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This question already has an answer here:

The question is the title. But I'm quite doubtful if this question is meaningful or not. Since this constant is obtained by experiment, we can never know its exact value, unlike $π$ or $e$. Is it correct?

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marked as duplicate by Ruslan, Mozibur Ullah, John Rennie, Kyle Kanos, M. Enns Jan 23 at 19:52

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

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It depends on your system of units. In SI units G is an experimentally determined number known only to a certain precision. In Planck units it is 1, which is clearly rational. Other systems of units will vary.

This is generally true of most universal dimensionful constants.

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  • $\begingroup$ Can G be calculated by mathematical method? $\endgroup$ – user220683 Jan 23 at 5:04
  • $\begingroup$ No, it can't be calculated. It gets measured experimentally. $\endgroup$ – G. Smith Jan 23 at 5:42
  • $\begingroup$ Again, that depends on your system of units. In SI units G is measured experimentally, but in Planck units it is defined to be 1 $\endgroup$ – Dale Jan 23 at 11:43
  • $\begingroup$ If you don’t measure $G$, you have no idea how large the Planck units are so they are meaningless as units. $\endgroup$ – G. Smith Jan 23 at 16:33
  • $\begingroup$ Sure, but in that case you are not measuring G in Planck units, you are measuring the Planck mass. In Planck units G cannot be measured. This is similar to Planck’s constant in the new SI. In the new SI Planck’s constant cannot be measured, instead you measure the kilogram. $\endgroup$ – Dale Jan 23 at 16:44
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According to our current theoretical paradigms (general relativity, quantum field theory), it's an arbitrary real number.

In string theory, in each model that resembles the real world, it should have a uniquely determined value. I don't know what kind of numbers those values are, but they are likely to contain contributions from square roots and transcendental functions, and not just ratios of integers.

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