	O
general-purpose	O
computer	B-General_Concept
algebra	I-General_Concept
systems	I-General_Concept
.	O
</s>
<s>
Significant	O
systems	O
include	O
Axiom	B-Language
,	O
Maxima	B-Language
,	O
Magma	B-General_Concept
,	O
Maple	B-Language
,	O
Mathematica	B-Language
,	O
and	O
SageMath	B-Application
.	O
</s>
<s>
Computer	B-General_Concept
algebra	I-General_Concept
systems	I-General_Concept
began	O
to	O
appear	O
in	O
the	O
1960s	O
and	O
evolved	O
out	O
of	O
two	O
quite	O
different	O
sources	O
—	O
the	O
requirements	O
of	O
theoretical	O
physicists	O
and	O
research	O
into	O
artificial	B-Application
intelligence	I-Application
.	I-Application
</s>
<s>
A	O
prime	O
example	O
for	O
the	O
first	O
development	O
was	O
the	O
pioneering	O
work	O
conducted	O
by	O
the	O
later	O
Nobel	O
Prize	O
laureate	O
in	O
physics	O
Martinus	O
Veltman	O
,	O
who	O
designed	O
a	O
program	O
for	O
symbolic	B-Algorithm
mathematics	I-Algorithm
,	O
especially	O
high-energy	O
physics	O
,	O
called	O
Schoonschip	B-Application
(	O
Dutch	O
for	O
"	O
clean	O
ship	O
"	O
)	O
in	O
1963	O
.	O
</s>
<s>
Another	O
early	O
system	O
was	O
FORMAC	B-Language
.	O
</s>
<s>
Using	O
Lisp	O
as	O
the	O
programming	O
basis	O
,	O
Carl	O
Engelman	O
created	O
MATHLAB	B-Application
in	O
1964	O
at	O
MITRE	O
within	O
an	O
artificial-intelligence	B-Application
research	O
environment	O
.	O
</s>
<s>
Later	O
MATHLAB	B-Application
was	O
made	O
available	O
to	O
users	O
on	O
PDP-6	O
and	O
PDP-10	O
systems	O
running	O
TOPS-10	O
or	O
TENEX	O
in	O
universities	O
.	O
</s>
<s>
Today	O
it	O
can	O
still	O
be	O
used	O
on	O
SIMH	B-Application
emulations	O
of	O
the	O
PDP-10	O
.	O
</s>
<s>
MATHLAB	B-Application
(	O
"	O
mathematical	O
laboratory	O
"	O
)	O
should	O
not	O
be	O
confused	O
with	O
MATLAB	B-Language
(	O
"	O
matrix	B-Language
laboratory	I-Language
"	O
)	O
,	O
which	O
is	O
a	O
system	O
for	O
numerical	O
computation	O
built	O
15	O
years	O
later	O
at	O
the	O
University	O
of	O
New	O
Mexico	O
.	O
</s>
<s>
The	O
first	O
popular	O
computer	B-General_Concept
algebra	I-General_Concept
systems	I-General_Concept
were	O
muMATH	B-Language
,	O
Reduce	B-Language
,	O
Derive	B-Language
(	O
based	O
on	O
muMATH	B-Language
)	O
,	O
and	O
Macsyma	B-Language
;	O
a	O
popular	O
copyleft	B-License
version	O
of	O
Macsyma	B-Language
called	O
Maxima	B-Language
is	O
actively	O
being	O
maintained	O
.	O
</s>
<s>
Reduce	B-Language
became	O
free	O
software	O
in	O
2008	O
.	O
</s>
<s>
As	O
of	O
today	O
,	O
the	O
most	O
popular	O
commercial	O
systems	O
are	O
Mathematica	B-Language
and	O
Maple	B-Language
,	O
which	O
are	O
commonly	O
used	O
by	O
research	O
mathematicians	O
,	O
scientists	O
,	O
and	O
engineers	O
.	O
</s>
<s>
Freely	O
available	O
alternatives	O
include	O
SageMath	B-Application
(	O
which	O
can	O
act	O
as	O
a	O
front-end	B-Architecture
to	O
several	O
other	O
free	O
and	O
nonfree	O
CAS	O
)	O
.	O
</s>
<s>
In	O
1987	O
,	O
Hewlett-Packard	O
introduced	O
the	O
first	O
hand-held	O
calculator	O
CAS	O
with	O
the	O
HP-28	B-Device
series	I-Device
,	O
and	O
it	O
was	O
possible	O
,	O
for	O
the	O
first	O
time	O
in	O
a	O
calculator	O
,	O
to	O
arrange	O
algebraic	O
expressions	O
,	O
differentiation	O
,	O
limited	O
symbolic	B-Algorithm
integration	I-Algorithm
,	O
Taylor	O
series	O
construction	O
and	O
a	O
solver	O
for	O
algebraic	O
equations	O
.	O
</s>
<s>
In	O
1999	O
,	O
the	O
independently	O
developed	O
CAS	O
Erable	B-Application
for	O
the	O
HP	O
48	O
series	O
became	O
an	O
officially	O
integrated	O
part	O
of	O
the	O
firmware	O
of	O
the	O
emerging	O
HP	O
49/50	O
series	O
,	O
and	O
a	O
year	O
later	O
into	O
the	O
HP	B-Device
40	I-Device
series	I-Device
as	O
well	O
,	O
whereas	O
the	O
HP	B-Device
Prime	I-Device
adopted	O
the	O
Xcas	B-Language
system	O
in	O
2013	O
.	O
</s>
<s>
The	O
Texas	O
Instruments	O
company	O
in	O
1995	O
released	O
the	O
TI-92	B-Device
calculator	O
with	O
a	O
CAS	O
based	O
on	O
the	O
software	O
Derive	B-Language
;	O
the	O
TI-Nspire	B-Device
series	I-Device
replaced	O
Derive	B-Language
in	O
2007	O
.	O
</s>
<s>
The	O
TI-89	B-Device
series	I-Device
,	O
first	O
released	O
in	O
1998	O
,	O
also	O
contains	O
a	O
CAS	O
.	O
</s>
<s>
Casio	O
released	O
their	O
first	O
CAS	O
calculator	O
with	O
the	O
CFX-9970G	B-Device
and	O
succeeded	O
it	O
with	O
the	O
Algebra	B-Device
FX	I-Device
Series	I-Device
in	O
1999-2003	O
and	O
the	O
current	O
ClassPad	B-Device
Series	I-Device
.	O
</s>
<s>
More	O
recently	O
,	O
computer	B-General_Concept
algebra	I-General_Concept
systems	I-General_Concept
have	O
been	O
implemented	O
using	O
artificial	B-Architecture
neural	I-Architecture
networks	I-Architecture
.	O
</s>
<s>
change	O
of	O
form	O
of	O
expressions	O
:	O
expanding	O
products	O
and	O
powers	O
,	O
partial	O
and	O
full	O
factorization	O
,	O
rewriting	O
as	O
partial	O
fractions	O
,	O
constraint	B-Application
satisfaction	I-Application
,	O
rewriting	O
trigonometric	O
functions	O
as	O
exponentials	O
,	O
transforming	O
logic	O
expressions	O
,	O
etc	O
.	O
</s>
<s>
Some	O
computer	B-General_Concept
algebra	I-General_Concept
systems	I-General_Concept
focus	O
on	O
specialized	O
disciplines	O
;	O
these	O
are	O
typically	O
developed	O
in	O
academia	O
and	O
are	O
free	O
.	O
</s>
<s>
They	O
can	O
be	O
inefficient	O
for	O
numeric	B-Algorithm
operations	O
as	O
compared	O
to	O
numeric	B-Application
systems	I-Application
.	O
</s>
<s>
)	O
;	O
arbitrary	O
functions	O
of	O
expressions	O
;	O
optimization	O
;	O
derivatives	O
,	O
integrals	O
,	O
simplifications	O
,	O
sums	O
,	O
and	O
products	O
of	O
expressions	O
;	O
truncated	O
series	O
with	O
expressions	O
as	O
coefficients	O
,	O
matrices	B-Architecture
of	O
expressions	O
,	O
and	O
so	O
on	O
.	O
</s>
<s>
Numeric	B-Algorithm
domains	O
supported	O
typically	O
include	O
floating-point	B-Algorithm
representation	I-Algorithm
of	I-Algorithm
real	I-Algorithm
numbers	I-Algorithm
,	O
integers	O
(	O
of	O
unbounded	O
size	O
)	O
,	O
complex	O
(	O
floating-point	B-Algorithm
representation	I-Algorithm
)	O
,	O
interval	B-Algorithm
representation	I-Algorithm
of	I-Algorithm
reals	I-Algorithm
,	O
rational	O
number	O
(	O
exact	O
representation	O
)	O
and	O
algebraic	O
numbers	O
.	O
</s>
<s>
There	O
have	O
been	O
many	O
advocates	O
for	O
increasing	O
the	O
use	O
of	O
computer	B-General_Concept
algebra	I-General_Concept
systems	I-General_Concept
in	O
primary	O
and	O
secondary-school	O
classrooms	O
.	O
</s>
<s>
The	O
primary	O
reason	O
for	O
such	O
advocacy	O
is	O
that	O
computer	B-General_Concept
algebra	I-General_Concept
systems	I-General_Concept
represent	O
real-world	O
math	O
more	O
than	O
do	O
paper-and-pencil	O
or	O
hand	O
calculator	O
based	O
mathematics	O
.	O
</s>
<s>
Computer	B-General_Concept
algebra	I-General_Concept
systems	I-General_Concept
have	O
been	O
extensively	O
used	O
in	O
higher	O
education	O
.	O
</s>
<s>
The	O
companies	O
that	O
develop	O
computer	B-General_Concept
algebra	I-General_Concept
systems	I-General_Concept
have	O
pushed	O
to	O
increase	O
their	O
prevalence	O
among	O
university	O
and	O
college	O
programs	O
.	O
</s>
<s>
Symbolic	B-Algorithm
integration	I-Algorithm
via	O
e.g.	O
</s>
<s>
Limit	B-Algorithm
computation	I-Algorithm
via	O
e.g.	O
</s>
