A quick question concerning constants and planck units

[FysixFox]

Recently I have taken an interest in physical constants (and, through it, an interest in SI units and the upcoming redefinition of the kilogram). I actually have a list of almost all of them now, standard uncertainties included. After a bit, I decided to have a bit of fun messing around with the constants just to see what popped out.

Upon dividing Planck length by Planck time, I was surprised and intrigued to discover that the answer is the speed of light. I then began to wonder if there could possibly be other relations like this that, using Planck units, give such constants. Planck energy times Planck time gave me the reduced Planck constant, 1.0545717×10^-34 J×s. Planck length cubed divided by the product of Planck mass and Planck time squared gave me the gravitational constant, 6.67384×10^-11.

But oddly, upon dividing Planck energy by Planck temperature in an attempt to arrive at the Boltzmann constant, I hit a... strange roadblock. The Boltzmann constant DID technically come out the other side... but it came out 10⁶⁴ times larger than it should have. Instead of 1.38065×10^-23, I was getting 1.38065×10⁴¹. Why is this?

 

[dauto]

You must've made a silly mistake.
Planck energy: E_P = 1.9561 × 10⁹ J
Planck Temperature :T_P = 1.416 8 × 10³² K

E_P/T_P = 1.9561 × 10⁹/1.416 8 × 10³² J/K = 1.38 × 10^-23 as expected. By the way, this is not surprising at all. Planck's units are carefully defined with that exact property in mind. It is that way by design.

 

[FysixFox]

You must've made a silly mistake.
Planck energy: E_P = 1.9561 × 10⁹ J
Planck Temperature :T_P = 1.416 8 × 10³² K

E_P/T_P = 1.9561 × 10⁹/1.416 8 × 10³² J/K = 1.38 × 10^-23 as expected. By the way, this is not surprising at all. Planck's units are carefully defined with that exact property in mind. It is that way by design.

Ah! I see! It seems that the × 10³² part of the denominator migrated to the numerator somehow, likely an error on my part rather than the calculator's. Silly me.

And interesting that the constants would come out like that. It kind of makes me wonder if some of the more complex constants could also be created using Planck's units in some manner. I'll continue to mess around with these constants for a while longer.

Thanks!

 

[dauto]

Ah! I see! It seems that the × 10³² part of the denominator migrated to the numerator somehow, likely an error on my part rather than the calculator's. Silly me.

And interesting that the constants would come out like that. It kind of makes me wonder if some of the more complex constants could also be created using Planck's units in some manner. I'll continue to mess around with these constants for a while longer.

Thanks!

As I said, the constants come out like that by design. Planck length, mass, time ,etc were chosen with that very property in mind. What that means is that all these constants are equal to 1 (exactly, by definition) in Planck units. To be surprised or amazed by that is equivalent to being surprised that the heat capacity of water is 1 cal/(g.°C). The unit calorie was defined that way.

 

[FysixFox]

As I said, the constants come out like that by design. Planck length, mass, time ,etc were chosen with that very property in mind. What that means is that all these constants are equal to 1 (exactly, by definition) in Planck units. To be surprised or amazed by that is equivalent to being surprised that the heat capacity of water is 1 cal/(g.°C). The unit calorie was defined that way.

I'm kind of a noob at this whole physics thing, I was surprised that the definition of the meter is why the speed of light is an exact number instead of an estimate. :P