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

In summary, in dimensional analysis, the quantity L/L, where L represents length, is considered a dimensionless number because when dividing two quantities with the same dimension, the units cancel each other out. This concept may seem counterintuitive, but it follows the rules of algebra and is necessary in order to solve problems involving different units of measurement. Additionally, angles, such as degrees and radians, are not dimensionless but are considered dimensionless units of angle. This is because they are ratios of two lengths and do not have their own algebra, unlike lengths in different directions.
  • #1
ChARMELeOn
6
0
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 which 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.
 
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  • #2
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.
 
  • #3
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.
 
  • #4
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.
 
  • #5
Once you start using steradians, you recognize the need for dimensionality in radians/degrees.

Mass was originally a meaurement of distance ratios.
 
  • #6
AlephZero said:
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.
 
  • #7
AlephZero said:
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?
 
  • #8
If radians were dimensionless, then it would be the same thing as a steradian... which it's not.
 
  • #9
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.
 
  • #10
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.
 
  • #11
Angles are dimensionless but not unitless. The radian is a unit of angle.
 
  • #12
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?
 
  • #13
Pythagorean said:
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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  • #14
I was hoping for a definition that excluded angle on principle. I don't see it if it's there.
 
  • #15
The only significant difference I can think of is the bounds (2pi rad = 0 rad)
 
  • #16
That is irrelevant and not always true.
 
  • #17
Dickfore said:
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'?
 
  • #18
Dickfore said:
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.
 
  • #19
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.
 
  • #20
Rap said:
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.
 
  • #21
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.
 
  • #22
Rap said:
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.
 
  • #23
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.
 
  • #24
Rap said:
... 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:

[tex]\cos{x}+\sin{x}=\ \sqrt{2} \ \sin(x+\frac{\pi}{4})[/tex]

I must really be missing the boat here...

Edit - fix LaTex
 
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  • #25
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.
 
  • #26
Not missing the boat as much as you think. This is a good challenge to the Siano extension. But the sum-of-angle formula is

[tex]
\sin(a+b) = \sin(a) \cos(b) + \sin(b) \cos(a)
[/tex]

which IS dimensionally consistent (sin(x) has dimension radians, cos(x) is dimensionless). So if b=π/4 then

[tex]
\sin(a+\pi/4) = \sin(a) \cos(\pi/4) + \sin(\pi/4) \cos(a)
[/tex]

which means [tex]\cos(\pi/4)=\sqrt{1/2}[/tex] is dimensionless, while [tex]\sin(\pi/4)= \sqrt{1/2}[/tex] radians. The dimensional analysis of a physically meaningful equation may give the result A*sin(a)+B*cos(a) where B/A has the dimensions of radians, and the numerical values of A and B may both be unity.
 
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  • #27
Measurements are nothing more than ratios of objects to other objects. They are there to facilitate comparisons between two different types of things. We can't compare a unit of length to a unit of momentum or a unit of magnetic field to a unit of time, so we use dimensions such as meters, teslas, coulombs, seconds, etc. to give us a common base for comparison.

Our base measurement systems are defined by certain objects or physical phenomena anyway. It only provides a reference point so we can compare a set of objects. Really, we say that a 2 kg billet of steel weighs 2 times as much as a weight locked up in a vault in France. If we compare that to a 4 kg rock, we are saying that the rock weighs twice as much as the steel. That's all we're really doing when we measure things.

We can cancel out units because it's basically stating a ratio anyway. Strain is a great way to see that. We measure elongation of a bar with a nominal length and we are dividing the former by the latter. The lengths cancel and we get a dimensionless unit. Angles do this too. We define an angle (in radians) as the ratio of a portion of a circle's arc length to its radius (s=r*theta). Since length divided by length cancels, the angle is dimensionless.

As far as the steradian and the radian, they are not the same thing because they are different ratios. The radian is used to compare arc length of a circle to its radius whereas the steradian is used to compare areas of a sphere to the square of the radius.
 
  • #28
Rap said:
...sin(x) has dimension radians, cos(x) is dimensionless...

This is something strange to me, I can't remember if I ever heard, or thought about this before. Does this follow from the series:

[tex]\sin{x}=x-...[/tex]

whereas,

[tex] \cos{x}=1-...[/tex]

That's kind of weird, let me think about that for awhile...
 
  • #29
Question:

What is the dimensionality of:

[tex]
\exp{(x)}
[/tex]

if [itex]x[/itex] is in radians?
 
  • #30
Dickfore said:
Question: What is the dimensionality of: [tex]\exp{(x)}[/tex] if [itex]x[/itex] is in radians?

Using the Siano extension, expanding [tex]\exp{(x)}[/tex] you get 1+x+x^2/2 + ... and these terms are incompatible (1 is dimensionless, x is radians, x^2 is dimensionless ...)

That means that no physically meaningful equation will yield this expression. On the other hand, [tex]\exp{(ix)}[/tex] IS meaningful, if x is in radians. That means that "ix" is dimensionless, as is exp(ix). According to the Wikipedia article "Dimensional analysis / Siano's extension", that means that "the complex quantity i has an orientation equal to that of the angle it is associated with". I think maybe a better way to say it is that expressions of the form exp(Ax), where x is in radians, are physically meaningful as long as A has an imaginary numerical value and has units of radians, oriented in the same direction as x. Note that exp(ix)=cos(x)+i sin(x) is dimensionally consistent, yielding a dimensionless value.
 
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  • #32
i is an interesting number. Particularly useful to this discussion is it's geometric definition: as an orthogonal transform.

Still not sure what it means for i to have a spatial orientation. Does that mean the imag axis ceases to be orthogonal to the real axis?
 
  • #33
Dickfore said:

The coefficient of static friction is [tex]\mu=F_r/F_t[/tex] where Fr is the force directed normal to the surface (i.e radially) and Ft is the frictional force, directed parallel to the surface (i.e. tangentially). Since these forces act in different directions, they have incompatible dimensions, and so the coefficient of friction is not dimensionless. Let's call the direction units 1r, 1t, and 1n for radial, tangential, and normal (normal to the plane of 1r and 1t), and let's call 10 dimensionless. Using Siano's extension, the angle direction is perpendicular to the plane of the angle, that is 1n. By Siano, the coefficient of static friction is directed as 1r/1t = 1n and so the product of the coefficient of friction times the angle is directed as 1n 1n = 10 (i.e. dimensionless), and so the term in the exponential is dimensionless and everything is ok.

Pythagorean said:
i is an interesting number. Particularly useful to this discussion is it's geometric definition: as an orthogonal transform.

Still not sure what it means for i to have a spatial orientation. Does that mean the imag axis ceases to be orthogonal to the real axis?

No, I think it means that it is oriented in the same direction as the angle it is multiplying, which is perpendicular to the plane of the angle. I don't have a clear mental picture of how it works yet. In the case of directed quantities, I think I understand the Siano extension, but in this case, I am just using the rules and noting that I cannot find an inconsistency.
 
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  • #34
Rap said:
The coefficient of static friction is [tex]\mu=F_r/F_t[/tex] where Fr is the force directed normal to the surface (i.e radially) and Ft is the frictional force, directed parallel to the surface (i.e. tangentially). Since these forces act in different directions, they have incompatible dimensions, and so the coefficient of friction is not dimensionless. Let's call the direction units 1r, 1t, and 1n for radial, tangential, and normal (normal to the plane of 1r and 1t), and let's call 10 dimensionless. Using Siano's extension, the angle direction is perpendicular to the plane of the angle, that is 1n. By Siano, the coefficient of static friction is directed as 1r/1t = 1n and so the product of the coefficient of friction times the angle is directed as 1n 1n = 10 (i.e. dimensionless), and so the term in the exponential is dimensionless and everything is ok.

How about the result of the exponential? The left hand side is a ratio of the magnitudes of two forces directed at an angle [itex]\theta[/itex].
 
  • #35
Sorry, I don't understand "result of the exponential". [tex]exp(\mu \theta)[/tex] is dimensionless because [tex]\mu \theta[/tex] is dimensionless. Also, I don't understand, the left side of what equation? [tex]\mu=F_r/F_t[/tex]? The left hand side of that is the directed radial force divided by the directed tangential force, which are at right angles.
 
<h2>1. Why is length/length equal to 1 in dimensional analysis?</h2><p>Dimensional analysis is a mathematical method used to convert between different units of measurement. In this method, we use the fundamental dimensions of a physical quantity (such as length, time, mass, etc.) to create a dimensional equation. When we divide a quantity by itself, the resulting value is always 1. Therefore, length/length is equal to 1 in dimensional analysis.</p><h2>2. What is the significance of a dimensionless number in dimensional analysis?</h2><p>A dimensionless number is a numerical value that does not have any units associated with it. In dimensional analysis, we use dimensionless numbers to compare and analyze physical quantities without being affected by their units. This allows us to focus on the underlying relationships between different quantities and make accurate predictions and calculations.</p><h2>3. Can we use any units for length in dimensional analysis?</h2><p>No, we cannot use any units for length in dimensional analysis. The units used for length must be consistent throughout the entire equation. For example, if we are using meters for length, then all other quantities in the equation must also be expressed in meters. This ensures that the final result is in the correct units and has a dimensionless value.</p><h2>4. How is dimensional analysis used in scientific research?</h2><p>Dimensional analysis is used in scientific research to simplify complex equations and make predictions about physical phenomena. It is particularly useful in fields such as physics, chemistry, and engineering, where precise measurements and unit conversions are crucial. By using dimensional analysis, scientists can identify relationships between different quantities and make accurate predictions without the need for extensive experimentation.</p><h2>5. Are there any limitations to dimensional analysis?</h2><p>While dimensional analysis is a powerful tool, it does have some limitations. It can only be used for physical quantities that have a measurable dimension, such as length, time, mass, etc. It also cannot account for any non-dimensional factors that may affect the outcome of an experiment. Additionally, dimensional analysis is only applicable to linear relationships between quantities and may not be suitable for more complex systems.</p>

1. Why is length/length equal to 1 in dimensional analysis?

Dimensional analysis is a mathematical method used to convert between different units of measurement. In this method, we use the fundamental dimensions of a physical quantity (such as length, time, mass, etc.) to create a dimensional equation. When we divide a quantity by itself, the resulting value is always 1. Therefore, length/length is equal to 1 in dimensional analysis.

2. What is the significance of a dimensionless number in dimensional analysis?

A dimensionless number is a numerical value that does not have any units associated with it. In dimensional analysis, we use dimensionless numbers to compare and analyze physical quantities without being affected by their units. This allows us to focus on the underlying relationships between different quantities and make accurate predictions and calculations.

3. Can we use any units for length in dimensional analysis?

No, we cannot use any units for length in dimensional analysis. The units used for length must be consistent throughout the entire equation. For example, if we are using meters for length, then all other quantities in the equation must also be expressed in meters. This ensures that the final result is in the correct units and has a dimensionless value.

4. How is dimensional analysis used in scientific research?

Dimensional analysis is used in scientific research to simplify complex equations and make predictions about physical phenomena. It is particularly useful in fields such as physics, chemistry, and engineering, where precise measurements and unit conversions are crucial. By using dimensional analysis, scientists can identify relationships between different quantities and make accurate predictions without the need for extensive experimentation.

5. Are there any limitations to dimensional analysis?

While dimensional analysis is a powerful tool, it does have some limitations. It can only be used for physical quantities that have a measurable dimension, such as length, time, mass, etc. It also cannot account for any non-dimensional factors that may affect the outcome of an experiment. Additionally, dimensional analysis is only applicable to linear relationships between quantities and may not be suitable for more complex systems.

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