From: dmitry@elros.cbb-automation.de (Dmitry A. Kazakov)
Subject: Re: Dimensionality Checking (Ada 20XX)
Date: Fri, 14 Dec 2001 10:16:44 GMT
Date: 2001-12-14T10:16:44+00:00 [thread overview]
Message-ID: <3c19c13f.3311687@News.CIS.DFN.DE> (raw)
In-Reply-To: 3C18DF39.9BB2F868@mail.chem.sc.edu
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
next prev parent reply other threads:[~2001-12-14 10:16 UTC|newest]
Thread overview: 78+ messages / expand[flat|nested] mbox.gz Atom feed top
2001-12-07 0:09 Dimensionality Checking (Ada 20XX) Snodgrass, Britt (NM75)
2001-12-07 16:15 ` Ian
2001-12-09 17:58 ` Nick Roberts
2001-12-09 22:58 ` Nick Roberts
2001-12-10 0:17 ` Mark Lundquist
2001-12-10 1:51 ` James Rogers
2001-12-10 3:33 ` Nick Roberts
2001-12-10 19:09 ` Nick Roberts
2001-12-11 8:20 ` Thomas Koenig
2001-12-11 15:37 ` Nick Roberts
2001-12-11 20:18 ` Thomas Koenig
2001-12-12 0:58 ` Mark Lundquist
2001-12-12 8:19 ` Wilhelm Spickermann
2001-12-12 14:21 ` Stephen Leake
2001-12-12 19:10 ` Nick Roberts
2001-12-13 19:04 ` Stephen Leake
2001-12-13 22:56 ` Nick Roberts
2001-12-14 0:11 ` Nick Roberts
2001-12-14 22:14 ` Mark Lundquist
2001-12-15 1:30 ` Nick Roberts
2001-12-10 20:22 ` Thomas Koenig
2001-12-10 17:21 ` Wes Groleau
2001-12-10 19:51 ` Mark Lundquist
2001-12-10 19:56 ` Wes Groleau
2001-12-10 20:37 ` Mark Lundquist
2001-12-10 18:56 ` Nick Roberts
2001-12-11 15:05 ` Wes Groleau
2001-12-11 16:39 ` Stephen Leake
2001-12-11 19:05 ` Nick Roberts
2001-12-11 22:50 ` Mark Johnson
2001-12-12 1:59 ` Nick Roberts
2001-12-11 23:01 ` Stephen Leake
2001-12-12 2:21 ` Nick Roberts
2001-12-12 14:16 ` Stephen Leake
2001-12-13 19:52 ` Nick Roberts
2001-12-13 22:22 ` Nick Roberts
2001-12-14 6:40 ` Robert C. Leif, Ph.D.
2001-12-14 17:30 ` Stephen Leake
2001-12-14 17:38 ` Stephen Leake
2001-12-11 22:45 ` Mark Lundquist
2001-12-12 1:42 ` Nick Roberts
2001-12-12 15:17 ` Mark Lundquist
2001-12-12 14:03 ` Stephen Leake
2001-12-12 9:35 ` Dmitry A. Kazakov
2001-12-12 14:26 ` Stephen Leake
2001-12-13 17:02 ` daniele andreatta
2001-12-13 19:06 ` Stephen Leake
2001-12-14 10:16 ` Dmitry A. Kazakov [this message]
2001-12-14 22:01 ` Nick Roberts
2001-12-17 11:10 ` Dmitry A. Kazakov
2001-12-17 12:16 ` Thomas Koenig
2001-12-17 14:30 ` Dmitry A. Kazakov
2001-12-27 17:18 ` Steven Deller
2001-12-15 7:07 ` Steven Deller
2001-12-17 12:31 ` Dmitry A. Kazakov
2001-12-17 13:46 ` Thomas Koenig
2001-12-17 15:00 ` Dmitry A. Kazakov
2001-12-17 16:38 ` Thomas Koenig
2001-12-17 21:07 ` Britt Snodgrass
2001-12-20 13:44 ` Dmitry A. Kazakov
2001-12-13 19:33 ` Mark Lundquist
2001-12-13 22:15 ` Nick Roberts
2001-12-14 20:20 ` Mark Lundquist
2001-12-10 23:31 ` Mark Lundquist
2001-12-10 13:57 ` Ian
2001-12-10 17:24 ` Wes Groleau
2001-12-10 20:38 ` Britt Snodgrass
-- strict thread matches above, loose matches on Subject: below --
2001-12-11 13:11 Mike Brenner
2001-12-11 17:03 ` Mark Lundquist
2001-12-02 16:01 Another Idea for Ada 20XX James Rogers
2001-12-03 14:56 ` Mark Lundquist
2001-12-03 15:12 ` Lutz Donnerhacke
2001-12-03 21:13 ` Dimensionality Checking (Ada 20XX) Nick Roberts
2001-12-04 14:00 ` Dmitry A. Kazakov
2001-12-06 19:52 ` Britt Snodgrass
2001-12-06 20:55 ` Mark Lundquist
2001-12-06 22:38 ` Wes Groleau
2001-12-06 23:12 ` Mark Lundquist
2001-12-07 14:36 ` Wes Groleau
2001-12-07 9:37 ` Dmitry A. Kazakov
2001-12-07 22:51 ` Mark Lundquist
replies disabled
This is a public inbox, see mirroring instructions
for how to clone and mirror all data and code used for this inbox