[CrudeHypothesis]

I came up with this hypothesis:
You know how all equations have to be dimensionally homogeneous, as in, all terms of an equation have to have the same dimensions in order to be added together. Think of the implications of that. If you knew the dimensions of every variable, you could infer the equations to calculating them without having previously memorized the formulae, simply by thinking about the dimensions of the variable in terms of the other variables. This is my hypothesis on why one can get 100% on a maths based test without studying, simply by thinking about what the question is actually asking.

[PlatoHagel]

Hmmmm......I just came back to start a new thread and I seen yours. I was going to name it,
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, because of course of the link that I was going to provide for examination.

	Originally Posted by CrudeHypothesis
	If you knew the dimensions of every variable, you could infer the equations to calculating them without having previously memorized the formulae, simply by thinking about the dimensions of the variable in terms of the other variables.

So with the perspective of your hypothesis how would this link provide a truth for what you are saying?

This question of dimensional relevance can have many con nations and of course I titled some of these as well as tried to explore how artistic rendition could have examined the idea around them. Cubists. A Salvador Dali in painting "the
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" helped me to ponder what he may of thought about the tesseract as a means to question
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?

[CrudeHypothesis]

	Originally Posted by PlatoHagel To view links or images in this forum your post count must be 2 or greater. You currently have 0 posts.
	Hmmmm......I just came back to start a new thread and I seen yours. I was going to name it, To view links or images in this forum your post count must be 2 or greater. You currently have 0 posts. , because of course of the link that I was going to provide for examination. So with the perspective of your hypothesis how would this link provide a truth for what you are saying?

My internet is slow right now, so I'll check it out later. I got high in the class on a recent test I didn't study for, and I had no idea what the test was about, I just remembered the symbols associated to the words in the question, wrote them down, wrote down their dimensions, wrote an equation and solved the unknowns.

[PlatoHagel]

I know a few others on this forum who can benefit from, besides the expression of your question about and development as a hypothesis. Good educators can correct as we go along here to help make the process of your hypothesis a good one.

But while in context of this thread, I would also like to consider more about dimensional examination and do it in context of your thread. I can break away from it if you like?

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Can we say that the resulting reality here we are all engaged in is a condense feature of something much more then the objectifications, as Matter defined?

[Daoist]

I started a thread recently on the same question just within the realm of economics.
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This is usually a good approach...but it becomes less useful when you are adding up different assorted things - for instance, if you have a batch of x, another batch of y, another batch of z, batches might become less useful as a unit. But it can be sort of subjective. Units are very useful for any sort of algebra or calculus-related problem; less so for the sorts of things you tend to see in computer science.

[roninpro]

Originally Posted by CrudeHypothesis To view links or images in this forum your post count must be 2 or greater. You currently have 0 posts.

I came up with this hypothesis: You know how all equations have to be dimensionally homogeneous, as in, all terms of an equation have to have the same dimensions in order to be added together. Think of the implications of that. If you knew the dimensions of every variable, you could infer the equations to calculating them without having previously memorized the formulae, simply by thinking about the dimensions of the variable in terms of the other variables. This is my hypothesis on why one can get 100% on a maths based test without studying, simply by thinking about what the question is actually asking.

Here is a counterexample: suppose you are trying to determine the equation for the area of a circle. You know that it has to contain the square of a length. Which length is it? The radius? Some special chord of the circle (that is not the diameter)? What about the constant of proportionality? How would you go about determining that exactly?

The point here is that dimensional analysis can give you some intuition about what the formula might contain, but it does not replace honest experimentation, deduction, and an actual proof.

[Warrior]

Originally Posted by CrudeHypothesis To view links or images in this forum your post count must be 2 or greater. You currently have 0 posts.

I came up with this hypothesis: You know how all equations have to be dimensionally homogeneous, as in, all terms of an equation have to have the same dimensions in order to be added together. Think of the implications of that. If you knew the dimensions of every variable, you could infer the equations to calculating them without having previously memorized the formulae, simply by thinking about the dimensions of the variable in terms of the other variables. This is my hypothesis on why one can get 100% on a maths based test without studying, simply by thinking about what the question is actually asking.

"I cannot give credit to an answer that is dimensionaly incorrect."

That was the favorite saying of one of my college professors and he would, in fact, count a problem entirely wrong if you had the wrong dimensions on your answer. He would, however, give you some credit if you got the dimensions right and everything else wrong.

It helps and can give you some insight into a problem. The basic concept is taught in third grade. My son did a math section on it last year.

[Latro]

There is actually a very precise theorem that tells you how much information dimensional analysis can give. It's called the Buckingham Pi Theorem. It shows that a lot can be found, but not everything can be found. In particular it shows that even if there are few enough variables compared to dimensions that you can determine exactly how the variables can appear in the formula (which is only the case in fairly trivial situations), you cannot determine the constant of proportionality without doing some actual math and/or physics.

[CrudeHypothesis]

Originally Posted by Latro To view links or images in this forum your post count must be 2 or greater. You currently have 0 posts.

There is actually a very precise theorem that tells you how much information dimensional analysis can give. It's called the Buckingham Pi Theorem. It shows that a lot can be found, but not everything can be found. In particular it shows that even if there are few enough variables compared to dimensions that you can determine exactly how the variables can appear in the formula (which is only the case in fairly trivial situations), you cannot determine the constant of proportionality without doing some actual math and/or physics.

So, take (n) parameters and subtract (j) number of dimensions yields (k) pi groups. Choose scaling variables...

[Monte314]

CH's approach will work if the formula is a ratio of sums-of-products where all the product factors have the same multi-index of exponents. It will also work in certain other similar cases.

It will not work if the functional form is a power series or infinite product, e.g., is a transcendental function such as sin(x) or log(x). For example, sin(30 degrees) is 0.5, and the 0.5 is regarded as being unitless. So, what happened to the "degrees"? Degrees is a bona fide unit, as you will discover if you forget that your calculator is in "radian mode" and punch in 30.

What CH is saying here is that dimensional analysis can be used to infer candidate models in a finite-dimensional model space. This is how Johannes Rydberg determined his formula for the wavelengths of spectral emission lines (published in 1888). The result is an empirical formula that appears to fit the data, but has no underlying theoretical basis... like macro-evolutionary theories, for example.

[The Dan Keizer]

Could you demonstrate this because I don't get it.

[Latro]

Let L be the dimension of length, T the dimension of time, and M the dimension of mass. Now for some examples:
The area of a circle is some function of its radius, and no other parameters. (This is not part of the analysis, it is an assumption.) The dimension of area (we will write [A]) is L^2, while [r] is L. Since the only combination of products/exponents of the variables that results in the dimension L^2 is r^2, we conclude that A = C*r^2 for some constant C.
The period of a pendulum is some function of its length, the mass at the end of it (we are assuming the actual rod is massless), the gravitational acceleration g (assumed constant), and no other parameters. (This is actually not quite true, but it is true to a first approximation.) [P] = T; [g] = L T^-2; [l] = L; and [m] = M. The only combination of the latter three that results in T is (l/g)^(1/2), so P = C*(l/g)^(1/2) for some constant C. Notably under these assumptions it is impossible for C to depend on m at all.
This becomes less useful in circumstances where you have more variables than dimensions. In these cases you generally write down an equation for the exponents on each term that makes everything balance. For an example of such equations, we can use the previous case, where I'd write P = C g^a l^b m^c, so T = L^(a+b) T^(-2a) M^c, giving us:
a+b = 0
-2a = 1
c = 0
which has solution a=-1/2, b=1/2, c=0, which is the solution I wrote down above.

[PlatoHagel]

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As a Layman I also found it useful to follow the Professors orientations that lead to link above. It has also been very difficult process for me.

Meanwhile I’m continuing to develop the Extra Dimensions series of articles, and I’ve now followed up my To view links or images in this forum your post count must be 2 or greater. You currently have 0 posts. with a next installment, To view links or images in this forum your post count must be 2 or greater. You currently have 0 posts. The new article describes one of the key clues that would indicate their presence. But this is far from the end of the story: I owe you more articles, explaining why extra dimensions would generate this clue, outlining how we try to search for this clue experimentally, and mentioning other possible clues that might arise. All in due course… To view links or images in this forum your post count must be 2 or greater. You currently have 0 posts. by Theoretical Physicist Matt Strassler

[SirJac]

The only problem with dimentional analysis is that you must also consider the dimension of all parameters as well as variables. Unfortunately, the dimensions of parameters are often only determined by the equations themselves. Thus relying on dimensional analysis to determine the functional form the equations of motion that describe a system will in general fail because it does not consider the dimensions of the relevant parameters (physical constants). For example, Newton's gravitational constant has units of m^3 kg^-1 s^-2 which is required so that gravitational force actually has units of force.

Dimentional analysis is an important check, but it cannot be used to actually find equations of motion on it's own except in the simplest of systems.

[Latro]

Originally Posted by SirJac To view links or images in this forum your post count must be 2 or greater. You currently have 0 posts.

The only problem with dimentional analysis is that you must also consider the dimension of all parameters as well as variables. Unfortunately, the dimensions of parameters are often only determined by the equations themselves. Thus relying on dimensional analysis to determine the functional form the equations of motion that describe a system will in general fail because it does not consider the dimensions of the relevant parameters (physical constants). For example, Newton's gravitational constant has units of m^3 kg^-1 s^-2 which is required so that gravitational force actually has units of force. Dimentional analysis is an important check, but it cannot be used to actually find equations of motion on it's own except in the simplest of systems.

It is considerably more powerful in taking systems for which they already exist equations of motion which predict what the parameters should be, in which case often a great deal can be determined, especially if there aren't very many parameters involved. This was used famously to estimate certain parameters (energy, blast radius, etc.) having to do with the early atomic bomb explosions.

[PlatoHagel]

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?

It is a learning process for me so there is bound to be some confusion on my part. I reveal the scientific analysis below that I refer too, to help me along, as well as, the comments that are being put here in this thread.

See Also:
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Some one also mention "
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." Used, as an approximation?

[The Dan Keizer]

I guess I don't see how this isn't just regular math.

[Latro]

	Originally Posted by Dan Keizer To view links or images in this forum your post count must be 2 or greater. You currently have 0 posts.
	I guess I don't see how this isn't just regular math.

It basically is; dimensional analysis in its full form is essentially just an application of linear algebra. It is pretty useful, though, for getting some initial insight into the structure of a problem. One particularly useful thing about it is that you can find out parameters, which are generally some kind of ratio (since they are always dimensionless), which determine the qualitative behavior of the system. Fluid dynamics has several of these, the Reynolds number (which is a measure essentially of turbulence) being one of the most well known.

The reason they have to be dimensionless is because it is impossible for a dimensioned parameter to be the argument of a "nonlinear" (that is, non-monomial, in this context) function. An intuitive reason for this, though it doesn't work in every case, is that it would require you to add numbers that have different units when you write the Taylor series of the nonlinear function. If you do, say, sin(1 meter), you get 1 m - 1/6 m^3 + 1/120 m^5 + ... which is absurd.