How do you find the repeating decimal 0.1 with 1 repeated as a fraction?

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Apr 15, 2018

Answer:

#0.1111111... = 0.bar1 = 1/9#

Explanation:

#0.111111.....#is a recurring decimal which can be written as #0.bar1#

It is useful to know that the decimals which have only 1 digit repeating all come from the fractions which are ninths.

#1/9 = 0.1111..." "2/9 = 0.22222...." "3/9 = 0.333333...#

#4/9 = 0.444444...#

and so on... But why is this?

Let #" "x= 0.111111111." "larr# one digit recurs, multiply by 10.
#" ":.10x = 1.11111111..." "larr# subtract them. #10x-x=9x#

#" "9x = 1.0000000..." "larr# all the way to infinity

Now: #" "9x = 1 hArr x = 1/9#

Dividing will confirm this: #1 div 9 = 0.1111111.....= 0.bar1#

What do you make of #0.99999..... ?#

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