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Computer Algebra
Symbols as well as numbers can be manipulated by a computer.
New, general-purpose algorithms can undertake a wide variety of routine
mathematical work and solve intractable problems by Richard Pavelle,
Michael Rothstein and John Fitch
Of all the tasks to which the computer can be applied none is
more daunting than the manipulation of complex mathematical expres-
sions. For numerical calculations the digital computer is now thor-
oughly established as a device that can greatly ease the human burden
of work. It is less generally appreciated that there are computer pro-
grams equally well adapted to the manipulation of algebraic expres-
sions. In other words, the computer can work not only with numbers
themselves but also with more abstract symbols that represent numeri-
cal quantities.
In order to understand the need for automatic systems of alge-
braic manipulation it must be appreciated that many concepts in sci-
ence are embodied in mathematical statements where there is little
point to numerical evaluation. Consider the simple expression 3π
2
/π.
As any student of algebra knows, the fraction can be reduced by can-
celling π from both the numerator and the denominator to obtain the
simplified form 3π. The numerical value of 3π may be of interest, but it
may also be sufficient, and perhaps of greater utility, to leave the ex-
pression in the symbolic, nonnumerical form. With a computer pro-
grammed to do only arithmetic, the expression 3 π
2
/π must be evalu-
ated; when the calculation is done with a precision of 10 significant
figures, the value obtained is 9,424777958. The number, besides being
a rather uninformative string of digits, is not the same as the number
obtained from the numerical evaluation (to 10 significant figured) of 3π.
The latter number is 9,424777962; the discrepancy in the last two
decimal places results from round-off errors introduced by the com-
puter. The equivalence of 3 π
2
/π and 3π would probably not be recog-
nized by a computer programmed in this way.
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Computer Software in Science and Mathematics
Computation offers a new means of describing and investigating scientific
and mathematical systems. Simulation by computer may be the only way
to predict how certain complicated systems evolve by Stephen Wolfram
Scientific laws give algorithms, or procedures for determining
how systems behave. The computer program is a medium in which the
algorithms can be expressed and applied. Physical objects and mathe-
matical structures can be represented as numbers and symbols in a
computer, and a program can be written to manipulate them according
to the algorithms. When the computer program is executed, it causes
the numbers and symbols to be modified in the way specified by the
scientific laws. It thereby allows the consequences of the laws to be
deduced.
Executing a computer program is much like performing an ex-
periment. Unlike the physical objects in a conventional experiment,
however, the objects in a computer experiment are not bound by the
laws of nature. Instead they follow the laws embodied in the computer
program, which can be of any consistent form. Computation thus ex-
tends the realm of experimental science: it allows experiments to be
performed in a hypothetical universe. Computation also extends theo-
retical science. Scientific laws have conventionally been constructed in
terms of a particular set of mathematical functions and constructs, and
they have often been developed as much for their mathematical sim-
plicity as for their capacity to model the salient features of a phenome-
non. A scientific law specified by an algorithm, however, can have any
consistent form. The study of many complex systems, which have re-
sisted analysis by traditional mathematical methods, is consequently
being made possible through computer experiments and computer
models. Computation is emerging as a major new approach to the sci-
ence, supplementing the long-standing methodologies of theory and
experiment.
Text 2 Text 3
Translate the Text with dictionary in written form (time 45 Translate the Text with dictionary in written form (time 45
minutes) minutes)
Computer Algebra Computer Software in Science and Mathematics
Symbols as well as numbers can be manipulated by a computer. Computation offers a new means of describing and investigating scientific
New, general-purpose algorithms can undertake a wide variety of routine and mathematical systems. Simulation by computer may be the only way
mathematical work and solve intractable problems by Richard Pavelle, to predict how certain complicated systems evolve by Stephen Wolfram
Michael Rothstein and John Fitch
Scientific laws give algorithms, or procedures for determining
Of all the tasks to which the computer can be applied none is how systems behave. The computer program is a medium in which the
more daunting than the manipulation of complex mathematical expres- algorithms can be expressed and applied. Physical objects and mathe-
sions. For numerical calculations the digital computer is now thor- matical structures can be represented as numbers and symbols in a
oughly established as a device that can greatly ease the human burden computer, and a program can be written to manipulate them according
of work. It is less generally appreciated that there are computer pro- to the algorithms. When the computer program is executed, it causes
grams equally well adapted to the manipulation of algebraic expres- the numbers and symbols to be modified in the way specified by the
sions. In other words, the computer can work not only with numbers scientific laws. It thereby allows the consequences of the laws to be
themselves but also with more abstract symbols that represent numeri- deduced.
cal quantities. Executing a computer program is much like performing an ex-
In order to understand the need for automatic systems of alge- periment. Unlike the physical objects in a conventional experiment,
braic manipulation it must be appreciated that many concepts in sci- however, the objects in a computer experiment are not bound by the
ence are embodied in mathematical statements where there is little laws of nature. Instead they follow the laws embodied in the computer
point to numerical evaluation. Consider the simple expression 3π2/π. program, which can be of any consistent form. Computation thus ex-
As any student of algebra knows, the fraction can be reduced by can- tends the realm of experimental science: it allows experiments to be
celling π from both the numerator and the denominator to obtain the performed in a hypothetical universe. Computation also extends theo-
simplified form 3π. The numerical value of 3π may be of interest, but it retical science. Scientific laws have conventionally been constructed in
may also be sufficient, and perhaps of greater utility, to leave the ex- terms of a particular set of mathematical functions and constructs, and
pression in the symbolic, nonnumerical form. With a computer pro- they have often been developed as much for their mathematical sim-
grammed to do only arithmetic, the expression 3 π2/π must be evalu- plicity as for their capacity to model the salient features of a phenome-
ated; when the calculation is done with a precision of 10 significant non. A scientific law specified by an algorithm, however, can have any
figures, the value obtained is 9,424777958. The number, besides being consistent form. The study of many complex systems, which have re-
a rather uninformative string of digits, is not the same as the number sisted analysis by traditional mathematical methods, is consequently
obtained from the numerical evaluation (to 10 significant figured) of 3π. being made possible through computer experiments and computer
The latter number is 9,424777962; the discrepancy in the last two models. Computation is emerging as a major new approach to the sci-
decimal places results from round-off errors introduced by the com- ence, supplementing the long-standing methodologies of theory and
puter. The equivalence of 3 π2/π and 3π would probably not be recog- experiment.
nized by a computer programmed in this way.
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