Proving the impossibility

uhh uhh

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:- letx=0.68888888888....(1)let x=0.68888888888....-(1) Multiplying both sides by 100:- 100x=68.88888888.....(2)100x=68.88888888.....-(2) Subtracting equation one from equation two:- 99x=68.299x=68.2 On canceling comen divisers:- x=3145x=\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 ab\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:- letx=0.999999...........(1)let x=0.999999...........-(1) Multiplying both sides by 10:- 10x=9.99999999.......(2)10x=9.99999999.......-(2) Subtracting (1) from (2):- 9x=9==>x=19x=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
6 years, 10 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 1/171/17.

Michael Mendrin - 6 years, 10 months ago

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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 - 6 years, 10 months ago

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hard works makes man perfect

Ram Charan - 6 years, 3 months ago

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

.99999999 or .12367456456456

so both these no. Are rational and can be convert into a:b...

Rishabh Neekhara - 6 years, 10 months ago

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Can you convert 0.99999......... into a:b

Aman Sharma - 6 years, 10 months ago

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