PUMPA - SMART LEARNING

எங்கள் ஆசிரியர்களுடன் 1-ஆன்-1 ஆலோசனை நேரத்தைப் பெறுங்கள். டாப்பர் ஆவதற்கு நாங்கள் பயிற்சி அளிப்போம்

Book Free Demo
The decimal expansion of a rational number is terminating or non-terminating and recurring. Conversely, the decimal expansion of a number is terminating, or non-terminating recurring is a rational number.
Example:
1. Prove that 0.77777... = 0.7¯ is a rational number. That is, show that 0.7¯ can be expressed in p/q, where 'p' and 'q' are integers with q0.
 
Solution:
 
Let us take the provided number as 'x'.
 
That is x\ =\ 0.77777...
 
Now observe the number 'x'. The only repeated digit is 7.
 
Here we have to make multiples of 'x' in such a way that its decimal part will be the same as the given number.
 
Let us multiply 'x' by 10.
 
10x\ =\ 7.77777...
 
Now subtract 'x' from 10x,
 
10x - x = 7.77777... - 0.777777...
 
9x = 7
 
x = 7/9
 
Therefore, the fractional form of the rational number 0.7¯ is 7/9.
 
 
2. Prove that 0 .2363636...\ =\ 0.236¯ is a rational number. That is, show that 0.236¯ can be expressed in p/q, where 'p' and 'q' are integers with q0.
 
Solution:
 
Let us take the provided number as 'x'.
 
That is x = 0.2363636...
 
Now observe the number 'x'. There are two repeated digits. That is 36.
 
Here we have to make multiples of 'x' in such a way that its decimal part will be the same as the given number.
 
Also, we have one non-repeating number 2.
 
Let us multiply 'x' by 10 to get the repeated decimals separately.
 
10x\ =\ 2.363636...
 
Now multiply 'x' by 1000 to get the same decimal pars of 10x,
 
1000x\ =\ 236.363636...
 
Subtract 10x from 1000x.
 
1000x - 10x = 236.363636... - 2.363636...
 
990x\ =\ 234
 
x\ =\ 234/990
 
Therefore, the fractional form of the rational number 0.236¯ is 234/990.