r/askmath • u/Loose-Balance3225 • May 06 '26

Discrete Math Can infinity contain infinity

If pi has no end it has to have every combination of numbers but could it hold an infinite combination? Like 1 2 3 4... To infinity

Please help this is missing my brain up

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u/tbdabbholm Engineering/Physics with Math Minor May 06 '26

Pi could be 3.1415...123456789101112... yes. It would be very strange for that to be the case but there's nothing that would make it impossible

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u/2ndcountable May 06 '26

Surprisingly, we can prove that it is impossible. The base-10 champernowne constant 0.123456789101112... has irrationality measure(https://en.wikipedia.org/wiki/Irrationality_measure) equal to 10. By the definition of the irrationality measure, any number with base-10 decimal expansion terminating in 1234567... has the same irrationality measure of 10. Hence, if pi were to be 3.1415...123456..., it too would have irrationality measure 10, but it is known that this is not the case(https://arxiv.org/abs/1912.06345).