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0 points6gvONxR4sf7o2y ago0 comments

Sure, but that's orthogonal to the "angles don't really have units" assertion and the "it does make sense to add an angle to its cube" assertion, which are the ones I'm responding to.

As another example for the second assertion, you can compute e(-t) via power series too, adding seconds to seconds squared and seconds cubed, etc, which comes up all the time. But that doesn't mean `dimensionless + seconds + seconds^2` implies seconds are dimensionless any more than sin's series with `angle + angle^3` implies that angles are dimensionless.

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orangecat2y ago

you can compute e(-t) via power series too, adding seconds to seconds squared and seconds cubed

The argument of exp does have to be dimensionless, exactly because adding seconds to seconds squared doesn't work. If t has units of time, there has to be another factor with units of inverse time, for example continuous compound interest is exp(rate*time).

eigenket2y ago

If you say "angles have units" I agree with you - obviously you can measure them in degrees or radians or whatever you want. I was responding to the claim

> Angles aren't dimensionless any more than lengths are dimensionless

They are dimensionless, but they still have units. The concepts are orthogonal.

As for the question about adding an angle to its cube, I would say the enormous usefulness of computing trig functions by power series suggests strongly that this is meaningful.

adrian_b2y ago

> They are dimensionless, but they still have units. The concepts are orthogonal.

The concepts are not orthogonal, they are incompatible.

A dimensionless quantity (which angles are not) is by definition the ratio of two quantities that are measured with the same unit.

When you compute the ratio by division, the two identical units disappear from the result, therefore the result is indeed dimensionless.

There is no way to choose a unit for a dimensionless quantity in the usual sense.

At most you could define a new different dimensionless quantity, as the ratio of two dimensionless quantities, i.e. as a ratio of ratios, but because it needs a different definition this should better be viewed as a different quantity, not as the same quantity with a different unit.

eigenket2y ago

Ok, you've replied to quite a lot of my comments, with some fairly confusingly worded responses. Some of your claims appear to be incompatible as far as I read them.

For example you claim that angles are not dimensionless, but that dimensionless quantities are formed by the ratio of two quantities with the same unit. Since the angle subtended by an arc in a circle is the ratio of the arc length and radius, it would seem that these two claims contradict each other.

I do agree that without some aditional work the power series argument I wrote above does seem to be wrong.

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6gvONxR4sf7oOP2y ago

> I would say the enormous usefulness of computing trig functions by power series suggests strongly that this is meaningful.

That argument holds just as true for dimensional quantities frequently computed by power series though, which means it can't be valid.

eigenket2y ago

I would argue that dimensionful quantities are never used as arguments to power series, however you are correct that this does not imply that angles are dimensionless, since (like we do with other quantities, we can divide by whatever unit we like to get something dimensionless). I withdraw that argument.

A better argument that angles are dimensionless is that dimensionless quantities are formed by the ratio of two quantities with the same dimension, and the angle subtended by an arc in a circle is given by the ratio of the arc length and the radius.

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orangecat2y ago

you can compute e(-t) via power series too, adding seconds to seconds squared and seconds cubed

eigenket2y ago

If you say "angles have units" I agree with you - obviously you can measure them in degrees or radians or whatever you want. I was responding to the claim

> Angles aren't dimensionless any more than lengths are dimensionless

They are dimensionless, but they still have units. The concepts are orthogonal.

As for the question about adding an angle to its cube, I would say the enormous usefulness of computing trig functions by power series suggests strongly that this is meaningful.

adrian_b2y ago

> They are dimensionless, but they still have units. The concepts are orthogonal.

The concepts are not orthogonal, they are incompatible.

A dimensionless quantity (which angles are not) is by definition the ratio of two quantities that are measured with the same unit.

When you compute the ratio by division, the two identical units disappear from the result, therefore the result is indeed dimensionless.

There is no way to choose a unit for a dimensionless quantity in the usual sense.

eigenket2y ago

Ok, you've replied to quite a lot of my comments, with some fairly confusingly worded responses. Some of your claims appear to be incompatible as far as I read them.

I do agree that without some aditional work the power series argument I wrote above does seem to be wrong.

1 more reply

6gvONxR4sf7oOP2y ago

> I would say the enormous usefulness of computing trig functions by power series suggests strongly that this is meaningful.

That argument holds just as true for dimensional quantities frequently computed by power series though, which means it can't be valid.

eigenket2y ago

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