MODERATE EXERCISES
(3) Convert 0._
(3) Convert 0.9 to rational form.
SOLUTION:
Let x = 0._
Let x = 0.9, so that x = 0.9999... .
Note that 10x = 9.9999... ,
so then 10x - x = - ( 9.9999... )
so then 10x - x = - ( 0.9999... )
so then 10x - x = - ( 9.0000...
so then 10x - x = - ( 9
But 10x - x = 9x, so 9x = 9 implies
( 1/9 )( 9x ) = x = ( 1/9 )( 9 )
( 1/9 )( 9x ) = x = 9/9 = 1/1.
Thus, as odd as it may seem, 0._
Thus, as odd as it may seem, 0.9 has the rational form 1/1, or just 1.
On the other hand, this result should not be too surprising (just think about what the difference between 1 and 0.9999... actually is).
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