"Strong typing vs. strong testing"

[Dann Corbit]

In article <i930ek$uvp$1 at news.eternal-september.org>, bc at freeuk.com 
says...
> 
> "RG" <rNOSPAMon at flownet.com> wrote in message 
> news:rNOSPAMon-20651E.17410012102010 at news.albasani.net...
> > In article <i92dvd$ada$1 at news.eternal-september.org>,
> > "BartC" <bc at freeuk.com> wrote:
> >
> >> "Thomas A. Russ" <tar at sevak.isi.edu> wrote in message
> 
> >> > But radians are dimensionless.
> >>
> >> But they are still units
> >
> > No, they aren't.
> >
> >> so that you can choose to use radians, degrees or gradians
> >
> > Those aren't units either, any more than a percentage is a unit.  They
> > are just different ways of writing numbers.
> >
> > All of the following are the same number written in different notations:
> >
> > 0.5
> > 1/2
> > 50%
> >
> > Likewise, all of the following are the same number written in different
> > notations:
> >
> > pi/2
> > pi/2 radians
> > 90 degrees
> > 100 gradians
> > 1/4 circle
> > 0.25 circle
> > 25% of a circle
> > 25% of 2pi
> >
> > See?
> 
> But what exactly *is* this number? Is it 0.25, 1.57 or 90?
> 
> I can also write 12 inches, 1 foot, 1/3 yards, 1/5280 miles, 304.8 mm and so 
> on. They are all the same number, roughly 1/131000000 of the polar 
> circumference of the Earth.

Those are all units of length.

> This does depend on the actual size of an arbitrary circle, but that seems 
> little different from the choice of 0.25, 1.57 or 90 for your quarter 
> circle.

Radians are arc length divided by radius.  Both of those can have units 
attached (e.g. centimeters).  But when you divide two lengths, the 
length units cancel out and you are left with a pure number.

However, you can think of it as units of rotation measure if you find it 
pleasing.

Suppose that you work for $50/Hr.  If you work for 10 Hours the wage is 
$50 / Hr * 10 * Hr = $500.  Notice that the hours have dissappeared from 
the equation because they cancelled out.  

In a similar way, that's why radians is commonly called a dimentionless 
unit.  The two lengths have cancelled out.

But in a very real sense it is a measure of rotation.  We could call it 
a special measure, sort of like the way that e is a special base 
compared to all others.

Tempest in a teapot, at any rate.