subexpressions (OT: math)

[Steven D'Aprano]

On Sat, 02 Jun 2007 08:29:59 -0700, Steve Howell wrote:

> 
> --- Steven D'Aprano
> <steve at REMOVE.THIS.cybersource.com.au> wrote:
> 
>> On Sat, 02 Jun 2007 05:54:51 -0700, Steve Howell
>> wrote:
>> 
>> >> 
>> >>    def f(x): y = x*x; return sin(y)+cos(y);
>> >> 
>> > 
>> > Although I know valid trigonometry is not the
>> point of
>> > this exercise, I'm still trying to figure out why
>> > anybody would ever take the square of an angle. 
>> > What's the square root of pi/4 radians?
>> 
>> Approximately 0.886 radians. It corresponds to the
>> angle of a point on the
>> unit circle quite close to (sqrt(2/5), sqrt(3/5)),
>> or if you prefer
>> decimal approximations, (0.632, 0.775).
>> 
>> Angles are real numbers (in the maths sense), so
>> sqrt(pi/4) radians is
>> just as reasonable an angle as pi/4 radians. Both
>> are irrational numbers
>> (that is, can't be written exactly as the ratio of
>> two integers).
>> 
> 
> Yes, I understand that, but what is the geometrical
> meaning of the square root of an arc length?  

That's a different question to your original question, which was asking
about the square root of an angle.

> And what would the units be?  

Angles are a ratio of two lengths, and are therefore dimensionless units.
So the square root of an angle is just another angle, in the same units,
and it requires no special geometric interpretation: the square root of 25
degrees (just an angle) is 5 degrees (just another angle). 

(Note: I see that the Unix program "units" does not agree with me. It
tries to use angles as dimensionless in some contexts, but taking roots is
not one of those cases.)

Arc lengths are dimensional lengths. While you can take the square root of
a length, it doesn't have any (obvious) geometrical or physical
interpretation. One might even say it is meaningless, e.g. you can always
say that the square root of (say) 9 feet is 3 feet**(1/2), but there is no
physical meaning to that -- it doesn't correspond to anything measurable.


> If you take the square root of an
> area, the units change from acres to feet, or from
> square meters to meters.

Yes, because areas have dimension L**2, so square-rooting them has the
obvious geometrical interpretation of asking "what length, when squared,
has this area?". 

I've seen interpretations of fractional powers of length as scaling
factors for fractals. It's a nice interpretation, but not meaningful in
this context.


-- 
Steven.