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From: dmitry@elros.cbb-automation.de (Dmitry A. Kazakov)
Newsgroups: comp.lang.ada
Subject: Re: Dimensionality Checking (Ada 20XX)
Date: Fri, 14 Dec 2001 10:16:44 GMT
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Date: 2001-12-14T10:16:44+00:00
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On Thu, 13 Dec 2001 12:02:49 -0500, daniele andreatta
<andreatta@mail.chem.sc.edu> wrote:

>
>
>Stephen Leake wrote:
>
>> dmitry@elros.cbb-automation.de (Dmitry A. Kazakov) writes:
>>
>> > On 11 Dec 2001 11:39:30 -0500, Stephen Leake
>> > <stephen.a.leake.1@gsfc.nasa.gov> wrote:
>> >
>> > >It is simply wrong to try to define units for trig and exponential
>> > >functions. Remember the Taylor expansion for Sin:
>> > >
>> > >Sin (x) = x - 1/6 x**3 ...
>> > >
>> > >If x has dimensions of meters (shudder :), then what are the
>> > >dimensions of Sin (x)? This is why angles must be dimensionless, as
>> > >radians are.
>> >
>> > It is true, but the proof is wrong. Consider sqrt (x). It also has a
>> > Taylor expansion in 1, which is also endless, yet sqrt (m**2) exists
>> > and is m.
>>
>> Hmm. Good point. I guess infinite series are trickier than I remember;
>> it has been a long time :).
>>
>> --
>> -- Stephe
>
>Taylor series keeps the unit of the function. Basically the series is (as
>you recall)
>f(x) = f(x_0) + f'(x_0) (x-x_0) +f''(x_0) (x-x_0)**2 + ...
>and the unit of the derivatives are (indicating with [] the unit of the
>quantity)
>[f'(x)] = [f]/[x];    [f''] = [f]/[x]**2;    ....
>In the end
>[f(x)] = [Taylor expansion]

Right. Taylor series keep the unit just *per definition*, because
otherwise it would be impossible to have f(x)=Taylor expansion (:-)). 

However this fact does not ensure that the function *exists*. In
contrary, IF the function exits AND .. AND .. AND .. THEN there is an
expansion.

Let consider exp(x).

exp(x) * exp(x)=exp(x+x) =>

[exp(x) * exp(x)] = [exp(2x)] =

(surely, we don't want to have units depending on the magnitude)

= [exp(x)]

At the same time:

[exp(x) * exp(x)] = [exp(x)] * [exp(x)] =>

=> [exp(x)] = [exp(x)] * [exp(x)] =>

=> [exp(x)] = [1]

Now for any x, [log (exp (x))] = [x] => for any [x], [log (number)] =
[x]  ?!!

This is why exp of meters is incompatible with the requirement that
for any X, Y (real numbers), [exp (Xm)] = [exp (Ym)]. The same is
valid for sin, of course.

Regards,
Dmitry Kazakov