How can we know that the decimal expansion of an irrational number will never repeat?

kramberol

kramberol

Answered

2022-06-29

How can we know that the decimal expansion of an irrational number will never repeat?

Answer & Explanation

Salma Bradley

Salma Bradley

Expert

2022-06-30Added 13 answers

If there appears a period in the decimal expansion of a number then You have after subtracting a rational term an expression of the form
n = k ( d 1 . . . d r ) 10 r n
where d j { 0 , . . . , 9 } are the digits in the period and r its length. Clearly this expression is rational if and only if n = 0 ( d 1 . . . d r ) 10 r n is rational and
n = 0 ( d 1 . . . d r ) 10 r n = n = 0 ( d 1 . . . d r 10 r ) n = 1 1 d 1 . . . d r 10 r
is rational as a geometric series. Note d 1 . . . d r 10 r < 1. Since there are (quite sophisticated) proofs that π is irrational (even transcendental, i.e. no root of any polynomial with rational coefficients,) one concludes that its decimal expansion cannot lead to any period.

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