I'm in $9th$ class and I was wondering how to solve this problem. I only know how to prove that $0.1010010001\dots$ is irrattional.
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http://math.stackexchange.com/questions/191386/how-to-can-i-show-irrational-numbers?rq=1 – R.N Sep 20 '15 at 18:38
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4possible duplicate of Rational or irrational. Another possible duplicate is this – Winther Sep 20 '15 at 18:40
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The decimal representation of a rational ends by repeating the same digits periodically.
The number 0.1234567891011… includes arbitrarily large sequences of zeroes (it inculdes 10, 100, 1000, 10000, ...). But it also includes arbitrarily large sequences of 1's (it includes 1, 11, 111, 1111, ...). This is contradictory with the fact that it ends with a finite period.
Florian F
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It has many proofs. Okay I can give you one hint this is much interesting any rational number can have only finite decimal representation or recurrence decimal representation.
For start just take $\frac pq$ where $gcd(p,q)=1$ then see after using division algorithm what are the remainders that will come.
Do it then you will get generalised statement.
Ri-Li
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1Can you reword your answer; it is currently almost incoherent. You might also want to give a proof, or point to one, for your claim. – Simon S Sep 20 '15 at 18:32
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Yeah I know I have stated little bit more and wondering the OS will be motivated from that. – Ri-Li Sep 20 '15 at 18:34