From: Mark Allums <mark_at_allums.com>
Date: Mon, 14 Jan 2008 04:36:19 -0600

Linus Nielsen Feltzing wrote:
> Mike Holden wrote:
>> But that's precisely what "proportional" means - linearly proportional!
>>
>> To be proportional, the two values have to be always at exactly the same
>> ratio, such as y = x * 2.
>
> Well, it can also be exponentially or logarithmically proportional, as
> far as I know.
>
> Linus


A proportion is usually written in the form

y = kx + c

where k is called the "constant of proportionality".

A "proportionality" can be expressed by almost any function; the
definition of "proportional" however implies a linear function. One
possible more "general" equation might be

y = F(x) = Px + c

where P is some polynomial in some other variable, with P generally a
constant, or close to constant, for the range of values we are
interested in. This is really a "function" in two variables:

e.g.,

P(q) = s^2 + 2s + 3

y = xs^2 + 2xs + 3x + c

If s is close to 1.0 and we can assume it *stays* there, then it becomes

y = x + 2x + 3x + c

y = 6x + c


If it can be represented by an exponential, logarithmic, harmonic or
some other function, it is not strictly a "proportion", but that is just
nitpicking. It is still useful to make statements like "a is
proportional to the square root of b".

a = k(b^0.5) + c, where c == 0


And if we *know* the function that approximates the value, we can use
it, whatever it is.

At any rate, we know what you mean when you say "proportional".

:)

--Mark Allums
Received on 2008-01-14