# [om] DefMP elements

Richard Fateman fateman at cs.berkeley.edu
Wed Dec 10 02:52:09 CET 2003

```And is pi a numerical value?  How about  pi^2+e ?
thus   sin(1)  means   sin(1)    looks ok to computer algebra systems.
so does tan(x) means sin(x)/cos(x).  Is x a numerical value?

In a CAS, there is typically a definition for a function Q that
looks like this:

Q(x):=   case
is x a special argument like pi, infinity, ... for which Q
has a special value?
is x a peculiar kind of argument like a matrix, an interval
etc.  ..do something
is x an explicit numerical quantity for which Q(x) can be
computed approximately,
and is the context such that the approximation is
expected to be computed?
is x nothing special, but Q(x) can be decomposed to some
other basis functions
like sin/cos into complex exponentials,  and does the
context require this?
is x nothing special and no other transformations apply,
so leave Q(x) alone.

Q could be sin.

I'm sure that one could elaborate on the cases above; I am just writing this
freehand.

If you insist than sin be given an explicit numerical value, you are
describing
something like BASIC.

RJF

Bill Naylor wrote:

>>>>As to the fourth:
>>>>d) use signatures in definitions: this might prevent a definition from
>>>>being misapplied in an application that doesn't know any better. This
>>>>may well be done in the simple way of using universal quantifiers and
>>>>set memberships inside the defining FMPs, so that this may simply be
>>>>added as a recommendation to CD writers.
>>>>
>>>>
>>But the symbol being defined has a signature in STS, which is why the sin
>>on matrices is not the sin from transc1.
>>James
>>
>>
>
>Excuse me for appearing a bit ignorant here. The signature for sin in STS
>is defined as sin: NumericalValue -> NumericalValue, where NumericalValue
>is defined as:
>
>"NumericalValue
>
>Denotes an OpenMath object that is to be thought of as something that
>represents a numerical value, or a numerical value."
>
>this seems very circular to me, where do we define a 'numerical value'? I
>mean integers are clearly numerical values, so are reals. Almost as
>clearly complex numbers are numerical values. What about Quaternions? I
>would have thought so? What is it that makes them numerical and matrices
>not?
>
>thoughts?
>
>Bill.
>
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