Re: [boost] [PQS] Conjecture: The rules appled to the SI in PQS can be applied to other units sysrtems.

From: Carlo Wood (carlo_at_[hidden])
Date: 2006-06-15 09:10:00

On Thu, Jun 15, 2006 at 12:09:44PM +0100, Andy Little wrote:
> >> OK . This  seems to confirm your use of the term 'unitless' as equivalent to
> >> dimensionless or numeric. If the speed of light is a numeric type in the
> >> relativistic system it leads me to wonder what math constants such as pi
> >> would mean (if anything) in this system?
> >
> > Pi, and e and all the various others mean exactly the same things
> > they have always meant, and have exactly the same values. All I'm doing
> > is defining relationships between dimensions and selecting scales to
> > measure those dimensions. No matter what scale I choose for the
> > measurement, the ratio of the circumference of a circle to the diameter
> > doesn't change. The values of things like pi and e are in some sense
> > that I think we should avoid trying to describe in detail on this list
> > more fundamental than any unit system.
> 
> OK :-) ... I'm happy to accept that. I am not a physicist! It strikes me that
> what we are discussing is a conjecture.
> The conjecture is(something like) The rules applied to the SI system in PQS can
> be applied to the relativistic units system.
> What you say above makes the alarm bells ring for me, that the conjecture is
> false, because PQS has no mechanism for distinguishing dimensionless types.
> However not being a physicist I dont want to go there and I probably  have it
> all wrong....
> 
> ...so I'm happy not to try and prove or disprove the conjecture. I'll leave that
> to someone else :-)

Proving is a bit much :p. Although it can easily be made plausible... it's almost
axiomatic. The "rules" are that the exponent (dimension) of each unit of any
such system is the same on either side of the equal sign of any equation. This
will always be true when the units of that system are independ (and they have
to be, or it wasn't a unit system). A rule is also that you cannot add two
quantities that have different dimensions.

Relativistics or not-- those rules simply apply.

Below I write square brackets around quantities with a dimension, and
nothing around normal numbers.

[m] = m * [kg] = [m_0 / sqrt(1 - ([v]/[c])^2)]

note that '1' is dimensionless, so this should compile if ([v]/[c])^2
isn't dimensionless, and thus if [v]/[c] isn't dimensionless, and thus
if the dimension of [v] and [c] weren't the same. But they are:
[v] = v * [m/s], [c] = c * [m/s] --> [v]/[c] = v/c, thus

[m] = m * [kg] = [m_0 / sqrt(1 - (v/c)^2)] = [m_0] / sqrt(1 - (v/c)^2)

And because the dimension on either side of the '=' has to be the same,
we need a 'kg' to the power 1 on the right-hand side as well for this
to be legal. Obviously, [m_0] = m_0 * [kg], so we have: [kg] = [kg]. Check.

The only thing that is important for systems that those rules are applied
to is that it's different units cannot be expressed into eachother.

For example, if a system has units U and V, then there shouldn't be ANY
way to write: U = f(V). If that *is* possible, then you can completely
get rid of U: either U or V is not a unit, and the "system" with both
U and V isn't a unit system.

> false, because PQS has no mechanism for distinguishing dimensionless types.

All the built-in types? I'd use double (or Modular_Integer etc etc),
for them :)

-- 
Carlo Wood <carlo_at_[hidden]>