Number Systems-2


This article focuses on conversion of  numbers such as 1.2323.........,2.33333333..........,3.67777777777777777...........(non-terminating) into rational numbers(p/q).Later we will discuss a CAT level problem on this concept
So lets start with (a) 1.2222222222....(often written as1.2bar)
Standard Procedure
Assume x=1.2bar.....(1)
     Multiply by 10 ,we get
         10 x=12.2bar
Well what has happened ???There are so many 2's(infinte) after decimal so u can multiply  the equation by 10 but still decimal part would remain unchanged .This is the concept.Whatever be the number you have to form equations in such a manner (by multiplying by different powers of 10)so that decimal part gets cancelled after subtraction of the 2 equations so formed.
Now u can easily subtarct 2equations. Decimal part will get cancel
9x=11
x=11/9 is the same number which was expressed in decimals

(b) 1.23333333333333333....non -terminating
Well try to think by thinking upon my lines written above :)
Here is the solution
x=1.2333333.......
Multiply by 10
10x=12.333333333333
Now can you subtact to get x?Nope!!!!...Because decimal part is not same .(you can proceed in that manner too but my aim is to give you a clear thought process )
Multiply one more time
100x=123.333333333333........
Subtarct 10x from 100x
90x=111
x=111/90
or x=37/30


(c) 0.98989898   
Let x=.9898..........
Multiply by 100
100x=98.9898...........
See there is nothing hard and fast about whether we are nultiplying by 10,100,.............
Aim is to make decimal part same
Subtarct 99x=98
==>x=98/99
I hope i have been able to explain the above concept.
Happy Problem Solving


1.  2.345345345..............
2.  3.73455555555..........
3.  7.244444444444.......
4.  1.7777777777777.................
5. Given that x=.abababababab............................
What is the minimum possible number which when multiplied by x would  will make certain that result is a natural number?