Radian measure and real numbers

[f9CSERS]

TL;DR Summary

Radian measure of any angle is a real number and vice-versa. How?

Formula used : arc length = radius × angle (in radian).

I interpreted this as:
•Taking a unit circle, we get "angle (in radian) = arc length".

This means radian measure of an angle is arc length, which can be represented on a real number line. Hence, it is a real number.

Is this way to interpret correct? (I think it's not!)

 

[Office_Shredder]

Radians are a real number, even without that stuff about arc lengths. Like, $2\pi$ is both real number and the radians of a complete circle. I'm confused by what your question

 

[BvU]

Hello @f9CSERS, !

This means radian measure of an angle is arc length

No it is not: an arc length has a dimension length, the radian is a ratio and therefore a dimensionless unit.

A ratio of quantities with dimension length can be zero or negative if we allow quantities of dimension length to be negative, which we do (cf x-coordinates).

 

[f9CSERS]

Radians are a real number, even without that stuff about arc lengths. Like, $2\pi$ is both real number and the radians of a complete circle. I'm confused by what your question

I got confused when I saw this prove in a textbook. I didn't realize that radian is ratio of quantities of same dimension and so is real number. Thanks!

 

[BvU]

Note that there are some complications in dimensionless units

The radian is defined as 1.[5] There is controversy as to whether it is satisfactory in the SI to consider angles to be dimensionless.[6] This can lead to confusion when considering the units for frequency and the Planck constant.[4][7]

 

[f9CSERS]

Note that there are some complications in dimensionless units

If I myself do not encounter any issue, till then the ratio point of view is fine. Thanks!

 

[BobF]

I believe that angles have a dimension. The ratio theory is simply wrong. While I have a longer description, here is a summary. I'd like feedback on whether this makes sense, and whether giving it a dimension works with the various formulas that you might use in your work.
First a question: what is a quick mental approximation to sin(1.57)? If an angle were dimensionless there would be an answer. If we add a label such as radian or degree then you might be able to know the answer. But these are measurements, so what they are measuring has a dimension.
An angle in radians is determined by the length of an arc of a circle divided by the radius of the circle

If I myself do not encounter any issue, till then the ratio point of view is fine. Thanks!

Summary:: Radian measure of any angle is a real number and vice-versa. How?

Formula used : arc length = radius × angle (in radian).

I interpreted this as:
•Taking a unit circle, we get "angle (in radian) = arc length".

This means radian measure of an angle is arc length, which can be represented on a real number line. Hence, it is a real number.

Is this way to interpret correct? (I think it's not!)

I encountered a problem in my work, which is why I worked on this. In automating dimensional analysis, angles wound up with no units. That was a problem. Adding a unit was awkward because other units in SI (that I was using) have dimensions, so this unit would need to be implemented differently from others. This approach creates a dimension with the usual SI properties.

The formula m/m has not retained the notion that the numerator is an arc.
A Circle is a dimension in geometrical terms. To make it part of the SI, give it a base unit of measure. How about the "arc meter", or am? Measure 1 meter along a circle of radius 1 meter, and the angle with a vertex at the center and radii touching the arc's end points is 1 radian. The formula for this is am/m.

 

[PeroK]

I believe that angles have a dimension. The ratio theory is simply wrong.

Mathematics doesn't have theories, it has definitions.

 

[BvU]

Note that there are some complications in dimensionless units

That was in november 2020. The fluid nature of Wikipedia makes the links completely worthless and confusing.

I know next to nothing about the inner workings of Wikipedia, but I can see that there have been numerous edits since then.

Since I find this an intriguing subject: Plodding through the edits one actually can dig up the whole drama (kudos to Wikipedia on that one!) Sorry to rant on and on but here goes:

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

New text (or better: currect text)

As the ratio of two lengths, the radian is a pure number. In fact, the radian is defined as 1.[4]

Still with a reference to ISO 80000-3:2006 ?!?

[​

reference ISO 80000-3:2006 is outdated and withdrawn. Now ISO 80000-3:2019 (-- probably identical for this topic--)​

page 137-138 here:​

]​

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

Old text

The radian is defined as 1. $\qquad]$[reference ISO 80000-3:2006 ]

There is controversy as to whether it is satisfactory in the SI to consider angles to be dimensionless.

[​

"SI units need reform to avoid confusion". Editorial. Nature. 548 (7666): 135. 7 August 2011.​

(lower half of this editorial)​

doi:10.1038/548135b. PMID 28796224. https://pubmed.ncbi.nlm.nih.gov/28796224​

(there seem to be multiple links to the same stuff)​

​

Deemed unpopular and insignficant​

​

​

]​

This can lead to confusion when considering the units for frequency and the Planck constant.[4][7]

[​

Mohr, J. C.; Phillips, W. D. (2015). "Dimensionless Units in the SI". Metrologia. 52 (1): 40–47. arXiv:1409.2794. Bibcode:2015Metro..52...40M. doi:10.1088/0026-1394/52/1/40. S2CID 3328342.​

​

Deemed flawed​

​

]​

[​

Mills, I. M. (2016). "On the units radian and cycle for the quantity plane angle". Metrologia. 53 (3): 991–997. Bibcode:2016Metro..53..991M. doi:10.1088/0026-1394/53/3/991.​

]​

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

These changes are dated January 23 , 2021. No more controversy.
Integrity of post #5 restored, I hope. Phew...

In short: the radian is a unit with a special name for a derived quantity that has dimension m/m

To boot: all this fuss about complications has nothing to do with the confusion of the OP

But the problems of @BobF remain. Introducing am/m renders a Taylor series for a sine illegal

$\$

 

[Mark44]

The ratio theory is simply wrong.

As already stated by @PeroK, this isn't a "theory." It's a definition.

The formula m/m has not retained the notion that the numerator is an arc.

Why is that relevant? If you take a length of string of length 6 inches, does it matter whether the string is laid out straight or wrapped along a curve?