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Comment by hasley | Hacker News Reader

I basically agree, at least for standard functions like sin, cos, tan, exp etc. It is even possible to see mistakes in equations just by checking that all the units to standard functions cancel out making the arguments dimensionless.

On the other hand I am still unhappy with calling the ratio of two quantities, that happen to have the same units, "dimensionless". This way, you could create any two "dimensionless" quantities and try to compare them or use one in place of the other which might be meaningless.

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This is a great paper from NIST that gets into some of the problems with the limit of ‘dimensionlessness’ in metrology and the SI, and in particular issues like the fact that Hertz is considered a coherent SI unit but radian isn’t.

https://www.nist.gov/publications/dimensionless-units-si

A key takeaway:

> one conclusion that is not optional is that the unit hertz cannot be regarded as a coherent unit of the SI, in contrast to its designation in the current form of the SI, where cycles are ignored and Hz may be replaced by s^−1

Really interesting read, thank you!

One thing I found especially interesting: They argue that you can express the (complex) exponential function exp(x) as a power series with powers x^k. They do not say it explicitly, but if we assume the power series comes from a Taylor series, then the k-th factor 1/(k!) is the derivative evaluated at x=0. And the k-th derivative has exactly the unit needed to cancel the unit of x^k. So, all summands of the series are unitless and hence the exponential function's argument is unitless.

This argument would hold for any function which can be written as a series like this. I am wondering whether this is actually a "problem" of the derivative operator.

> This way, you could create any two "dimensionless" quantities and try to compare them or use one in place of the other which might be meaningless.

Case in point:

https://trac.ffmpeg.org/ticket/11279

https://trac.ffmpeg.org/ticket/11284

Yeah, "dimensionless" would mean they have equal dimension, which would mean they are comparable, which isn't necessarily the case. E.g. both radians and degrees are called "dimensionless".

Edit: Apparently "same dimension" doesn't imply "same unit".

You can also create meaningless dimensionless quantities by blindly mashing the number keys on your keyboard. Should we stop using keyboards?

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