The meaning of symbolic computation
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The meaning of symbolic computation
> What exactly does symbolic manipulation mean?
> It means that if you enter sin(pi/4), the calculator returns
> sqrt(2)/2 instead of 0.7071...
> Similarly, if you enter something like x+x+y+3y, the calculator
> returns 2x + 4y, unless x and y are defined, in which case the
> calculator returns the value for the specific values of x.
"Symbolic manipulation" is of course a general term, and could refer to a
computer sorting phone-book data. In the context of hand-held calculators,
however, it means a radically different way of doing business.
Traditionally calculators have worked with floating-point numbers, twelve or so
places of precision. It only _looked_ to the user that integer calculations
were being performed exactly -- there was always the lurking possiblility of
"round-off error". Any variables (such as x) entered by the user would
immediately be "looked up" by the calculator and replaced by the floating-point
value stored for that variable. Any result returned could only be a
floating-point number.
Real computer languages, in richer environments, could handle symbols in
different ways, distinguishing between floating-point numbers and actual
integers, numeric expressions stored as text, etc. and calculating with these
latter in other ways. Gradually, "computer algebra" systems (Maple, Derive,
Mathematica, etc.) developed to run on larger computers and provide the user
with convenient literal or exact-integer calculation became more and more
inexpensive. Finally, the breakthrough was made of building this sort of
capability into a real calculator -- the TI-92 (since then, the TI-89 also). In
order to provide continuity, this calculator could also operate in the old
floating-point mode if the user wished (plus do all to the other things that a
graphing calculator was expected to do).
The essence of symbolic manipulation of input is a set of smart routines that
implement standard mathematical conventions. If the user has _exact_ mode set
and enters an expression such as 2/3-1/5 the TI-92 does not _evaluate_ this
expression, but rather recognizes that this is a difference of fractions, and
follows standard rules to come up with a result of 7/15 (this result is also
_stored_ as a fraction rather than as a floating-point equivalent).
Input may also contain _variables_. As with earlier calculators, the TI-92
looks up any values that may have been stored for the variables encountered in
an entered expression. However, if it doesn't _find_ any stored value, it
merely preserves the variables it finds and operates with them according to the
rules of algebra. Of course, stored values can also be _other_ algebraic
expressions, and so on... This can be very useful, for example in simplifying
huge expressions, in solving literal equations, in developing trigonometric
identities, etc. The power of this environment is just beginning to be explored.
Just as in the old days, however, what appears to the user to be happening when
the calculator processes input may be subtly different from what is _really_
happening. The TI-92 doesn't really do mathematics the way a human would, and
will sometimes go different ways than you or I would in performing a
calculation. Sometimes the calculator even (gasp!) makes mistakes. As always,
it is up to the human user to monitor the situation and provide checking and
correction.
Of course this all implies a huge shift in what a person needs to learn in a
mathematics class. But that is a topic for a different list...
RWW Taylor
National Technical Institute for the Deaf
Rochester Institute of Technology
Rochester NY 14623
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