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

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"
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.

## Answers and Replies

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sophiecentaur
Gold Member
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.

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

Mass was originally a meaurement of distance ratios.

sophiecentaur
Gold Member
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.
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.

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

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.

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

Pythagorean
Gold Member
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?

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?
From the Wikipedia article on http://en.wikipedia.org/wiki/Dimensional_analysis" [Broken]:

The dimension of a physical quantity is the combination of the basic physical dimensions (usually mass, length, time, electric charge, and temperature) which describe it; for example, speed has the dimension length / time, and may be measured in meters per second, miles per hour, or other units. Dimensional analysis is based on the fact that a physical law must be independent of the units used to measure the physical variables.

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

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

That is irrelevant and not always true.

sophiecentaur
Gold Member
Angles are dimensionless but not unitless. The radian is a unit of angle.
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'?

Pythagorean
Gold Member
That is irrelevant and not always true.
Well yes, that's my point! What makes an angle not a dimension?

Just like x,y are an eigenbasis of the Cartesian coordinate system, r,theta represent the eigenbasis of the polar coordinate system.

That's what I thought dimensionality was all about.

Rap
Keep reading the Wikipedia article on dimensional analysis, especially the Siano extensions. In this formulation, lengths in different directions are incompatible, and have their own algebra. A radian is the ratio of two differently directed lengths and is therefore not dimensionless, but a radian squared IS dimensionless. Thus we can say, for example, that sin(θ)=θ + θ3/6 + ... in which each term is compatible, and cos(θ)=1+θ2/2+... in which each term is compatible. The extension is self-consistent and may be used to derive more information from the dimensional analysis of a problem than simply assuming the radian to be dimensionless. For example, you will never find a physically meaningful equation which contains cos(θ)+sin(θ) because the two are dimensionally incompatible.

Pythagorean
Gold Member
Keep reading the Wikipedia article on dimensional analysis, especially the Siano extensions. In this formulation, lengths in different directions are incompatible, and have their own algebra. A radian is the ratio of two differently directed lengths and is therefore not dimensionless, but a radian squared IS dimensionless. Thus we can say, for example, that sin(θ)=θ + θ3/6 + ... in which each term is compatible, and cos(θ)=1+θ2/2+... in which each term is compatible. The extension is self-consistent and may be used to derive more information from the dimensional analysis of a problem than simply assuming the radian to be dimensionless. For example, you will never find a physically meaningful equation which contains cos(θ)+sin(θ) because the two are dimensionally incompatible.
Actually, I think this is what might make it dimensionless. That the length's are independent.

In my argument for dimensionality above, I neglected that r and theta are coupled (you can't integrate over theta without defining some r, for instance), whereas in the cartesian coordinate system, x and y are completely independent.

and the infinite sum argument of course: sin(x) = x -x^3/6 + ...

x^m + x^n with m != n, x must be dimensionless.

I don't know, I'm beginning to regret not double-majoring with the maths.

Rap
Actually, in the infinite sum argument for the sine, the x need not be dimensionless, all you need is that x^(2n+1) have the same dimension for any n. By the Siano extension, they do, yet x is not dimensionless.

Actually, in the infinite sum argument for the sine, the x need not be dimensionless, all you need is that x^(2n+1) have the same dimension for any n. By the Siano extension, they do, yet x is not dimensionless.
This can only be true if the dimension of the angle is zero, -1 or 1.

Rap
That's assuming that dimensions multiply according to a group algebra, similar to integers. If Lx is a length in the x direction, then successive powers are (Lx, Lx2, Lx3 Lx4 ...) According to the Siano extension, lengths have a "directed dimension" and combine according to a different group (Klein group) i.e. sucessive powers are (Lx, 1, Lx, 1,... ). Its sort of like the cross product of vectors: Lx Ly -> Lz, but Lx Lx -> 1 and Lx Lx Lx -> (1) Lx -> Lx.

... you will never find a physically meaningful equation which contains cos(θ)+sin(θ) because the two are dimensionally incompatible.
Can someone help me understand this? seems like cos(θ)+sin(θ) is just another sine function:

$$\cos{x}+\sin{x}=\ \sqrt{2} \ \sin(x+\frac{\pi}{4})$$

I must really be missing the boat here...

Edit - fix LaTex

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It's even more generally valid.
Any combination of sin and cos of an angle can be written in terms of a phase shift of a single function.

a Sin(x) + b Cos(x) = A Sin(x+f)

where f is a phase shift.
You can do it with Cos as well.
In order to find the relationship between (a,b) and (A,f) you can use the formula for the Sin of the sum.