In Dimensional analysis why is Lenght/Lenght=1 (a dimensionless number)?

ChARMELeOn

In Dimensional analysis if L=Lenght why is L/L=1 (a dimensionless number), and not just nothing more than L/L ?

It started with that I was thinking of why degrees do not have any dimension, and then I thought of the definition of radians:
q = "angle"
s = "a length on the circles circumference"
r = "radius"
then:
s/r=q

But now we have the dimensions L/L in the equation s/r=q wich then must be dimensionles if degrees is dimensionles. So to solve this problem I need to solve that.

It may have been a stupid question but I want to know, so I appreciate any help.

sophiecentaur

If you accept that dimensional analysis follows the rules of algebra then L/L is 1 - so no dimension.
If you can't accept that then how would you get on with resistivity, energy, stress, strain and all the others?

It's true that one person in one Lab would get the same value for a strain measurement on a given spring with an identical load, using mm, as another person, in another Lab, would get, using inches. That's what 'dimensionless' means.

Leperous

L is a number multiplied by a unit. The units cancel each other out when you divide because of simple algebra* and convenience - if you asked "how many metres in 100 metres", the answer "100 metres/metres" doesn't make any sense. It's even more insane to claim it is "100 degrees".

* You can and should treat units in an equation like algebraic terms - it avoids confusion. For example, speed = distance / time = x metres / y seconds = x/y metres/second.

AlephZero

Degrees and radians are not dimensionless. They are two different measures of something (angles), just like inches and meters are different measures of something (lengths).

The fact that you have chosen to define a radian by drawing a diagram that contains two equal lengths is irrelevant. You could just as well define a radian to be 1/(2 pi) of a complete revolution, just like you define a degree to be 1/360 of a revolution.

Pythagorean

Once you start using steradians, you recognize the need for dimensionality in radians/degrees.

Mass was originally a meaurement of distance ratios.

sophiecentaur

So can you tell me how you can have a quantity with dimension if it's put into an exponential when defining the sine of that quantity?
You'll be telling me that you can have Exp(£35.50) next.

cdotter

How can that be if dθ=s/r. Arc lengths have units of meters, radii have units of meters.

If radians weren't dimensionless then how could you, for example, take the sine of an angle in radians?

Pythagorean

If radians were dimensionless, then it would be the same thing as a steradian... which it's not.

nasu

The fact that radian and steradian are both dimensionless does not necessarily mean they are "the same". There are many things that are dimensionless. It doe not follow that they all must be the same thing.
Work and torque have both units of N m and still are not the same.

K^2

It's the definition of division. L/L = L * (L^-1) = 1 by definition of inverse element or is undefined.

Keep in mind that this isn't a number 1. It's a unit. An element that you multiply by anything and get that same anything back. That's all it means.

Dickfore

Angles are dimensionless but not unitless. The radian is a unit of angle.

Pythagorean

Hrm... How's a dimension defined? If I have a model in which one of variables is an "angle" (not necissarily spatial) isn't it one of the dimensions of my model?

Dickfore

From the Wikipedia article on dimensional analyis:

Pythagorean

I was hoping for a definition that excluded angle on principle. I don't see it if it's there.

Pythagorean

The only significant difference I can think of is the bounds (2pi rad = 0 rad)

Dickfore

That is irrelevant and not always true.

sophiecentaur

A radian is just a ratio. Would you also say that a Sine (which is still a ratio of two lengths (cm / inches / furlongs / chains) is a unit?

Is there any difference between the two 'units'?