How can 0.99999... be converted into a proper fraction. Why is that the answer?

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justaguide's profile pic

justaguide | College Teacher | (Level 2) Distinguished Educator

Posted on

We have to convert 0.99999..... into a proper fraction.

Let 0.99999... = f

10f = 9.9999...

10f - f = 9.99999... - 0.99999

=> 9f = 9

=> f = 9/9

=> f = 1/1

The representation of .9999... as a proper fraction is 1/1 because in 0.999... as the number of 9s in the decimal tends to infinity its value tends to 1. Here the number of 9s in the decimal part is tending to infinity; this makes 0.999... equal to 1.

The proper fraction representation of 0.999... is 1/1.

neela's profile pic

neela | High School Teacher | (Level 3) Valedictorian

Posted on

Let 0.99999... = p be the proper fraction.

p = 0.999.....                      (1).

Multiply both sides of equation at (1) by 10.

10p = 9.999....                    (2)

(2) - (1):

10p - p = (9.999...) - (0.999.....)

9p = 9.

p = 9/9 = 1/1.

Therefore p = 1/1. Or p = 1.

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