# Dimensionless units

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## Main Question or Discussion Point

The topic has been discussed before at PF, in particular @haruspex's Insight Can angles be assigned a dimension?

Nature has now published an editorial on the subject: SI units need reform to avoid confusion
Nature said:
For example, radians could be made a new SI unit, and the unit 1 could be formally coupled with notation that includes the type of quantity that it represents.

## Answers and Replies

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.Scott
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Let's see...
The length ##l## (in meters) of an arc of angle ##a## (in radians) and radius ##d## (in meters) would then be:
## l (meters) = d (meters) \times a (radians) \times (1/radians) ## with the ##(1/radian)## being the conversion factor from angle to circumference.
The sine, cosine, and tangent of an angle would still be unitless values, since they are ultimately ratios. But they would need to operate on only radian values. So ##\sin(\sin(x))## would be meaningless - you would need to provide a radian conversion value - presumably one evident from the application.

Sounds like a potentially useful discipline. But not one that I am going to adopt any time soon.

This sounds to me like somebody with way too much time on their hands. Why meddle with something that works just fine as it is? I fail to see any problem with the present situation.

Specifically regarding SI, I do see room for improvement there. The meter is a bit to large for many dimensional measurements while way too small for many others (yes, I now about powers of 10 and all that). The Newton is too small for many purposes. We see the problem really come to light in derived units like pressure or stress; 1 Pa = 1 N/m^2 is really, really small, so once again, we are back to megaunits, or bastard units like 1 bar = 10^5 Pa.

The philosophical origins of the meter and other SI units may have an appeal, but the results are not very handy. Tying the meter to a quadrant of the earth's circumference was a nifty idea, but who measures on such grand scales for ordinary, everyday dimensions? A foot or an inch are not so grand, but they do approximate the lengths of thing we see every day.

I have no illusions that anyone would dare to challenge anything so well established as the SI system, but it is still horribly unhandy.