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I recently learned about Benfords law, "an observation about the leading digits of the numbers found in real-world data sets". Since every possible sequence of number exists in the digits of irrational number, and every number can theoretically have a scenario in real life, if I were to group every 4 digits of Pi: 3141 5926 ...

then tally the leading digit of the generated 4 digit numbers, will it follow benford's law as well? thx.

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    There are places where you can find millions of digits of $\pi$, so you can experiment and see for yourself what happens when you look at every fourth digit.2017-01-28

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No -- Benford's law applies not to any random collection of digits, but specifically to numbers that represent magnitudes from the real world (with the additional caveat that it doesn't work if the unit the magnitudes are measured in had a significant influence on how those magnitudes arose).

Groupings of digits of $\pi$ don't represent the magnitudes of anything sampled from the real world, so there's no reason to expect Benford's law to apply to them.


By the way, it is not true that "every possible sequence of number exists in the digits of irrational number" -- for example the number $$ 1.010010001000010000010000001\ldots $$ is irrational yet doesn't even contain any digits other than $0$ or $1$.

$\pi$ in particular is suspected (but not proved!) to have the stronger property of being normal, which means that not only does every digit sequence appear in it, but every digit sequence appears just as often as it would in an infinite random sequence of digits, going to the limit. In a certain technical sense "most" irrational numbers have this property, but not all of them.

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The reason for Benson's law: If you measure something say in meters, you could have measured it in any other measurement instead. Say in miles, feet, yards, inches, femtoparsecs, and so on. If it's physical constants, aliens on another planet would measure it in totally unrelated units. The units used would obviously change the numerical values, but shouldn't change the digit distribution of all numbers.