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Proving the impossibility



Yesterday i was puzzling with some problems related to fractions ,for example:- Find out the coprime possitive integers (a,b) such that when a is divided by b yields the fraction 0..688888888....... Problem can be easily solved as follows:- \[let x=0.68888888888....-(1)\] Multiplying both sides by 100:- \[100x=68.88888888.....-(2)\] Subtracting equation one from equation two:- \[99x=68.2\] On canceling comen divisers:- \[x=\frac{31}{45}\] Since 31 and 45 are coprime so answer is (31,45) Mathod can be used to convert all rational fractions into the form \[\frac{a}{b}\] After solving five or six such problems by using this mathod i tried to convert 0.99999999.... into the form a÷b but got a contradiction as follows:- \[let x=0.999999...........-(1)\] Multiplying both sides by 10:- \[10x=9.99999999.......-(2)\] Subtracting (1) from (2):- \[9x=9 ==> x=1\] But from equation (1) x=0.99999..... So does this means that 0.99999....=1 . . Well i think this could be probebly because 0.99999.... is an irrational number (i think so),but problem is that i am not much fimiler with mumber theory so i don't know any mathod to prove that this number is irrational...what do you think about it..

Sorry for all grammer and spelling mistakes i am not good in english :p

Note by Aman Sharma
2 years, 1 month ago

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hard works makes man perfect Ram Charan · 1 year, 6 months ago

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Anytime you have a repeating decimal, it's a rational number. So, 0.9999999.... etc is pretty rational to me.

Another example:


looks irrational, but it actually repeats (look CAREFULLY). This one is \(1/17\). Michael Mendrin · 2 years, 1 month ago

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@Michael Mendrin Well reccuring decimal numbers are always rational i read it somewhere..........but sir i am realy confused with 0.9999999....'s case, i am realy intrested in its solutions..i am trying to find numbers a and b such that a/b=0.99999........ I also tried to solve it by converting number into infinite series but in vain

Also number given by you is aldo rational i solved it by mathod of differences Aman Sharma · 2 years, 1 month ago

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See.. no. Is irrational only when it is neither recurring nor repeating....like

.99999999 or .12367456456456

so both these no. Are rational and can be convert into a:b... Rishabh Neekhara · 2 years, 1 month ago

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@Rishabh Neekhara Can you convert 0.99999......... into a:b Aman Sharma · 2 years, 1 month ago

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