On Tue, Apr 05, 2022 at 04:36:24AM +1000, Chris Angelico wrote:

> Some unit cancellations really do result in pure scalars. The ratio of
> a circle's circumference to its diameter isn't a unit of m/m any more
> than the ratio of a circle's area to that of a circumscribed square is
> a unit of m²/m². They're both just numbers.

Of course it is a ratio. You said it yourself: it is a ratio of 
circumference to diameter.

That ratio is only numerically equal to π if the units you measure the 
cicumference and diameter are the same. Otherwise it has units 
"inches/cm" (or whatever units you used) and a completely different 
numerical value.

> On the other hand, a
> radian is a very real unit of distance/distance (based on its
> definition of arc length), and it's a unit of angle.

The SI system defines both radians and steradians as dimensionless 
derived units, previously known as "supplementary units".


-- 
Steve
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