what is dimensional analysis?

[univeral theory]

The foundation of dimensional analysis comes from the need to compare two seemingly different phenomena that describe the same phenomena in relative terms and see whether they are equivalently the same in absolute terms. In mathematics, physics or any other scientific discipline,
analyzing a formula with absolutely same variables at both sides of its equation makes no meaning; reason being that the formula is dimensionless and thus does not qualify to be a true equation. For example; any formula in the form of 9=9 or E=E (suppose that E represents energy) is dimensionless because variables that describe the same phenomena have been described in their
absolute term from both sides of the equation.

A true equation is subject to dimensional analysis and that is why it is called an equation and because of dimensional analysis; any meaningful formulation of the equation must be dimensional. A dimensional formulation is where variables from one or both sides of the equation that describe the same phenomena are dimensional, for example; E=MC² is a formula that relates energy in

both its absolute and relative terms. That is to say, E is a variable that describes energy in its absolute terms and its formulation is dimensionless, while MC² are variables that describe energy in relative terms from the reference frame of mass and the speed of light and their formulation is
dimensional. Thus E=MC² is a true equation in its formulation sense and we can only check whether it is correct by converting MC² to E variable(which is the basic variable of energy described in absolute terms). And it can only be correct if E=E, remember that E=E is just the proof of the correctness of the equation but not its true equation sense.

Remember that we have been using the E=MC² example which is a physical phenomena based equation. But its scenario applies equally to the arithmetical quantity based equations, for example; 6+3=9 or 5+4=6+3. In fact, 6+3=9 or 5+4=6+3 they are just different dimensional framework of describing the same phenomena and our task in dimensional analysis is to convert these relative

variables into the basic variable of the formula that describe the phenomena in absolute terms and see whether both sides of the equation equal. And here, 5+4 or 6+3 will be converted into 9 and see whether 6+3=9 or 5+4=6+3.

A dimensional equation differs from a dimensionless equation in that; the former describes a

given absolute phenomena in terms of its relative variables that constitute its basic dimensional framework, while the later is just an absolute variable that describes the phenomena. But they do not differ because one can not represent physical units or the other can not describe arithmetical quantity.

The above description is my personal point of view as far as dimensional analysis

is concerned. It differs from that of my tutor and I have found problems with him as far as my approach is concerned. If am right; please make a comment or a supplement. And if am wrong please correct me with some mathematically explained and consistent examples.

[Klaynos]

You're comment that E is dimensionless is wrong, as explained to you in a different thread energy is measured in Joules, which has dimensions.

 

Equations you describe as dimensionless, E=mc² is not dimensionless but the dimensions balance.

 

Also as explained to you other places an equation that passes dimensional analysis is not definitely correct but if it fails then it is definitely not correct.

[Iggy]

A true equation is subject to dimensional analysis and that is why it is called an equation and because of dimensional analysis; any meaningful formulation of the equation must be dimensional. A dimensional formulation is where variables from one or both sides of the equation that describe the same phenomena are dimensional, for example; E=MC² is a formula that relates energy in

both its absolute and relative terms. That is to say, E is a variable that describes energy in its absolute terms and its formulation is dimensionless, while MC² are variables that describe energy in relative terms from the reference frame of mass and the speed of light and their formulation is

dimensional. Thus E=MC² is a true equation in its formulation sense and we can only check whether it is correct by converting MC² to E variable(which is the basic variable of energy described in absolute terms). And it can only be correct if E=E, remember that E=E is just the proof of the correctness of the equation but not its true equation sense.

 

I'm afraid that isn't how the terms are understood. Energy is not dimensionless. A dimensionless quantity is the same regardless of what units you decide to measure it in. So, for example, mass (specifically say the mass of a boat) is dimensionful because you could use units of kilograms or you could use units of bananas. A boat weighs more bananas than kilograms.

 

Energy is dimensionful, you know, because the energy (let's specifically say the gravitational potential energy gained by lifting a banana 2 feet off the floor) is different if you express it in joules as it is when you express it in ergs.

 

The dimensionality of energy is mass * length * length / time / time. If energy were dimensionless then those would cancel.

 

An example of a dimensionless quantity is Mach number (let's say the Mach number of a North Korean missile targeting a banana).

 

[math]M=\frac{v}{a}[/math]

 

where v is the velocity of the missile relative to the air and a is the speed of sound in the air. Both are measured in speed:

 

[math]M=\frac{\mbox{speed}}{\mbox{speed}}[/math]

 

and speed is length divided by time

 

[math]M=\frac{\mbox{length / time}}{\mbox{length / time}}[/math]

 

You know that M is dimensionless because M stays the same no matter what units you pick. You could measure length with a meter stick or a banana, and you could measure time in seconds or banana growing seasons. The number you get for M will be the same either way.

 

The last thing...

 

 

For example; any formula in the form of 9=9 or E=E (suppose that E represents energy) is dimensionless because variables that describe the same phenomena have been described in their absolute term from both sides of the equation.

 

Imagine that for every distance you could run, I could only run half that distance. X is however far you move and Y is however far I move. X=2Y would always be true. X and Y have the same dimensionality (length) but you don't cancel them and say that the equation is dimensionless. That wouldn't make sense to cross the equals sign in that way. You would just say that both quantities have the same dimensionality.

 

EDIT:

 

Sorry to repeat you Klaynos. I was typing when you posted.

Edited February 2, 2013 by Iggy