Is G^(ε0μ0) Exactly Equal to 1 in MATLAB?

[lightfire]

I've got MATLAB set to several(about 20) decimal places and I noticed something.

G^(E0M0)=1 seemingly exactly.

That is the gravitational constant G raised to the quantity of the permittivity constant multiplied by the constant of permeability.

That is approximately
(6.67428E-11)^((1.2566370614359E-6)*(8.854187815E-12))=1.0000000000.

Now obviously I know that that doesn't necessarily mean anything because the numbers are so small, but has anyone ever noticed it or commented on it or at least investigated it?

Keep in mind that those are just approximations for the physical constants.

 

[lightfire]

Are those constants irrational and transcendental!

That would be the major question. Since any rational number raised to a power OTHER THAN ZERO cannot be 1, than it is just a case of running out of decimal places. But a transcendental number raised to a transcendental exponent could hypothetically equal 1 I believe. If someone is aware of a proof showing otherwise, please tell me.

 

[D H]

G^(E0M0)=1 seemingly exactly.

This expression doesn't make a bit of sense. An exponent must be unitless for an expression to be valid. That the result is approximately one is just a fluke of the of the choice of units. In other systems (e.g. "natural units", $\epsilon_0=1, \mu_0=1$) the result of this expression is not even close to unity.

To show that the number is not one, just take the log of the expression:
$$\log_{10}\left((6.67428\cdot10^{-11})^{1.2566370614359\cdot10^{-6}*8.854187815\cdot10^{-12}}\right)= -9.64327467\cdot10^{-17}$$

To set Matlab to twenty decimal places you need to do a lot more than set the format. You need to use variable-precision arithmetic throughout. In particular, you need to convert the double precision numbers used the expression to variable-precision values.

 

[D H]

But a transcendental number raised to a transcendental exponent could hypothetically equal 1 I believe. If someone is aware of a proof showing otherwise, please tell me.

You want to find two numbers $a$ and $b$ such that $a^b=1$. Take the log of this expression: $b\log a = 0$. The only way to arrive at such a result is if either $b=0$ or $a=1$.

 

[Mech_Engineer]

Congratulations, you have successfully unified gravity with electromagnetic theory


Seriously though, what are you going to do with 20 decimal places?! Sig figs are an important rule to keep in mind.

 

[lightfire]

HaHa

Yeah, I saw that on the web awhile ago and I was startled at first, but quickly realized that it was just a calculator thing and the claim . I was just hoping that a few "wows" would post or something. But I guess only alert people posted. I forget what the site is, but it has a whole list of phony unified field theory and Riemann hypothesis proofs and so forth.

Oh one more thing, just about any exponential relation between numbers that small will surely yield one on nearly all calculators.

 

[chroot]

If you already know it's crap, please don't post it here. Thanks.

- Warren