Computer Algebra

Date: 2015-10-07; view: 503.

Text 7. Make the written translation into Russian (time 90 minutes)

(1670 characters)

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 expressions. For numerical calculations the digital computer is now thoroughly established as a device that can greatly ease the human burden of work. It is less generally appreciated that there are computer programs equally well adapted to the manipulation of algebraic expressions. In other words, the computer can work not only with numbers themselves but also with more abstract symbols that represent numerical quantities.

In order to understand the need for automatic systems of algebraic manipulation it must be appreciated that many concepts in science are embodied in mathematical statements where there is little point to numerical evaluation. Consider the simple expression 3π²/π.

As any student of algebra knows, the fraction can be reduced by cancelling 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 expression in the symbolic, nonnumeric form. With a computer programmed to do only arithmetic, the expression 3π²/π must be evaluated; 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 computer. The equivalence of 3 π²/π and 3π would probably not be recognized by a computer programmed in this way.