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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5Its value is an artifact of the units we choose to measure it in. It's even possible to choose a system of units where its value is exactly 1. – The Photon Jan 23 '19 at 04:59
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Related: https://physics.stackexchange.com/q/127752/2451 and links therein. – Qmechanic Jan 23 '19 at 06:34
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3Possible duplicate of Is speed of light and sound rational or irrational in nature? That one asks not about $G$, but the answer is the same. – Ruslan Jan 23 '19 at 06:42
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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.
Dale
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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 – Dale Jan 23 '19 at 11:43
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If you don’t measure $G$, you have no idea how large the Planck units are so they are meaningless as units. – G. Smith Jan 23 '19 at 16:33
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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. – Dale Jan 23 '19 at 16:44
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By the way, I am not arguing that G can’t be defined as 1. It can. I am arguing that in currently accepted theory there is no mathematical formula that explains why gravity has the strength it has. You have to measure gravity to find that out. You can measure G, you can measure the Planck mass, or whatever. You have to measure something, not just calculate. BTW, my original comment was a respnse to the OP. I do not dispute your answer at all. – G. Smith Jan 23 '19 at 18:17
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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.
Mitchell Porter
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