On Wed, Apr 06, 2022 at 12:28:49AM +1200, Greg Ewing wrote:
> On 5/04/22 6:05 am, Ethan Furman wrote:
> >It seems to me that these "unitless' units actually have units, even if
> >they *appear* to cancel each other out.
>
> I think it's more that units alone don't capture everything
> that's important about the physical situation.
+1
This, a thousand times this!
Neither dimensional analysis nor units capture all the semantics of
calculations of real quantities. Both are powerful tools, but they do
not capture everything interesting in the universe.
For example, I'm going to try to draw some ASCII art of a rectangle with
a diagonal line:
+---+
| /|
| / |
|/ |
+---+
We can capture the shape of the rectangle by giving the aspect ratio,
let's say 1:2 using the convention width:height, which is another way of
writing the fraction 1/2 = 0.5. Or we can give the gradient of the
diagonal (rise over run), which is height/width, or 2. Or the angle made
by the diagonal to the base, which would be arctan(2) in radians. Or the
angle made by the diagonal to the vertical. In degrees.
All these things describe the same rectangle, but they are numerically
distinct, and tracking units "cm/cm" isn't going to tell you whether the
number you have is an aspect ratio, gradient, angle or something else.
--
Steve
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