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The following real numbers have decimal expansions as given below. In each case, decide whether they are rational or not. If they are rational, and of the form, p q what can you say about the prime factors of q? 0.120120012000120000. . .

The following real numbers have decimal expansions as given below. In each case, decide whether they are rational or not. If they are rational, and of the form, p q what can you say about the prime factors of q? 
 0.120120012000120000. . . 

Grade:12

2 Answers

Harshit Singh
askIITians Faculty 5870 Points
2 months ago
Dear Student

0.120120012000120000. . .
Since, it has non-terminating and non- repeating decimal expansion, it is an irrational number.

Thanks
Ram Kushwah
108 Points
one month ago
Dear student ,
 
The decimal expansion of a real number can be of three types
1.Terminating 
    For exaple 0.125=1/8, 0.375=3/8,such numbers are always rational numbers
2.Non-terminating and repeating
Example : 0.333333...........=1/3
0.555555555555555555....=1/9
such numbers are also always rational numbers
2.Non-terminating and non-repeating
Example :
3.141592653589793.........................= ( irrational)
2.30300400050000.............................Irrational
such numbers are always irrational numbers
 
Now the given number is:
 
0.120120012000120000.......................................
 
Here we see that decimal expansion is not terminating( Not ending)
But here is not follwing any patter so it also nor repeating
 
Hence this number is non terminating and non repeating 
Thus the number is AN  IRRATIONAL NUMBER 

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