Re: Units (and operators.hpp, too)

From: Deane_Yang_at_[hidden]
Date: 2001-07-02 09:05:39

Amen. Thank you for a very clear explanation for what's going on.

But that argument about Taylor series is not completely rigorous. 
A skeptic might ask why you couldn't just apply the argument to 
the Taylor series of sin(x), where x is in degrees.
But the cool thing is that (assuming you buy into the Taylor 
series argument) when you look at this Taylor series,
you see that the coefficients have the right units (namely, 
degrees raised to the right negative power) to cancel out the 
powers of x. I understood the principle that "pure" transcendental 
functions take and return unitless quantities only after I saw this.

I think the justification that radians are not units coomes from the 
fact that it can be defined as a ratio of lengths. Since the 
definition does not rely on any predefined unit of angle and is 
completely independent of the unit of length, it is unitless.

--- In boost_at_y..., k.hagan_at_t... wrote:
> There still seems to be some discussion to the effect that 
decibels 
> and pH values or the arguments to exp() and log() might have 
units, 
> so I'd like to make two points.
> 
> Firstly, I would put money on the fact that any physicist who 
saw a 
> units library that handled decibels as a unit would smile and 
surf 
> over to Fermilab to use a library written by someone who "knew 
what 
> they were doing". For anyone trained as a physicist, it's a no-
> brainer. (Looking to any future standardisation, you'll never get 
> decibels past all the national bodies.)
> 
> Secondly, someone has already posted to explain why. One 
can expand 
> exp(x) as a Taylor series in powers of x. Therefore, exp(x) has 
units 
> of x + x^2 + x^3 + ... The only "x" that fits the bill is 
> dimensionless. Consequently, log(x) must also be 
dimensionless 
> because log(exp(x))==x. Similary arguments apply to all 
> transcendental functions.
> 
> If economists habitually use formulae where exp(x) sits on the 
right 
> and some dimensioned value sits on the left then that's fine. 
The 
> true formula must be something like...
> 
>    LHS = c * RHS
> 
> ...where c is numerically equal to 1 but carries the units of the 
> left hand side. In such circumstances, I might not write "c" 
either, 
> but I would strongly maintain that it is there. If you disagree, 
just 
> measure the LHS in different units, and c no longer ==1.