Does sqrt 2 contain all the digits of pi in order? Does it contain all the digits of pi in order an infinite number of times? Does pi contain all the digits of sqrt 2 in order?

Uroskopieulm

Uroskopieulm

Answered question

2022-11-04

Does 2 contain all the digits of π in order? Does it contain all the digits of π in order an infinite number of times? Does π contain all the digits of 2 in order?

Answer & Explanation

Gwendolyn Alexander

Gwendolyn Alexander

Beginner2022-11-05Added 16 answers

It is not known if π is a normal number in base 10. If it is, then the answer to your question is ‘yes’, in the sense that infinitely many subsequences of the decimal expansion of π will be the decimal expansion of 2 . What I just said remains valid if we swap the roles of π and 2 .
That almost every real number is normal in base 10 w.r.t. the Lebesgue measure, but aside from constructions that have been designed on purpose to create normal numbers, almost no examples exist.
On the other hand, the decimal expansion of π cannot be exactly the decimal expansion of 2 after some point. Otherwise, we would have
π = q + r 2
for some q , r Q (I shall leave this as an easy exercise for you), thus implying that π is algebraic over Q , which is a contradiction.

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