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From: Kevin Lynch (krlynch_at_[hidden])
Date: 2001-07-03 14:10:38

> Message: 9
> Date: Mon, 02 Jul 2001 14:05:39 -0000
> From: Deane_Yang_at_[hidden]
> Subject: Re: Units (and operators.hpp, too)
> 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.

But in fact quite the opposite is true. The Taylor series argument is
the clearest demonstration that all of the exponential, circular, and
hyperbolic functions take dimensionless arguments. The coefficients of
a Taylor as you say are pure numbers, uncontaminated by dimensionless

exp(x) = Sum i=0->Infinity x^i / i!

which is analytic everywhere (converges for all values of x). There are
no dimensions anywhere, there is no coefficients taken to any powers,
and there is certainly no correction of the units here.... and since all
of the other transcendental functions of the C++ SL can be expressed in
terms of exp(x), the same applies to them.

To repeat again: If you are applying these functions to a quantity that
has units, you are doing something wrong, even if you don't immediately
realize it. And any economics or finance formula where you think you
are doing this, you really are not; there is an (incorrectly) unstated
"non-dimensionalizing factor" that appropriately removes the units and
another (incorrectly) unstate "dimensionalizing factor" that puts the
units back in after the fact. OR you are using a unit (such as radian)
that has no actual dimensions. The unit of the radian, for example, is
1. It is a unit, but not a dimensionful unit. Same for degrees, and
grads, and steradians, and bels, and pH. They are units without
dimensions, and as such you can do things like take their sin and their
log. But you CAN NOT take the sin of $5, for instance, or 9meters, or
any other dimensionful unit.

> 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.

As I mentioned above, a radian IS a unit, but it is DIMENSIONLESS. They
are not the same thing, even though they are often mixed.

Kevin Lynch				voice:	 (617) 353-6065
Physics Department			Fax: (617) 353-6062
Boston University			office:	 PRB-565
590 Commonwealth Ave.			e-mail:	 krlynch_at_[hidden]
Boston, MA 02215 USA

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