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

## Comments

Sort by:

TopNewesthard works makes man perfect – Ram Charan · 2 years, 1 month ago

Log in to reply

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

Another example:

\(0.05882352941176470588235294117647058823529....\)

looks irrational, but it actually repeats (look CAREFULLY). This one is \(1/17\). – Michael Mendrin · 2 years, 8 months ago

Log in to reply

aandbsuch that a/b=0.99999........ I also tried to solve it by converting number into infinite series but in vainAlso number given by you is aldo rational i solved it by mathod of differences – Aman Sharma · 2 years, 8 months ago

Log in to reply

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, 8 months ago

Log in to reply

– Aman Sharma · 2 years, 8 months ago

Can you convert 0.99999......... into a:bLog in to reply