Interesting property of MOND's constant

[IDMclean]

I have recently read up, somewhat, on MOND.

 

 

I have been playing with the equation:

[math]F = \frac{ma}{a_0}[/math]

 

Where [math]a_0 = 1.2*10^{-12} \frac{m}{s^2}[/math].

 

 

Using some tricks, based on dimensional analysis, I came to a much reduced form which said basically:

 

[math] ? = 1/1.2*10^{-12} m[/math]

Which can be given as:

[math] ? = 833333333333 m[/math]

which on a hunch I thought might be related to pi. so. Here's what I got.

[math] ? = 2\pi \cdot 132,629,119,243 m[/math]

Which indicates, to me, a circumfrence.

 

What do you think of this? I have some ideas myself, but I don't think many would agree with how I got that. I even question my method, because I lack the info of how this constant was found. however, assuming the constant is exact value, with no uncertainty, then this should be valid. I attempted factoring out greater quanta of pi, but 2 pi is the only one which results in a whole number.

 

Given that I used nothing but constants to arrive at this, I do not trust anything but whole, natural numbers.

 

Anyway just thought I would share this.

[CraigD]

[math] ? = 1/1.2*10^{-12} m[/math]

Which can be given as:

[math] ? = 833333333333 m[/math]

which on a hunch I thought might be related to pi. so. Here's what I got.

[math] ? = 2\pi \cdot 132,629,119,243 m[/math]

[math] \frac{1}{1.2 \cdot 10^{-12}} \neq 2 \cdot \pi \cdot 132629119243 [/math]

 

[math] \frac{1}{1.2} = 0.8[3][/math]. (The notation 0.8[3] indicates a repeating decimal, eg: 0.8333…) The reason KAC calculated it as 0.833333333333[0] is likely due to limited calculator precision.

 

There’s no rational number that can be multipled or divided by an irrational number to give a rational number. Care must be taken not to confuse irrational numbers, such as [math]\pi[/math], with rational number approximations of them, such as most calculators use.

[IDMclean]

Did I calculate as .8333? I took [math]1/1.2\cdot10^{-12}[/math] which resulted in 833,333,333,333. Not [math]833,333,333,33\overline{3}[/math]. Usually my calculator will round up the last digit to the next number up to indicate a repeating decimal place. so that would be in the form of .83334, or something similar.

 

I use a Ti-85, if that makes a difference. still, I know of a few numbers which use pi and come out fine. One of which is c.

 

Still. Perhaps your right and it is nothing. Then again I never bought that a number or pattern is insignificant. Math derives from the physical world, and like everything else, is intimately tied to it. Any pattern that emerges has a potential of being physically significant, in my humble opinion.

 

I would like to know more about the origin of the [math]a_0[/math] constant, how it is derived (divined) and what it's uncertainties are, if any. If anyone knows more about Milgrom, MOND, and TeVeS. I would love to know more about it.

 

 

That, Non-Communitive Geometry, and Renormalization. I recently was reading about these topics in Discover Magazine (August 2006), and Scientific American (Aug 2006).

[Jay-qu]

Did I calculate as .8333? I took [math]1/1.2\cdot10^{-12}[/math] which resulted in 833,333,333,333. Not [math]833,333,333,33\overline{3}[/math]. Usually my calculator will round up the last digit to the next number up to indicate a repeating decimal place. so that would be in the form of .83334, or something similar.

 

I use a Ti-85, if that makes a difference

 

In this case it would more than likely be a repeating dec. because the calculator wont round up .8333... recurring to .83334, only in the case of .66666... recurring would it round up to .666667

 

you can test this if you think Im wrong, just type in 1/3, im pretty sure youll get .3333... and not .33334

[IDMclean]

Ok, So it is a rounding part of my calculator. however, it still is interesting to me that they factor so nicely. I would bet that they would factor rather well to some arbitary number of decimal places.

 

Quick question, what happens if we put these into binary? base two and all that.

 

Anyway, one reply is encouraged, and then I wish to get on with discussing the topic of the thread. Which is, indirectly perhaps, MOND and Non-communitive Space-time Geometry, and other possible solution spaces for gravity and other such similar topics.0

[Jay-qu]

what is MOND?

[IDMclean]

Modified Newtonian Dynamics

 

TeVeS

 

Noncommunative Geometry

 

Mordehai Milgrom

Jacob Bekenstein

Alain Connes

 

Just to get everyone started and on the same page.

[Jay-qu]

thats a good idea! be back in a while to pick up from where I had no idea where I was going...

[CraigD]

Ok, So it is a rounding part of my calculator. however, it still is interesting to me that they factor so nicely. I would bet that they would factor rather well to some arbitary number of decimal places.

My pointing out the arithmetic error, and the limitations of a particular calculator, is not a minor quibble. There’s an important number theory principle at issue.

 

Irrational numbers such as [math]\pi[/math] don’t factor into a finite number of rational terms. The higher the precision you use to represent a rational approximation of [math]\pi[/math], or any simplified expression containing it, the more non-zero digits you’ll see.

 

They can factor nicely into infinite numbers of rational terms. For example,

[math]\pi = \sum_{n=1}^\infty \frac{(-1)^{n+1}}{2n-1} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + . . . [/math]

 

Most calculators aren’t well suited to expressing numbers as sums of infinite numbers of terms, but that’s just a minor technical limitation.

Quick question, what happens if we put these into binary? base two and all that.

In base 10, [math]\frac{1}{1.2} = 0.8[3][/math]

In base 2, the same equation is, [math]\frac{1}{1.[0011]} = 0.1[10][/math]

 

The choice of base is unimportant – a rational number can be exactly represented as a repeating positional numeral (eg: “repeating decimal”), while an irrational number, such as [math]\pi[/math], cannot be.

Still. Perhaps your right and it is nothing. Then again I never bought that a number or pattern is insignificant. Math derives from the physical world, and like everything else, is intimately tied to it. Any pattern that emerges has a potential of being physically significant, in my humble opinion.

I didn’t mean to suggest that it was nothing, only that the equation you appeared to be treating as exact was actually approximate.

 

Looking for unexpected patterns in nature is a lot of what being human is about, IMHO. It’s important to have the best possible understanding of the tools we use, however, Math (and its branch, number theory) being one of the best of them. Being aware of artifacts introduced by such aids as limited-precision calculators is important, lest one be misled into perceiving patterns that are not exactly present.

I would like to know more about the origin of the [math]a_0[/math] constant, how it is derived (divined) and what it's uncertainties are

Based on my limited knowledge of MOND, I believe that the 1.2*10^-10 m/s/s value for [math]a_0[/math] is not derived from pure mathematical analysis, but an empirical physical constant calculated from observation of the motion of galaxies (as measured by the Doppler shift of the spectra of known star types), and some reasonable assumptions about the distribution of matter in galaxies. Compared to similar empirical constants, such as G, which is known to about 4 significant digits, it’s uncertainty is high, at about 2 significant digits. Further observation, and perhaps direct experimentation, should reduce this uncertainty.

 

:thumbs_up An in-depth discussion of MOND is an excellent idea. I look forward to it. :)