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Date: Thu, 13 Dec 2001 12:02:49 -0500
From: daniele andreatta <andreatta@mail.chem.sc.edu>
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Subject: Re: Dimensionality Checking (Ada 20XX)
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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]

HTH,
Daniel